The heximal numbering system

Introduction

The decimal numbering system is commonly used in daily life across most countries. It serves various purposes, such as facilitating trade, enabling calculations, and simplifying measurements of lengths and weights. Nearly everyone is familiar with decimal numbers. However, many people tend to think that the decimal numbering system is the sole positional numbering system, while some think that the exceptions are binary and hexadecimal, which are primarily used by computers and may not appear as practical for everyday use.

One notable advantage of the decimal numbering system, when compared to some older systems, lies in its positional nature. It employs a finite number of digits and doesn't require additional digits to represent large numbers, unlike Roman numerals that demand more characters for higher values. Further benefits include its ability to represent rational numbers with ease, consistent digit counts (where a larger number cannot have fewer digits in its decimal representation), and the simplicity of performing multiplication and addition operations in the decimal system, especially when compared to the complexities of Roman numerals. Additionally, it aligns well with the fact that most people have ten fingers, making it natural to count up to ten in the decimal system.

While many assume that the decimal numbering system is the ultimate numerical system, historical evidence suggests otherwise. Numerous ancient civilizations employed numbering systems with bases other than ten, such as base 5, base 20, base 6, base 15, base 60, and various others. For instance, the base 6 numbering system found use in some regions of the Pacific, specifically in Indonesian New Guinea and Papua New Guinea. In these regions, civilizations adopted a base 6 system to measure various aspects of their lives and even devised names for specific powers of 6, ensuring that they had names for all numbers less than the highest power of 6. Notable languages employing the base 6 system included the Ndom language and the Yam languages, which assigned names to powers of 6, ranging from 6 to 6^6.

Representation of Numbers and Counting Heximal Numeral System

The decimal numbering system uses the digits 0 to 9 to represent all numbers, and these digits are used for multiplications, additions, and divisions. In contrast, the heximal numbering system uses the digits 0 to 5 to represent all numbers, placing more importance on each digit and making them more commonly used.

The sequence of integers in the decimal numbering system is familiar to most people: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, and so on. This sequence is deeply ingrained in the minds of those accustomed to the decimal numbering system and is often associated with integers.

In contrast, the sequence in the heximal numbering system is: 0, 1, 2, 3, 4, 5, 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25, 30, 31, 32, 33, 34, 35, 40, 41, 42, 43, 44, 45, 50, 51, 52, 53, 54, 55, 100…

Understanding the sequence in the heximal system is straightforward. It utilizes digits from 0 to 5 to represent all numbers, excluding the digits 6, 7, 8, and 9. Essentially, it's equivalent to the sequence of all numbers that don't contain the digits 6, 7, 8, or 9 in the decimal system, arranged from smallest to largest.

To gain a better grasp of the heximal system, it's helpful to compare it to the decimal numbering system. In the decimal system, the meaning of a number is evident. For instance, the number 43 signifies 4 × 10 + 3, while 4281 translates to 4 × 10^3+ 2 × 10^2+ 8 × 10 + 1. The same principle applies to the heximal numbering system. For example, the heximal number 243 means 2 × 6^2+ 4 × 6 + 3, which is equivalent to 99 in the decimal system. Similarly, the heximal number 30001 corresponds to 3 × 6^4+ 1 in the decimal system, which equals 3889.

Now that we understand how numbers are represented in the heximal numbering system, let's delve into basic arithmetic, including prime factorizations, additions, multiplications, and divisions. To distinguish between decimal and heximal numbers, I will use "dec" for decimal numbers and "hex" for heximal numbers. For example, 931 (dec) represents the decimal number 931, and 1054 (hex) stands for the heximal number 1054, which is equivalent to 250 in the decimal system. It's important to note that numbers less than six are represented the same way in both heximal and decimal systems.

Before diving into arithmetic, let's first learn how to count in the heximal numbering system. To do this, you can utilize the knowledge of counting in the decimal system and adapt it. Typically, you would raise the number of fingers on your hand to represent the desired number. Using this method, you can count from 0 (dec) to 10 (dec) / 0 (hex) to 14 (hex). There are 11 (dec) / 15 (hex) possible states for your fingers, indicating that if you were to use a logic based on finger counting, people might opt for a base 11 (dec) / 15 (hex) positional numbering system, where there are 11 (dec) / 15 (hex) digits from 0 (dec) to 10 (dec) / 0 (hex) to 14 (hex). However, it's important to note that any prime base greater than 3 has inherent disadvantages.

Taking inspiration from some tribes that spoke the Ndom language or Yam languages in the Pacific, a more intelligent approach involves treating each hand as a digit in the heximal positional numbering system. This method allows you to count from 0 (dec) to 35 (dec) / 0 (hex) to 55 (hex). In this approach, each finger on the right hand represents one unit, while each finger on the left hand represents six units. To represent numbers from 0 to 5, you can use the same method as in the decimal system, simply lift the corresponding number of fingers on your right hand. When you wish to represent larger numbers, use the heximal representation of the number, and raise the corresponding number of fingers on your left hand for the second digit and on your right hand for the first digit. For instance, for the number 25 (dec) / 41 (hex), you'd use the heximal representation, 41, and raise 4 fingers on your left hand and 1 finger on your right hand.

If a number has a zero in its heximal representation, such as 24 (dec) or 40 (hex), raise 4 fingers on your left hand and leave your right hand closed to represent a 0. Using this technique, you can represent any number from 0 to 35 (dec) / 0 to 55 (hex). However, it's important to note that you cannot go beyond these limits using this method. Numbers after 35 (dec) / 55 (hex) in the heximal numbering system require three digits, necessitating the use of additional hands for representation.

Simple Arithmetic in the Heximal Numbering System

To grasp multiplication and addition in the heximal numbering system, it is essential to commit a portion of the multiplication and addition tables to memory.

Addition:

Begin by memorizing a section of the addition table, which only requires remembering 15 (dec) or 23 (hex) sums, as opposed to the 45 (dec) or 113 (hex) sums required in the decimal numbering system.

+	0	1	2	3	4	5
0	0	1	2	3	4	5
1	1	2	3	4	5	10
2	2	3	4	5	10	11
3	3	4	5	10	11	12
4	4	5	10	11	12	13
5	5	10	11	12	13	14

In the table, the sums that need to be memorized are highlighted in bold. Since 0 + a = a, there's no need to memorize sums involving the number 0, as adding zero has no impact on the number being added. Additionally, the commutative property of addition (a + b = b + a) means that for repeated sums where the order of the terms changes, you only need to memorize one of the sums. In total, you only need to memorize 15 (dec) / 23 (hex) sums, which is quite manageable.

For addition involving numbers with more than one digit in the heximal numbering system, an algorithm similar to decimal addition is used. In the decimal system, you align the two numbers vertically, starting with the units' digits, add each aligned pair of digits, and account for any carries. If the result exceeds ten, you carry over to the next line. This process is analogous in heximal. First, pad the smaller number with zeroes on the left until it matches the number of digits in the larger number. Then, align the numbers vertically, starting with the units' digits, and add the aligned digits. If the result exceeds six, carry over one to the next line. Importantly, use the results from the previously memorized addition table for heximal sums rather than relying on the decimal sum.

Let's illustrate this with an example: 455 (hex) + 305 (hex). To calculate the sum, first align the numbers vertically, starting with the units' digits, as shown in the table below:

+	4	5	5
	3	0	5

Next add the digits as follow:

+	4	5	5
	3	0	5
	11	5	14

Now move the carries to the next digit:

+	4	5	5
	3	0	5
	11	5	14
0 + 1	1	5 + 1	4
1	1 + 1 = 2	0	4

So, 455 (hex) + 305 (hex) = 1204 (hex) in the heximal numbering system.

Subtraction:

To perform subtraction or find the difference between two heximal numbers, it's important to establish a representation for negative numbers in heximal, which is analogous to how it's done in decimal. You simply add a "–" sign after the number to indicate negativity. For example, 405 (hex) is positive, and –405 (hex) is negative.

When subtracting, always subtract the number with the highest absolute value from the one with the smaller absolute value and include the negative sign if necessary when performing the inverse operation. You can apply a similar technique as in decimal subtraction, using the heximal addition values. Additionally, there's a method to subtract two heximal numbers using only addition, which can be adapted to decimal calculations if you are unfamiliar with it.

For example, let's calculate 3134 (hex) – 2331 (hex):

1. Determine the number with the greatest number of digits, which is 3134 (hex) in this case. It has 4 digits. Use the value 5555 (hex), which is the highest 4–digit number in heximal. Calculate 5555 (hex) – 2331 (hex). Since all the digits in 5555 (hex) are greater than or equal to the corresponding digits in 2331 (hex), the subtraction is straightforward:

5555 (hex) – 2331 (hex) = 3224 (hex)

2. Now, calculate the sum of 3134 (hex) + 3224 (hex):

3134 (hex) + 3224 (hex) = 10402 (hex)

3. Subtract 10000 (hex), which is one more than 5555 (hex):

10402 (hex) – 10000 (hex) = 402 (hex)

4. Add 1 to the result:

402 (hex) + 1 (hex) = 403 (hex)

So, 3134 (hex) – 2331 (hex) = 403 (hex). This technique is based on the idea of performing x + (a – y) – (a + 1) + 1, where a = 5555 (hex). The result simplifies to x – y, but it can be easier to perform compared with normal subtraction. If the numbers had more digits, you'd use the highest value with that many digits in heximal.

For example, for 435121 (hex) – 201431 (hex), you'd use the value 555555 (hex), as it's the highest 6–digit value in heximal. Subtract the smaller number from the larger one, and if you are performing the inverse operation, multiply by –1. For example, 2331 (hex) – 3134 (hex) = –403 (hex).

Multiplication:

Similar to the decimal numbering system, the heximal numbering system has a multiplication table that needs to be memorized. However, because the table is six by six and there are some multiplication rules like 0 x a = 0, 1 x a = a, and a x b = b x a, you only need to memorize 10 (dec) or 14 (hex) distinct products. To put it in perspective, in the decimal multiplication table, you would need to memorize 36 (dec) or 100 (hex) products, making the heximal multiplication table more manageable. Additionally, since the 1–digit numbers in heximal are smaller, memorizing each product in the heximal multiplication table is easier compared to the decimal multiplication table.

Here is the multiplication table:

x	1	2	3	4	5
0	0	0	0	0	0
1	1	2	3	4	5
2	2	4	10	12	14
3	3	10	13	20	23
4	4	12	20	24	32
5	5	14	23	32	41

In bold are the products that need to be memorized. As you can see, only a small portion of the table needs to be memorized, which is less than one–third, and you only need to memorize 10 (dec) or 14 (hex) distinct products. To make memorization easier, you can identify patterns. For instance, in the column of 2’s, you can observe that numbers repeat in the cycle: 0, 2, 4, 0, 2, 4… Similarly, the column of 3’s repeats with the cycle: 0, 3, 0, 3… The pattern in the column of 4’s is the reverse of the column of 2’s: 0, 4, 2, 0, 4, 2… Lastly, the column of 5’s follows this repeating pattern: 0, 5, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1…, which is reminiscent of the integers in reverse order, similar to the column of 9’s in the decimal multiplication table.

After memorizing the multiplication table, you can use the long multiplication method to multiply larger numbers, incorporating heximal addition instead of decimal addition. Alternatively, you can employ an alternative method that enables you to multiply not only numbers in any positional numbering system but also polynomials.

For example, let's multiply 513 (hex) x 344 (hex):

1. Organize the numbers in a grid:

x	3	4	4
5
1
3

2. Multiply the digits using the values from the multiplication table:

x	3	4	4
5	5 x 3	5 x 4	5 x 4
1	1 x 3	1 x 4	1 x 4
3	3 x 3	3 x 4	3 x 4

3. Calculate the results:

x	3	4	4
5	23	32	32
1	3	4	4
3	13	20	20

4. Add the numbers vertically:

x	3	4	4
5	23	32	32
1	3	4	4
3	13	20	20

Add the numbers vertically, with the help of the colours: 20 = 20, 20 + 4 = 24, 13 + 4 + 32 = 53, 3 + 32 = 35, 23 = 23. Now, add n zeroes after each result, where n is the position:

20–>20; 24–>240; 53–>5300; 35–>35000; 23–>230000

5. Add the values:

230000 + 35000 + 5300 + 240 + 20 = 315000

So, 513 (hex) x 344 (hex) = 315000 (hex). Using this algorithm, you can multiply any number in the heximal numbering system. Furthermore, this algorithm can be adapted to the decimal numbering system by substituting heximal multiplication with decimal multiplication, and it can also be used to multiply polynomials.

Division:

Before diving into division in other positional numbering systems, it's essential to understand rational numbers and their representation in decimal. Fractions are typically denoted as a/b, where a and b are integers and coprime, representing the proportion of those integers. This representation can be adapted to the heximal numbering system. Another way to represent fractions is by using the radix point, as in 1/2 (dec) = 0.5 (dec) or 1/3 (dec) = 0.333333… (dec). The radix point signiffies that the digits are being multiplied by a negative power of that base. For example, 0.5 (dec) = 5 x 10^(–1) (dec), and 0.333333… (dec) = 3 x 10^(–1) + 3 x 10^(–2) + 3 x 10^(–3) + 3 x 10^(–4) + 3 x 10^(–5) + 3 x 10^(–6)… (dec). You can apply this notation to the heximal numbering system, where the radix point indicates that the digits are being multiplied by a negative power of six, and it's called the "heximal point," just like the radix point is called the "decimal point" in the decimal numbering system.

For example, 0.3 (hex) means 3 x 6^(–1), and 0.111111… (hex) means 1 x 6^(–1) + 1 x 6^(–2) + 1 x 6^(–3) + 1 x 6^(–4) + 1 x 6^(–5) + 1 x 6^(–6)… (hex). Just as in decimal division, the heximal point is used when the remainder is not 0 in a quotient.

To divide two numbers in the heximal numbering system, you use an adaptation of long division in the decimal numbering system. However, subtraction is done using heximal numbering system rules, and the digits range from 0 to 5 for the quotient. You can stop after obtaining the integer part of the quotient, getting a remainder, or continue using the heximal point (decimal point in the heximal numbering system) to approximate the quotient or identify repeating digits in the quotient.

Example: 434431 (hex) ÷ 501 (hex). To calculate the result, first write both numbers as you would in long division if they were decimal numbers:

4	3	4	4	3	1	÷	5	0	1

Now calculate all multiples of 501 (hex), the divisor, from 0 to 5: 0 (hex) x 501 (hex) = 0 (hex); 1 (hex) x 501 (hex) = 501 (hex); 2 (hex) x 501 (hex) = 1402 (hex); 3 (hex) x 501 (hex) = 2303 (hex); 4 (hex) x 501 (hex) = 3204 (hex); 5 (hex) x 501 (hex) = 4105 (hex). Use these values in the calculation. Since 4344 (hex) is greater than 501 (hex), start with that value. Subtract 4105 (hex) from 4344 (hex), resulting in 235 (hex). The first digit of the quotient is 5, as 4105 (hex) = 5 (hex) x 501 (hex). Drop the 3, leaving 2353 (hex). Repeat this process similarly to decimal long division until you obtain the remainder and the quotient.

4	3	4	4	3	1	÷	5	0	1
–4	1	0	5							5
0	2	3	5
	2	3	5	3
	–2	3	0	3						3
	0	0	5	0
			5	0	1
			–5	0	1					1
			0	0	0

After these calculations, it's concluded that 434431 (hex) ÷ 501 (hex) = 531 (hex) with a remainder of 0.

Here's another example when the remainder is not 0: 4424 (hex) ÷ 43 (hex). Calculate all multiples of 43 (hex) from 0 to 5: 0 (hex); 43 (hex); 130 (hex); 213 (hex); 300 (hex); 343 (hex). Start the calculation:

4	4	2	4.	0	0	0	÷	4	3
–4	3									1
0	1
	1	2
	–0	0								0
	1	2
	1	2	4
	–0	4	3							1
	0	4	1
		4	1	0						.
		–3	4	3						5
		0	2	3
			2	3	0
			–2	1	3					3
			0	1	3
				1	3	0
				–1	3	0				2
				0	0	0

So that means that 4424 (hex) ÷ 43 (hex) = 101.532 (hex). Also, since the remainder is 41 (hex), 4424 (hex) = 101 (hex) x 43 (hex) + 41 (hex). The quotient is 101 (hex), the remainder 41 (hex) and the exact value is 101.532 (hex).

Here is the last example: 1 (hex) ÷ 11 (hex). Once again, you start by calculating the multiples of 11 (hex), which are 0 (hex); 11 (hex); 22 (hex); 33 (hex); 44 (hex); 55 (hex). Since 1 (hex) is smaller than 11 (hex), the quotient is 0 (hex), and the remainder is 1 (hex). However, if you continue calculating the quotient after the remainder:

1.	0	0	0	0	0	÷	1	1
–0									0
1									.
1	0
–0	0								0
1	0
1	0	0
	–5	5							5
		0	1
		0	1	0
			–0	0					0
			1	0
			1	0	0
				–5	5				5
				0	1

As seen, the digits will cycle forever with a period of 2, similar to decimal fractions, since this property holds for all integer bases. In this case, 1 (hex) ÷ 11 (hex) = 0.05050505050505… (hex), or simply 0.05 (hex), the underline indicates the digits that repeat.

Prime Numbers and divisibility tests for some integers in heximal

The list of prime numbers in decimal is as follows:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, and so on…

Prime numbers are fundamental in arithmetic. Knowing the prime factorization of a number allows you to calculate its divisors and numbers that are coprime to it. Therefore, they are important to mention in the context of the heximal numeral system.

In other integer bases, the prime numbers remain the same as in decimal, but they need to be correctly converted to that base. In heximal, the prime numbers look as follows:

2, 3, 5, 11, 15, 21, 25, 31, 35, 45, 51, 101, 105, 111, 115, 125, 135, 141, 151, 155, 201, 211, 215, 225, 241, 245, 251, 255, 301, 305, 331, 335, 345, 351, 405, 411, 421, 431, 435, 445, 455, 501, 515, 521, 525, 531, 551…

Notably, every prime number in the heximal numbering system, except 2 and 3, ends with the digit 1 or 5. This pattern is related to the divisibility tests within the heximal base. The primality of a number and its factors do not depend on the representation of the positional numbering system. Converting a prime number from heximal to decimal will not change its primality, and similarly, converting a composite number from heximal to decimal will retain its factors.

Divisibility tests for some integers in heximal:

1 – A number is divisible by 1 if it terminates in the heximal numbering system. For example, 421 (hex) is divisible by 1 because it terminates, but 240.1 (hex) is not.

2 – A number is divisible by 2 if its last digit in the heximal numbering system is 0, 2, or 4. For example, 520 (hex) is divisible by 2 because it ends with 0, but 241 (hex) is not. This test is similar to checking if a number is even or odd in decimal.

3 – A number is divisible by 3 if its last digit in the heximal numbering system is 0 or 3. For example, 513 (hex) is divisible by 3 because it ends with 3, but 244 (hex) is not. This test is similar to the divisibility test for 5 in the decimal system.

4 – A number is divisible by 4 if its last 2 digits in the heximal numbering system form a multiple of 4. The combinations that indicate divisibility by 4 are: 00, 04, 12, 20, 24, 32, 40, 44, 52. For example, 552 (hex) is a multiple of 4 because the last 2 digits are 52, and 224 (hex) is also a multiple of 4 because the last 2 digits are 24. However, 242 (hex) is not divisible by 4 because 42 (hex) is not a multiple of 4.

5 – A number is divisible by 5 if the sum of its digits in the heximal numbering system is a multiple of 5. For example, 11111 (hex) is a multiple of 5 because 1+1+1+1+1 (hex) = 5 (hex), and 12345 (hex) is also a multiple of 5 because 1+2+3+4+5 (hex) = 23 (hex), and 2+3 (hex) = 5. On the other hand, 1541 (hex) is not divisible by 5 because 1+5+4+1 (hex) = 15 (hex), and 1+5 (hex) = 10 (hex), and 1+0 (hex) = 1, which is not a multiple of 5. This test is similar to casting out 9’s in decimal, but it only works for the prime number 5.

10 (hex) / 6 (dec) – A number is divisible by 10 (hex) if its last digit is 0, which is similar to the divisibility test for 10 (dec) in the decimal numbering system. For example, 430 (hex) is a multiple of 10 (hex), but 4424 (hex) is not a multiple of 10 (hex).

11 (hex) / 7 (dec) – A number is divisible by 11 (hex) if the difference of the alternating sum of digits is a multiple of 11 (hex). This is similar to the divisibility test of 11 (dec) in the decimal numbering system. Example: 44535 (hex) is divisible by 11 (hex) because 4 + 5 + 5 (hex) = 22 (hex), 4 + 3 (hex) = 11, and 22 – 11 (hex) = 11 (hex) , and 11 (hex) is a multiple of 11 (hex).

12 (hex) / 8 (dec) – A number is divisible by 12 (hex) if the last 3 digits form a multiple of 12 (hex), making the last 3 digits the determinant factor of divisibility by 12 (hex). The combinations of 3 digits that form a multiple of 12 are: 000, 012, 024, 040, 052, 104, 120, 132, 144, 200, 212, 224, 240, 252, 304, 320, 332, 344, 400, 412, 424, 440, 452, 504, 520, 532, 544. For example, 2200 (hex) is divisible by 12 (hex) because it ends in 200 (hex), which is a multiple of 12 (hex).

13 (hex) / 9 (dec) – A number is divisible by 13 (hex) if the last 2 digits form a multiple of 13 (hex), which means that the determinant factor of divisibility by 13 (hex) is the last 2 digits. The combinations are: 00, 13, 30, 43. For example, 543 (hex) is divisible by 13 (hex) because the last 2 digits are 43 (hex). This test is similar to the divisibility test of 25 (dec) in decimal.

14 (hex) / 10 (dec) – A number is divisible by 14 (hex) if it is divisible by 2 (hex) and 5 (hex), so the last digit is either 0, 2, or 4, and the sum of digits is a multiple of 5 (hex). For example, 244 (hex) is divisible by 14 (hex) because the last digit is 4, and the sum of digits is 14 (hex).

20 (hex) / 12 (dec) – A number is divisible by 20 (hex) if the last 2 digits form a multiple of 20 (hex). The combinations are: 00, 20, 40. For example, 3300 (hex) is divisible by 20 (hex) because the last 2 digits are 00 (hex).

22 (hex) / 14 (dec) – A number is divisible by 22 (hex) if it is divisible by 2 (hex) and 11 (hex), so the last digit is either 0, 2, or 4, and the difference of the alternating sum of digits is a multiple of 11 (hex). For example, 524 (hex) is divisible by 22 (hex) because the last digit is 4, and the alternating sum is 0 (hex), which is a multiple of 11 (hex).

23 (hex) / 15 (dec) – A number is divisible by 23 (hex) if it is divisible by 3 (hex) and 5 (hex), so the last digit is either 0 or 3, and the sum of digits is a multiple of 5 (hex). For example, 1013 (hex) is divisible by 23 (hex) because the last digit is 3, and the sum of digits is 5 (hex).

24 (hex) / 16 (dec) – A number is divisible by 24 (hex) if the last 4 digits form a multiple of 24 (hex). There are 213 (hex) / 81 (dec) combinations, and here are they: 0000, 0024, 0052, 0120, 0144, 0212, 0240, 0304, 0332, 0400, 0424, 0452, 0520, 0544, 1012, 1040, 1104, 1132, 1200, 1224, 1252, 1320, 1344, 1412, 1440, 1504, 1532, 2000, 2024, 2052, 2120, 2144, 2212, 2240, 2304, 2332, 2400, 2424, 2452, 2520, 2544, 3012, 3040, 3104, 3132, 3200, 3224, 3252, 3320, 3344, 3412, 3440, 3504, 3532, 4000, 4024, 4052, 4120, 4144, 4212, 4240, 4304, 4332, 4400, 4424, 4452, 4520, 4544, 5012, 5040, 5104, 5132, 5200, 5224, 5252, 5320, 5344, 5412, 5440, 5504, 5532. For example, 1223224 (hex) is divisible by 24 (hex) because the last 4 digits are 3224 (hex), which is a multiple of 24 (hex).

30 (hex) / 18 (dec) – A number is divisible by 30 (hex) if the last 2 digits form a multiple of 30 (hex). The combinations are 00 and 30. For example, 423430 (hex) is divisible by 30 (hex) because the last 2 digits are 30 (hex). This test is similar to the divisibility test of 50 (dec) in decimal.

33 (hex) / 21 (dec) – A number is divisible by 33 (hex) if it is divisible by 3 (hex) and 11 (hex), so the last digit is 0 or 3, and the difference of the alternating sum of digits is a multiple of 11 (hex). For example, 2013 (hex) is divisible by 33 (hex) because the last digit is 3, and the alternating sum is 0 (hex), which is a multiple of 11 (hex).

40 (hex) / 24 (dec) – A number is divisible by 40 (hex) if the last 3 digits form a multiple of 40 (hex). The combinations are: 000, 040, 120, 200, 240, 320, 400, 440, 520. For example, 324240 (hex) is divisible by 40 (hex) because the last 3 digits are 240 (hex).

43 (hex) / 27 (dec) – A number is divisible by 43 (hex) if the last 3 digits form a multiple of 43 (hex). The combinations are: 000, 043, 130, 213, 300, 343, 430, 513. For example, 4132343 (hex) is divisible by 43 (hex) because the last 3 digits are 343 (hex).

50 (hex) / 30 (dec) – A number is divisible by 50 (hex) if it is divisible by 10 (hex) and by 5 (hex), so the last digit is 0, and the sum of digits is a multiple of 5 (hex). For example, 15130 (hex) is divisible by 50 (hex) because it ends in 0, and the sum of digits is 5 (hex).

55 (hex) / 35 (dec) – A number is divisible by 55 (hex) if the number is divisible by 5 (hex) and 11 (hex). This can also be checked by summing all groups of 2 digits in the number.

100 (hex) / 36 (dec) – A number is divisible by 100 (hex) if the last 2 digits of that number are 00. For example, 34300 (hex) is divisible by 100 (hex) because it ends in 00. This test is similar to the divisibility test of 100 (dec) in decimal.

To find other divisibility tests for 2^n and 3^m, just check if the last n/m digits form a multiple of 2^n / 3^m, and for numbers that are of the form 2^n x 3^m, the calculate the maximum of m and n, which is r, and check if the last r digits form a multiple of 2^n x 3^m.

Generalized divisibility tests for some primes for heximal

The divisibility test for the remaining primes is very similar to all, so there are 2 similar generalized divisibility tests for primes for specific primes, so you first need to check the last digit of the prime to know which test to use. For primes ending with 5 in heximal:

1. Add 1 to the prime number.

2. Divide the result by 10 (hex) to get the multiplier

3. For the number being tested, separate the last digit from the rest.

4. Multiply the last digit by the multiplier.

5. Add the result to the rest of the number.

10 (hex). Repeat steps 3–5 until you either find a number divisible by the prime or determine that it's not divisible.

This method can be used to check the divisibility of primes ending with 5 and can also be applied to composite numbers. However, it's particularly useful for prime numbers, and perfect prime powers, since primes don’t have any factors other than themselves, so you can’t just check divisibility by all factors separately.

For primes ending with 1 in heximal, only 1 step changes:

1. Subtract 1 to the prime number.

2. Divide the result by 10 (hex) to get the multiplier

3. For the number being tested, separate the last digit from the rest.

4. Multiply the last digit by the multiplier.

5. Subtract the result to the rest of the number.

10 (hex). Repeat steps 3–5 until you either find a number divisible by the prime or determine that it's not divisible.

This method can be used to check the divisibility of primes ending with 1 and can also be applied to composite numbers. However, it's particularly useful for prime numbers, and perfect prime powers, since primes don’t have any factors other than themselves, so you can’t just check divisibility by all factors separately.

There is a faster divisibility test for some primes:

For Primes Ending with 55 in heximal:

1. Add 1 to the prime number.

2. Divide the result by 100 (hex) to obtain the multiplier.

3. For the number being tested, separate the last two digits from the rest.

4. Multiply the last two digits by the multiplier.

5. Add the result to the rest of the number.

10 (hex). Repeat steps 3–5 until you either find a number divisible by the prime or determine that it's not divisible.

This also works for composite numbers, but it is only useful for primes and prime powers.

For Primes Ending with 01 in heximal

1. Subtract 1 from the prime number.

2. Divide the result by 100 (hex) to obtain the multiplier.

3. For the number being tested, separate the last two digits from the rest.

4. Multiply the last two digits by the multiplier.

5. Subtract the result from the rest of the number.

10 (hex). Repeat steps 3–5 until you either find a number divisible by the prime or determine that it's not divisible.

This also works for composite numbers, but it is only useful for primes or prime powers.

Additionally, this trick can be generalized for primes where p + 1 or p – 1 ends with n zeroes. Divide the result by 10^n (hex) to get the multiplier and split the last n digits from the number to perform an even faster divisibility test.

This optimization allows for quicker divisibility tests for certain prime numbers, improving efficiency in calculations.

Prime factorizations and square roots in heximal

To do the prime factorization of a number you first need to calculate the square root of that number, so you know when you can be safe to say that the number is prime.

To calculate the square root of a heximal number, follow these steps:

1. Begin by separating the digits of the heximal number into groups of two. For instance, for √113343001 (hex), separating the digits makes the list: 01 13 34 30 01

2. Find the largest square that's at most the value of the first group of digits. In this case, 01 (hex) is the largest square that's less than or equal to 01 (hex), so 1 is the first digit of the square root.

3. Subtract the square found in step 2 (in this case, 1) from the first group of digits, which results in 0. The remaining part of the number is 13 34 30 01.

4. Add 1 to itself, resulting in 2 (hex), and then try to find a digit n such that when you multiply it by 2, you get a number smaller than 13 (hex) which is the next part of the number.

– For n=2, you get 22 x 2 (hex) = 44 (hex)

– For n=5, you get 25 x 5 (hex) = 221 (hex)

– For n=1, you get 21 x 1 (hex) = 21 (hex)

In this case, n must be 0 since 21 (hex) is greater than 13 (hex). Therefore, the next digit of the square root is 0.

5. Add 0 to 20 (hex) to get 20 (hex), and repeat the process. Find the largest digit n such that when you multiply it by 20 and add it to the current number 20 (hex), you get a number smaller than 1334 (hex).

– For n = 4, 204 x 4 (hex) = 1224 (hex), and 1334 – 1224 (hex) = 110 (hex).

The next digit is 4.

10 (hex). Continue this process until you reach the desired precision or the end of the number.

204 + 4 (hex) = 212 (hex)

2123 x 3 (hex) = 10413 (hex)

11030 – 10413 (hex) = 213 (hex)

The next digit is 3.

2123 + 3 (hex) = 2130 (hex)

21301 x 1 (hex) = 21301 (hex)

21301 – 21301 (hex) = 0 (hex)

The final digit is 0.

11 (hex). The square root of 113343001 (hex), √113343001 (hex) is 10431 (hex).

Here's a visual representation of the process:

							1	0	4	3	1

	√	0	1	1	3	3	4	3	0	0	1
1	–	0	1
2		0	0
2		0	0	1	3
20	–	0	0	0	0
20		0	0	1	3
20		0	0	1	3	3	4
204	–	0	0	1	2	2	4
212		0	0	0	1	1	0
212		0	0	0	1	1	0	3	0
2123	–	0	0	0	1	0	4	1	3
2130		0	0	0	0	0	2	1	3
2130		0	0	0	0	0	2	1	3	0	1
21301	–	0	0	0	0	0	2	1	3	0	1
21302		0	0	0	0	0	0	0	0	0	0

This method enables you to accurately calculate the square root of a heximal number up to the integer part.

To perform prime factorization of a number in heximal, follow these steps:

1. Begin by finding the square root of the number and round it down to the nearest integer. Let's consider the example of √550 (hex), which the integer part results in 22 (hex).

Since the largest prime smaller than or equal to 22 (hex) is 21 (hex), you only need to check for divisibility up to 21 (hex).

2. Check for divisibility by 2. Since the last digit is 0, the number is divisible by 2. In this case, divide 550 (hex) by 2 to get 253 (hex).

Continue the process by finding the square root of 253 (hex), which the integer part is 14 (hex). Since 253 (hex) ends with 3, it is divisible by 3. Dividing 253 (hex) by 3 gets the result results: 253 ÷ 3 (hex) = 55 (hex).

3. Continue this process until you find more factors. After each division, find the square root of the remaining number and check for divisibility with primes up to the square root.

In this example, 55 (hex) is divisible by 5 (hex), and, 55 ÷ 5 (hex) = 11 (hex).

4. Once you reach a point where the square root of the remaining number is smaller than the current factor being checked that number is a prime, and it is the last prime factor of the original number.

In this case, the integer part of √11 (hex) = 2 (hex), which is less than 5 (hex). Therefore, the factorization is complete.

In this case, 550 (hex) factors into 2 x 3 x 5 x 11 (hex).

Here's another example with the number 5001 (hex):

1. Calculate √5001 (hex), which the integer part results in 52 (hex).

Since the largest prime smaller than 52 (hex) is 51 (hex), you only need to check for divisibility up to 51 (hex).

2. Examine the last digit of 5001 (hex). In this case, it's 1, so it's not divisible by 2 or 3.

3. Continue the process, and you'll find that 5001 (hex) is divisible by 35 (hex), resulting in 5001 ÷ 35 (hex) = 115 (hex).

4. Calculate √115 (hex), which the integer part is 10 (hex).

Since 35 (hex) is greater than 10 (hex), 115 (hex) is prime.

So, 5001 (hex) factors into 35 x 115 (hex).

In summary, to perform prime factorization in heximal, you first find the square root of the number to determine how many factors you need to check. Utilize divisibility tests for primes smaller than the square root to identify if a number is prime or not. The prime factorization of negative integers is just the factorization of that integer with the ‘–‘ in the beginning, the factorization of 1 is the empty product, and the factorization of –1 is the empty product multiplied by –1. The prime factorization of 0 is undefined. For the numbers 2 and 3, they are considered prime since there are no smaller primes to check below the square root.

Nomenclature of Numbers in Heximal

In the decimal numbering system, we use the decimal representation of numbers to name them. I've created a similar naming system based on the heximal representation of numbers. For numbers between zero and twelve, the names remain the same:

· 0 (hex) is zero

· 1 (hex) is one

· 2 (hex) is two

· 3 (hex) is three

· 4 (hex) is four

· 5 (hex) is five

· 10 (hex) is six

· 11 (hex) is seven

· 12 (hex) is eight

· 13 (hex) is nine

· 14 (hex) is ten

· 15 (hex) is eleven

· 20 (hex) is twelve

For the remaining integers, I assigned different names. From 21 (hex) to 25 (hex), I used "dozen" followed by the respective units. This is in line with the common practice of using "dozen" to represent twelve when counting objects.

· 21 (hex) is "dozen–one"

· 22 (hex) is "dozen–two"

· 23 (hex) is "dozen–three"

· 24 (hex) is "dozen–four"

· 25 (hex) is "dozen–five"

For the next numbers, I introduced invented words:

· 30 (hex) is "thirsy" (three times six)

· 31 (hex) is "thirsy–one"

· 32 (hex) is "thirsy–two"

· 33 (hex) is "thirsy–three"

· 34 (hex) is "thirsy–four"

· 35 (hex) is "thirsy–five"

I changed "thirty" to "thirsy" to indicate that it's three times six, similar to how "thirty" in decimal represents three times ten. The second–to–last letter indicates whether the number is decimal or heximal.

The following numbers are also named following this logic:

· 40 (hex) is "forsy" (four times six)

· 41 (hex) is "forsy–one"

· 42 (hex) is "forsy–two"

· 43 (hex) is "forsy–three"

· 44 (hex) is "forsy–four"

· 45 (hex) is "forsy–five"

· 50 (hex) is "fifsy" (five times six)

· 51 (hex) is "fifsy–one"

· 52 (hex) is "fifsy–two"

· 53 (hex) is "fifsy–three"

· 54 (hex) is "fifsy–four"

· 55 (hex) is "fifsy–five"

For the next numbers, I used non–English words:

· 100 (hex) is "niff”, nif in the IPA,

· 101 (hex) is "niff–one"

· 102 (hex) is "niff–two"

· 103 (hex) is "niff–three"

· 104 (hex) is "niff–four"

· 105 (hex) is "niff–five"

· 110 (hex) is "niff–six"

· 111 (hex) is "niff–seven"

· 112 (hex) is "niff–eight"

· 113 (hex) is "niff–nine"

· 114 (hex) is "niff–ten"

· 115 (hex) is "niff–eleven"

· 120 (hex) is "niff–twelve"

In the Ndom language of Indonesian New Guinea, they used a word similar in sound to "nif" to represent six times six, which is 100 (hex). Since they used a heximal numbering system, having a word for thirty–six was essential, which is why I chose "niff" for thirty–six. From 120 (hex) to 155 (hex), the names are just “niff” followed by the name of the rest of the number.

· 124 (hex) is "niff dozen–four"

· 141 (hex) is "niff forsy–one"

· 155 (hex) is "niff fifsy–five"

Moving on, we have:

· 200 (hex) is "two niff"

· 201 (hex) is "two niff one"

· 202 (hex) is "two niff two"

This structure is similar to the hundredths structure in decimal where you state how many hundreds followed by the rest. For instance, 244 (hex) is "two niff forsy–four," 321 (hex) is "three niff dozen–one," 344 (hex) is "three niff forsy–four," 441 (hex) is "four niff forsy–one," 524 (hex) is "five niff dozen–four," 543 (hex) is "five niff forsy–three," and finally, 555 (hex) is "five niff fifsy–five."

For the larger numbers:

· 1,000 (hex) is "tarnexy”, tärneksi in the IPA,

· 1,001 (hex) is "one tarnexy and one"

· 1,002 (hex) is "one tarnexy and two"

· 1,003 (hex) is "one tarnexy and three"

· 1,004 (hex) is "one tarnexy and four"

· 1,005 (hex) is "one tarnexy and five"

· 1,010 (hex) is "one tarnexy and six"

I chose "tarnexy" since the Yam languages in Papua New Guinea had a word that sounded similar to "tarumba," which meant six cubed or 1,000 (hex). For numbers between 1,001 (hex) and 5,555 (hex), it's straightforward. For instance, 2,013 (hex) is "two tarnexy and nine," 2,241 (hex) is "two tarnexy two niff forsy–one," 2,521 (hex) is "two tarnexy five niff dozen–one," 3,213 (hex) is "three tarnexy two niff–nine," and 4,344 (hex) is "four tarnexy three niff forsy–four."

The pattern continues until 5,555 (hex), which is "five tarnexy five niff fifsy–five." Beyond this, the next name for a number only comes after six to the power of six:

· 10,000 (hex) is "six tarnexy"

· 11,000 (hex) is "seven tarnexy"

· 12,000 (hex) is "eight tarnexy"

· 13,000 (hex) is "nine tarnexy"

· 14,000 (hex) is "ten tarnexy"

· 15,000 (hex) is "eleven tarnexy"

· 20,000 (hex) is "twelve tarnexy"

· 30,000 (hex) is "thirsy tarnexy"

· 40,000 (hex) is "forsy tarnexy"

· 55,555 (hex) is "fifsy–five tarnexy five niff fifsy–five"

· 100,000 (hex) is "niff tarnexy"

· 110,000 (hex) is "niff–six tarnexy"

· 120,000 (hex) is "niff–twelve tarnexy"

· 130,000 (hex) is "niff thirsy tarnexy"

· 200,000 (hex) is "two niff tarnexy"

· 300,000 (hex) is "three niff tarnexy"

· 400,000 (hex) is "four niff tarnexy"

· 555,555 (hex) is "five niff fifsy–five tarnexy five niff fifsy–five"

· 1;000,000 (hex) is “one unwixion”, wən uːnuiksˈiːɔn in the IPA

I chose “unwixion”, since in the Yam languages had a word similar in sound to “wi”, that meant six to the power of six, so I put an “un” prefix, which you will see why later, and I put a “xion” suffix to have a similar end to the word “million”, since one unwixion is the heximal equivalent of the decimal million.

· 1;000,001 (hex) is “one unwixion and one”

· 1,001,000 (hex) is “one unwixion and one tarnexy”

· 1;111,111 (hex) is “one unwixion niff–seven tarnexy niff–seven”

· 10;000,000 (hex) is “six unwixion”

· 100;000,000 (hex) is “niff unwixion”

· 1,000;000,000 (hex) is “tarnexy unwixion”

· 10,000;000,000 (hex) is “six tarnexy unwixion”

· 100,000;000,000 (hex) is “niff tarnexy unwixion”

· 1;000,000;000,000 (hex) is “one biwixion”

· 10;000,000;000,000 (hex) is “six biwixion”

· 100;000,000;000,000 (hex) is “niff biwixion”

· 1,000;000,000;000,000 (hex) is “tarnexy biwixion”

· 10,000;000,000;000,000 (hex) is “six tarnexy biwixion”

· 100,000;000,000;000,000 (hex) is “niff tarnexy biwixion”

· 1;000,000;000,000;000,000 (hex) is “one triwixion”

· 1;000,000;000,000;000,000;000,000 (hex) is “one tetrwixion”

· 1;000,000;000,000;000,000;000,000;000,000 (hex) is “one pentwixion”

· 1;000,000;000,000;000,000;000,000;000,000;000,000 (hex) is “one unnilwixion”…

For larger numbers there are some rules to determine the name of that number. First you need to calculate the power of unwixion corresponding with that number. For example 1 biwixion = 10^20 (hex) is equal to unwixion squared, 1 triwixion = 10^30 (hex) is equal to unwixion cubed, and 1 pentwixion = 10^50 (hex) is equal to unwixion to the power of five. The prefixes are ‘nil’ for the digit 0, ‘un’ for the digit 1, ‘bi’ for the digit 2, ‘tri’ for the digit 3, ‘tetr’ for the digit 4, and finally ‘pent’ for the digit 5. 10^100 (hex) is unnilwixion, since it is the same as unwixion to the power of six, and six is 10 (hex), which has the digits 1 and 0 in that order; 10^140 (hex) is untetrwixion, since it is just unwixion to the power of ten, and ten is 14 (hex). You just remove the last zero in the number to get the power of unwixion, since 1 unwixion = 10^10 (hex). For numbers that are not powers of unwixion, like 10^253 (hex) use six, niff and tarnexy for the name, in this case 10^253 (hex) is tarnexy bipentwixion. There are special rules for the prefixes, and in the case a ‘un’ is followed by a ‘bi’ or ‘pent’, then it transforms into ‘um’, so 10^150 (hex) is umpentwixion, not unpentwixion, and if ‘pent’ is followed by ‘nil’ or ‘bi’ or ‘tri’ or ‘tetr’ or ‘pent’, then it transforms into ‘penta’, and if ‘tetr’ is followed by ‘nil’ or ‘bi’ or ‘tri’ or ‘tetr’ or ‘pent’ then it transforms into ‘tetra’. These prefixes are the same used for naming chemical elements that weren’t discovered, so that they can be referred to even if they haven’t been discovered. There are two other names for two large numbers. A goosol is defined to be 10^(unwixion) (hex), and a goosolplex is defined to be 10^(goosol) (hex), which correspond to the third and fourth tetrations of six respectively. I used the ‘,’ to separate each power of tarnexy, and a ‘;’ to separate each power of unwixion. To name numbers that are not positive integers here is how:

Negative Numbers

· Negative numbers are indicated simply by adding the word "negative" before the corresponding positive number's name.

· For example, –13 (hex) is named "negative nine," and –144 (hex) is called "negative niff forsy–four."

Rational/Real numbers

· Rational and Real numbers are named by pronouncing the digits after the heximal point sequentially, and in the case, there is only one digit after the heximal point, then just use the exact result.

· For instance, 23.1043 (hex) becomes "dozen–three point one zero four three,", –144.4 (hex) is "negative niff forsy–four and two thirds, and 1.5 (hex) is one and five sixths.

Complex Numbers

· Complex numbers follow the standard naming convention, where you first state the real part and then the imaginary part, or you can just state the absolute value followed by the argument, in case the complex number is in polar form.

Names of Ordinal numbers and polygons

Ordinal numbers:

The name of ordinal numbers is also very simple:

Ordinals from 0^th to 20^th (hex) have the same name:

· 0^th is ‘zeroth’

· 1^st is ‘first’

· 2^nd is ‘second’

· 3^rd is ‘third’

· 4^th is ‘fourth’

· 5^th is ‘fifth’

· 10^th (hex) is ‘sixth’

· 11^th (hex) is ‘seventh’

· 12^th (hex) is ‘eighth’

· 13^th (hex) is ‘nineth’

· 14^th (hex) is ‘tenth’

· 15^th (hex) is ‘eleventh’

· 20^th (hex) is ‘twelfth’

Ordinals between 21^st (hex) and 25^th (hex):

· 21^st (hex) is ‘dozen first’

· 22^nd (hex) is ‘dozen second’

· 23^rd (hex) is ‘dozen third’

· 24^th (hex) is ‘dozen fourth’

· 25^th (hex) is ‘dozen fifth’

Ordinals between 30^th (hex) and 55^th (hex):

· 30^th (hex) is ‘thirsieth’

· 31^st (hex) is ‘thirsy first’

· 32^nd (hex) is ‘thirsy second’

· 33^rd (hex) is ‘thirsy third’

· 34^th (hex) is ‘thirsy fourth’

· 35^th (hex) is ‘thirsy fifth’

· 40^th (hex) is ‘forsieth’

· 41^st (hex) is ‘forsy first’

· 42^nd (hex) is ‘forsy second’

· 43^rd (hex) is ‘forsy third’

· 44^th (hex) is ‘forsy fourth’

· 45^th (hex) is ‘forsy fifth’

· 50^th (hex) is ‘fifsieth’

· 51^st (hex) is ‘fifsy first’

· 52^nd (hex) is ‘fifsy second’

· 53^rd (hex) is ‘fifsy third’

· 54^th (hex) is ‘fifsy fourth’

· 55^th (hex) is ‘fifsy fifth’

Ordinals greater than 55^th (hex)

· 100^th (hex) is ‘niffth’

· 101^st (hex) is ‘niff first’

· 102^nd (hex) is ‘niff second’

· 103^rd (hex) is ‘niff third’

· 110^th (hex) is ‘niff sixth’

· 111^th (hex) is ‘niff seventh’

· 120^th (hex) is ‘niff twelfth’

· 122^nd (hex) is ‘niff dozen second’

· 130^th (hex) is ‘niff thirsieth’

· 144^th (hex) is ‘niff forsy fourth’

· 200^th (hex) is ‘two niffth’

· 212^th (hex) is ‘two niff eighth’

· 244^th (hex) is ‘two niff forsy fourth’

· 300^th (hex) is ‘three niffth’

· 400^th (hex) is ‘four niffth’

· 500^th (hex) is ‘five niffth’

· 1,000^th (hex) is ‘tarnexieth’

· 1,100^th (hex) is ‘tarnexy niffth’

· 1,200^th (hex) is ‘tarnexy two niffth’

· 1,300^th (hex) is ‘tarnexy three niffth’

· 2,000^th (hex) is ‘two tarnexieth’

· 3,000^th (hex) is ‘three tarnexieth’

· 4,000^th (hex) is ‘four tarnexieth’

· 5,000^th (hex) is ‘five tarnexieth’

· 10,000^th (hex) is ‘six tarnexieth’

· 100,000^th (hex) is ‘niff tarnexieth’

· 1;000,000^th (hex) is ‘unwixionth’

You can probably guess the name of the other numbers, you just add ‘th’ at the end. There is also the ordinal goosolth and goosolplexth, which you can probably know what they mean.

Polygons:

I invented a system to name polygons in a heximal way. The system offers a distinct way of naming polygons with various sides, providing a seamless transition from familiar shapes like triangles and quadrilaterals to polygons with a significant number of sides. The names are derived by combining prefixes, numerical representations, and a clever method for handling shapes with over 100 (hex) sides. Here are the names:

3 – triangle

4 – quadrilateral

5 – pentagon

10 – hexagon

11 – septagon

12 – octagon

13 – nonagon

14 – decagon

15 – elevagon

20 – dozagon

21 – Undozagon

22 – Bidozagon

23 – Tridozagon

24 – Tetradozagon

25 – Pentadozagon

30 – Tricosagon

31 – Untricosagon

32 – Bitricosagon

33 – Tritricosagon

34 – Tetratricosagon

35 – Pentatricosagon

40 – Tetracosagon

41 – Untetracosagon

42 – Bitetracosagon

43 – Tritetracosagon

44 – Tetratetracosagon

45 – Pentatetracosagon

50 – Pentacosagon

51 – Umpentacosagon

52 – Bipentacosagon

53 – Tripentacosagon

54 – Tetrapentacosagon

55 – Pentapentacosagon

100 – Nilniluncosagon

101 – Unnilcosagon

102 – Binilcosagon

103 – Trinilcosagon

104 – Tetranilcosagon

105 – Pentanilcosagon

110 – Nilununcosagon

111 – Unununcosagon

112 – Biununcosagon

For polygons with at least 100 (hex) sides, a special naming convention is employed, involving prefixes based on the digits of the number, the reversal of these digits, and a systematic approach to nomenclature. This fascinating system offers a creative and logical way to name geometric shapes within this heximal numerical framework. The only things you need to remember is to transform ‘un’ to ‘um’ in case the next prefix is ‘pent’ or ‘bi’, and when ‘pent’ and ‘tetr’ transform into ‘penta’ and ‘tetra’.

Conversion from decimal to heximal and heximal to decimal

Decimal to Heximal Conversion:

Converting numbers from decimal to heximal is a straightforward process. For instance, let's convert the decimal number 279,936 to heximal:

1. Start by dividing the decimal number by six in decimal: 279,936 ÷ 6 = 46,656 (with a remainder of 0).

2. The remainder indicates the last digit in heximal, so the rightmost digit of 279,936 in heximal is 0.

3. Repeat the process with the quotient from step 1: 46,656 ÷ 6 = 7,776 (with a remainder of 0). Thus, the second to last digit is also 0.

4. Continue the process: 7,776 ÷ 6 = 1,296 (with a remainder of 0), making the third to last digit 0.

5. Keep dividing: 1,296 ÷ 6 = 216 (with a remainder of 0), and 216 ÷ 6 = 36 (with a remainder of 0). Continue until the quotient is less than 6.

6 (dec). In the end, you'll have a series of remainders: 0, 0, 0, 0, 0, 0, 0, 1

7 (dec). Revert the list and use it to obtain the heximal representation: 279,936 (dec) = 10;000,000 (hex).

For negative numbers, simply add a negative sign at the beginning and perform the conversion as if it were positive. For example, –45,045 (dec) = –544,313 (hex).

Decimal to Heximal Fraction Conversion:

Converting the fractional part of a decimal number to heximal is also straightforward. For instance, consider 0.615234375 (dec):

1. Multiply the fractional part by six to obtain the first digit after the heximal point: 0.615234375 × 6 = 3.

2. Continue this process for subsequent digits: 0.615234375 × 6 = 4.1484375, so the second digit after the heximal point is 4.

3. Reiterate: 0.1484375 × 6 = 0.890625 (third digit), 0.890625 × 6 = 5.34375 (fourth digit), and so on.

4. Repeat the process until getting 0.615234375 (dec) = 0.340520213 (hex).

For repeating patterns, like in 3.2 (dec) = 3.111111... (hex), the pattern continues indefinitely.

Heximal to Decimal Conversion:

Converting a number from heximal to decimal follows a procedure similar to the one used for decimal to heximal conversion. However, in this case, calculations are performed within the heximal system, and the key difference is that instead of dividing and multiplying by six, we divide and multiply by ten. Here's how it works:

1. Begin with the heximal number you want to convert.

2. Start by dividing the number by ten to get the quotient and the remainder.

3. The remainder is the last decimal digit.

4. With the new number repeat step 2, to get the rest of the digits, the remainder always indicates the last digit.

Let's go through some examples:

Example: Converting 502,131 (hex) to decimal:

– Begin by dividing 502,131 (hex) by 14 (hex) and calculate the remainder.

– 502,131 ÷ 14 (hex) = 30,120 (hex) remainder 11 (hex), so the last digit in decimal is 7.

– Continue the process 30,120 (hex) ÷ 14 (hex) = 1,453 (hex) remainder 10 (hex), so the second to last digit is 6.

– Repeat the process until you get the list of remainders 7, 6, 3, 9, 3

Therefore, 502,131 (hex) = 39,367 (dec).

Heximal to decimal conversion for fractions:

To convert a number from heximal to decimal that has digits after the radix point follow these steps:

Example: 13.541513 (hex)

1. Start by converting the integer part, in this case 13 (hex) = 9 (dec)

2. Multiply the fractional part by 14 (hex), 0.541513 x 14 (hex) = 13.31043 (hex), so the first digit after the decimal point is 9.

3. Repeat step two using the new number, 0.31043 x 14 (hex) = 5.1513 (hex), so the second digit after the decimal point is 5…

4. The remainders will either follow a pattern or they will or the fractional part will eventually go to 0.

In this case 13.541513 (hex) = 9.953125 (dec).

Another example is 5.32 (hex), which if you follow the remainders they are always 5, so 5.32 (hex) = 5.55555555… (dec)

Exponential Notation, Orders of Magnitude and Rounding in Heximal

When dealing with large numbers in heximal, it can sometimes be challenging to read them. Exponential notation serves the same purpose as scientific notation in decimal but can be applied to all numerical bases. The exponential notation can be simplified using this formula: m x b^e, where m represents the mantissa, which always falls within the range [1, b), b being the base (in heximal, b is six), and e signifies the exponent, which can be any integer, including negative values. The mantissa is a real number between 1 and b but can never be equal to b, ensuring it's always smaller than b. A negative mantissa is indicated by a negative sign. The base represents the base of the exponential notation, while the exponent indicates the order of magnitude of the number, where a larger exponent corresponds to a larger number.

Here, we are working in heximal, where six is represented as 10 (hex). I'll denote b as 10 (hex), indicating that all variables are in heximal. Consider the following example: 2.3 x 10^12 (hex). This number is equivalent to 230;000,000 (hex), with the only difference being that it requires fewer digits to represent. Understanding the meaning of exponential notation is easier if you are already familiar with scientific notation since exponential notation is essentially scientific notation adapted for other bases.

To convert a number from normal notation to exponential notation, simply move the heximal point to the last significant digit and compensate for it with the exponent. For example, 113;243,101 (hex) is converted to 1.13243101 x 10^12 (hex), with the heximal point moving eight spaces. Since the first digit in heximal always falls between 1 and 5, the mantissa (m) can never be less than 1 or greater than 10 (hex).

Converting from exponential notation to normal notation involves shifting the heximal point by the number of positions indicated by the exponent. For instance, 4.3124521 x 10^21 (hex) becomes 43;124,521;000,000 (hex). When all digits belong to the integer part of the number, moving the heximal point to the right simply adds zeroes.

Exponential notation also accommodates negative exponents, denoting numbers with magnitudes less than 1. For example, 1.3421 x 10^(–3) (hex) means 0.0013421(hex). The negative exponent indicates how many spaces the heximal point should move, but in the opposite direction. If a number is less than 1, to convert it from normal notation to exponential notation, determine how many times you need to move the heximal point to reach a number greater than 1 and express it as a negative exponent.

Exponential notation is frequently employed to represent complex numbers in polar form. You can convert the magnitude of a complex number to heximal and then determine the argument to express the complex number in heximal polar form.

To convert a number expressed in exponential notation to another base, you can utilize logarithms. For example, to convert 3 x 10^101 (dec) to heximal, begin by taking the logarithm of 3 x 10^101 (dec). Then, divide it by log(6 (dec)) to obtain the integer part of the exponent in heximal. Finally, divide 3 x 10^101 by 6 (dec) raised to that exponent to derive the mantissa expressed in decimal, and subsequently convert it from decimal to heximal.

In cases where the exponent is negative, it's necessary to round up the magnitude. This is because the integer part of a non–integer negative number is defined to be one less than the rounded–down value. For example, consider 7 x 10^(–5) (dec). Calculating log(7 x 10^(–5) (dec)) / log(6(dec)) yields approximately –5.33945291219 (dec), indicating a negative exponent. Therefore, in heximal exponential notation, the exponent is –6 (dec). The mantissa, when converted, equals 3.1332344325304... (hex), making 7 x 10^(–5) (dec) equivalent to 3.1332344325304... x 10^(–10) (hex).

Orders of Magnitude:

To calculate orders of magnitude in heximal, simply determine the exponential notation of the given number. The exponent in the exponential notation represents the order of magnitude, particularly when the mantissa is small. If the mantissa falls within the range of 1 to √10 (hex) / √6 (dec), then the order of magnitude is equal to the exponent. However, if the mantissa lies between √10 (hex) / √6 (dec) and 10 (hex), then the order of magnitude becomes the exponent plus 1. When the mantissa is exactly equal to √10 (hex), the order of magnitude is also the exponent plus 1.

Since √10 (hex) is an irrational number, it's often easier to work with an approximation. In this case, √10 (hex) can be approximated as ≅ 2.241031220552145325004320404110552053… (hex). To determine the order of magnitude, if the mantissa is greater than 2.241 (hex), add 1 to the exponent; otherwise, use the exponent value as it is. The midpoint between these ranges is √10 (hex) / √6 (dec), given that 10^(1/2) (hex) = √10 (hex) / √6 (dec), and the exponent serves as the threshold.

Here are some examples to illustrate this concept:

– The order of magnitude of 1 x 10^31 (hex) is 31 (hex), as 1 is less than √10 (hex) / √6 (dec).

– The order of magnitude of 2.3 x 10^(–11) (hex) is –10 (hex), since 2.3 (hex) exceeds √10 (hex) / √6 (dec).

Rounding:

In heximal, you can approximate numbers using methods similar to those in decimal. You can round up to the nearest integer, round down to the nearest integer, or round to the nearest even number. Let's explore these methods:

1. Rounding Up to the Nearest Integer: To round up a real number to the nearest integer, simply ignore the fractional part and add 1 if the fractional part is not zero. For example, rounding up 203.1321 (hex) gives 204 (hex).

2. Rounding Down to the Nearest Integer: To round down a real number to the nearest integer, just ignore the fractional part. For example, rounding down 5,532.552134 (hex) results in 5,532 (hex).

3. Rounding to the Nearest Even Number: This method combines both rounding up and rounding down. Ignore the fractional part, and if the result is an even number, leave it as is. If the result is odd, add 1. For example, rounding 23,343.01234 (hex) to the nearest even number gives 23,344 (hex) because 23,343 (hex) is odd.

4. Rounding to the Nearest 'Round' Number: In this method, you round to the nearest 'round' number, which usually ends with 0 or 3 in heximal. A round number may have fewer decimal places. You can round to the nearest hundredth, nearest tenth, nearest half, nearest integer, nearest 5, nearest 10 (dec), nearest 50 (dec), nearest 100 (dec), and so on.

For heximal, keep in mind that round numbers typically end with 0 or 3, as opposed to 0 or 5 in decimal. For example, to round 4,344 (hex), you might choose either 4,400 (hex) or 4,300 (hex), the latter being a 3–smooth number, which can make it seem 'rounder'.

Additionally, rounding to the nearest integer works differently in heximal. If the fractional part is less than 0.3 (hex), rounding to the nearest integer means using only the integer part. If the integer part is greater than or equal to 0.3 (hex), you add 1 to the integer part. For instance, 2,521.2231 (hex) rounds to 2,521 (hex), while 333.333 (hex) rounds to 334 (hex).

In summary, rounding in heximal follows a similar concept to decimal but involves rounding from 3 and above instead of 5 and above.

3–Smooth Numbers and perfect powers in heximal

A 3–smooth number is defined as a number of the form ±2^j x 3^k, where both j and k are non–negative integers. These numbers hold significant importance in the context of heximal, as they are the only divisors of powers of six. This is due to the relationship: 10^e (hex) = 2^e x 3^e. It's worth noting that there are infinitely many 3–smooth numbers, and they extend the concept of powers of 2 (which are equivalent to 2–smooth numbers), just as they do for powers of 3 and powers of 10 (hex). 3–smooth numbers can be negative, but that is not that important, ignoring the negative ones, which are just the positives multiplied by –1, we can make a sequence of them in heximal.

The sequence of 3–smooth numbers in heximal follows this pattern: 1, 2, 3, 4, 10, 12, 13, 20, 24, 30, 40, 43, 52, 100, 120, 130, 144, 200, 213, 240, 300, 332, 400, 430, 520, 1,000, 1,043, 1,104, 1,200, 1,300, 1,440, 2,000, 2,130, 2,212, 2,400, 3,000, 3,213, 3,320, 4,000, 4,300, 4,424, 5,200, 10,000, 10,430, 11,040, 12,000, 13,000, 13,252, 14,043, 14,400, 20,000, 21,300, 22,120, 24,000, 30,000, 30,544, 32,130, 33,200, 40,000, 43,000, 44,240, 50,213, 52,000, 100,000, 101,532, 104,300, 110,400, 120,000, 130,000, 132,520, 140,430, 144,000, 200,000, 203,504, 213,000, 221,200, 231,043, 240,000, 300,000, 305,440, 321,300, 332,000, 400,000, 411,412, 430,000, 442,400, 502,130, 520,000, 1;000,000…

As you can observe, if a number is present in this list, appending any number of zeroes to the right of it will result in another number that also belongs to the list. This property makes the list quite versatile for various calculations and representations. In the future, I may refer to these 3–smooth numbers, and now you have an understanding of what they entail.

Now, let's delve into the concept of threeveness, which expands upon the notions of even and odd numbers. An even number is one that is divisible by 2, while an odd number is not divisible by 2. Building upon this foundation, I've introduced the terms "threeven," "throdd," "pre–threeven," and "post–threeven."

– A threeven number is a number divisible by 3 and takes the form 3n, where n is an integer.

– A throdd number is a number not divisible by 3 and can be expressed as 3n+1 or 3n–1.

– A pre–threeven number follows the pattern 3n–1.

– A post–threeven number is defined by the expression 3n+1.

Notably, no perfect square belongs to the category of pre–threeven numbers; instead, they are either categorized as threeven or post–throdd, since no square number can be expressed in the form 3n–1.

Perfect powers in heximal:

I will talk about some perfect powers in heximal, including how squares, cubes and sixth powers terminate in heximal.

Squares:

There is a pattern related to squares in heximal notation. I will show this pattern, which will help you identify possible squares more easily. The sequence of perfect squares in heximal notation is as follows:

0, 1, 4, 13, 24, 41, 100, 121, 144, 213, 244, 321, 400, 441, 524, 1,013, 1,104, 1,201, 1,300, 1,401, 1,504, 2,013, 2,124, 2,241, 2,400, 2,521, 3,044, 3,213, 3,344, 3,521, 4,100, 4,241, 4,424, 5,013, 5,204, 5,401, 10,000…

This sequence contains all squares from 0 to 100 (hex) represented in heximal. There's a pattern in this sequence related to the last digits. If a number 'a' is not divisible by 2 or 3, or in other words, a ends with 1 or 5 in heximal, then a^2 can be expressed as 40n + 1 (hex) in heximal, where n is an integer. For example:

· 1^2 (hex) = 40 x 0 (hex) + 1

· 5^2 (hex) = 40 x 1 (hex) + 1

· 11^2 (hex) = 40 x 2 (hex) + 1

· 15^2 (hex) = 40 x 5 (hex) + 1

· 21^2 (hex) = 40 x 11 (hex) + 1

· 25^2 (hex) = 40 x 20 (hex) + 1

· 31^2 (hex) = 40 x 23 (hex) + 1

· 35^2 (hex) = 40 x 34 (hex) + 1

· 41^2 (hex) = 40 x 42 (hex) + 1

· 45^2 (hex) = 40 x 55 (hex) + 1

· 51^2 (hex) = 40 x 104 (hex) + 1

· 55^2 (hex) = 40 x 123 (hex) + 1

This pattern holds because if a is odd, then a^2 = 1 mod 12 (hex), and if 'a' is throdd, then a^2 = 1 mod 3. Since a^2 = 1 mod 12 (hex) and a^2 = 1 mod 3, it follows that a^2 = 1 mod 40 (hex). This means that if a is an odd throdd number, then a^2 ends with these terminations: 001, 041, 121, 201, 241, 321, 401, 441, or 521, as these are all possible terminations for numbers of the form 40n + 1 (hex).

To continue this pattern, you can identify all 3–smooth numbers that are squares. After 1, comes 4. So, if a number b is of the form 20n – 2 (hex) or 20n + 2 (hex), then b^2 is of the form 240n + 4 (hex). For example:

· 2^2 (hex) = 240 x 0 (hex) + 4

· 14^2 (hex) = 240 x 1 (hex) + 4

· 22^2 (hex) = 240 x 2 (hex) + 4

· 34^2 (hex) = 240 x 5 (hex) + 4

· 42^2 (hex) = 240 x 11 (hex) + 4

· 54^2 (hex) = 240 x 20 (hex) + 4

This means that the possible terminations of b^2 are: 00004, 00244, 00524, 01204, 01444, 02124, 02404, 03044, 03324, 04004, 04244, 04524, 05204, 05444, 10124, 10404, 11044, 11324, 12004, 12244, 12524, 13204, 13444, 14124, 14404, 15044, 15324, 20004, 20244, 20524, 21204, 21444, 22124, 22404, 23044, 23324, 24004, 24244, 24524, 25204, 25444, 30124, 30404, 31044, 31324, 32004, 32244, 32524, 33204, 33444, 34124, 34404, 35044, 35324, 40004, 40244, 40524, 41204, 41444, 42124, 42404, 43044, 43324, 44004, 44244, 44524, 45204, 45444, 50124, 50404, 51044, 51324, 52004, 52244, 52524, 53204, 53444, 54124, 54404, 55044, 55324. These are all possible terminations of a number of the form 240n + 4 (hex).

The next 3–smooth square is 13 (hex). In this case, if c is of the form 30n + 3 (hex) or 30n – 3 (hex), then c^2 is of the form 1000n + 13 (hex). For example:

· 3^2 (hex) = 1000 x 0 (hex) + 13 (hex)

· 23^2 (hex) = 1000 x 1 (hex) + 13 (hex)

· 33^2 (hex) = 1000 x 2 (hex) + 13 (hex)

· 53^2 (hex) = 1000 x 5 (hex) + 13 (hex)

This means that c^2 terminates with the digits 013, as every number of the form 1000n + 13 in heximal ends with 013. You can continue this pattern by using more 3–smooth squares, the next square 3–smooth number is 24 (hex). Here are all possible terminations for a square for the last 1, 2, 3, and 4 digits: 0, 1, 3, 4; 00, 01, 04, 13, 21, 24, 41, 44; 000, 001, 004, 013, 024, 041, 044, 100, 104, 121, 124, 144, 201, 204, 213, 224, 241, 244, 300, 304, 321, 324, 344, 400, 401, 404, 424, 441, 444, 504, 521, 524, 544; 0000, 0001, 0004, 0013, 0024, 0041, 0100, 0121, 0124, 0144, 0201, 0213, 0241, 0244, 0304, 0321, 0400, 0401, 0404, 0424, 0441, 0521, 0524, 0544, 1001, 1013, 1041, 1044, 1104, 1121, 1201, 1204, 1224, 1241, 1300, 1321, 1324, 1344, 1401, 1441, 1444, 1504, 1521, 2001, 2004, 2013, 2024, 2041, 2100, 2121, 2124, 2201, 2241, 2244, 2304, 2321, 2400, 2401, 2404, 2424, 2441, 2521, 2524, 2544, 3001, 3013, 3041, 3044, 3104, 3121, 3201, 3204, 3213, 3224, 3241, 3321, 3324, 3344, 3401, 3441, 3444, 3504, 3521, 4001, 4004, 4013, 4024, 4041, 4100, 4121, 4124, 4144, 4201, 4241, 4244, 4304, 4321, 4400, 4401, 4404, 4424, 4441, 4521, 4524, 4544, 5001, 5013, 5041, 5044, 5104, 5121, 5201, 5204, 5224, 5241, 5321, 5324, 5344, 5401, 5441, 5444, 5504, 5521.

Cubes:

The sequence of cubes in heximal is as follows:

0, 1, 12, 43, 144, 325, 1,000, 1,331, 2,212, 3,213, 4,344, 10,055, 12,000, 14,101, 20,412, 23,343, 30,544, 34,425, 43,000, 51,431, 101,012, 110,513, 121,144, 132,155, 144,000, 200,201, 213,212, 231,043, 245,344, 304,525, 325,000, 345,531, 411,412, 434,213, 501,544, 530,255, 1;000,000…

This sequence contains all cubes from 0 to 100 (hex). There is a pattern in this sequence related to the last digits. If a is an odd throdd, then a^3 is of the form 30n + 1 or 30n – 1 (hex). Here are some examples:

· 1^3 (hex) = 30 x 0 (hex) + 1

· 5^3 (hex) = 30 x 11 (hex) – 1

· 11^3 (hex) = 30 x 31 (hex) + 1

· 15^3 (hex) = 30 x 202 (hex) – 1

· 21^3 (hex) = 30 x 322 (hex) + 1

· 25^3 (hex) = 30 x 1,133 (hex) – 1

· 31^3 (hex) = 30 x 1,433 (hex) + 1

· 35^3 (hex) = 30 x 3,044 (hex) – 1

· 41^3 (hex) = 30 x 4,004 (hex) + 1

· 45^3 (hex) = 30 x 10,135 (hex) – 1

· 51^3 (hex) = 30 x 11,355 (hex) + 1

· 55^3 (hex) = 30 x 15,010 (hex) – 1

If 'a' ends with 1 or 5 in heximal, then the possible endings for a^3 are: 01, 31 for numbers ending with 1, and 25, 55 for numbers ending with 5. This is because these are the possible endings for numbers of the form 30n + 1 or 30n – 1.

For the remaining cubes, you can simply multiply the 3–smooth cube numbers by the formula to obtain all the other possibilities. In this case, if a is of the form 20n + 2 or 20n – 2, then a^3 ends with the endings: 0012, 0412, 1212, 2012, 2412, 3212, 4012, 4412, 5212 if a ends with 2, and 0344, 1144, 1544, 2344, 3144, 3544, 4344, 5144, 5544 if a ends with 4. This is because 12 (hex) x (30n + 1) results in 400n ± 12 (hex), and these are all possible endings.

The next 3–smooth cube is 43 (hex). In this case, 43 (hex) x (30n ± 1) results in 2130n ± 43 (hex). The possible endings for this formula are: 00043, 02213, 04343, 10513, 13043, 15213, 21343, 23513, 30043, 32213, 34343, 40513, 43043, 45213, 51343, 53513 if the second to last digit of a is threeven, so a ends with the endings 03 or 33, while the endings: 02043, 04213, 10343, 12513, 15043, 21213, 23343, 25513, 32043, 34213, 40343, 42513, 45043, 51213, 53343, 55513, are for numbers where the second to last digit is pre–threeven, so they end with the endings 23 or 53.

In summary, the possible combinations for the last 1, 2, and 3 digits of cubes in heximal are: 0, 1, 2, 3, 4, 5; 00, 01, 12, 13, 25, 31, 43, 44, 55; 000, 001, 012, 025, 031, 043, 055, 101, 125, 131, 144, 155, 201, 212, 213, 225, 231, 255, 301, 325, 331, 343, 344, 355, 401, 412, 425, 431, 455, 501, 513, 525, 531, 544, 555. I didn't include a section for the last 4 digits, as I did for the squares, because assuming the last 3 digits are in the list, the 4th to last digit can be anything, and the next improvement is looking at the last 5 digits.

Sixth powers:

Now, let's delve into the details of perfect sixth powers. The sequence of perfect sixth powers in heximal is as follows:

0, 1, 144, 3,213, 30,544, 200,201, 1;000,000…

I've chosen to include all sixth powers up to 10 (hex) for consistency with the cubes. The odd, throdd sixth powers always take the form of 200n + 1 (hex), where n is an integer. By multiplying by a sixth power that's 3–smooth, we can derive formulas for all sixth powers. Additionally, there's the formula 33,200n + 144 (hex) if a is of the form 20m ± 2 (hex), and 1;043,000n + 3,213 (hex) if 'a' is of the form 30m ± 3 (hex). There are more formulas that can be obtained using 3–smooth numbers. The possible endings of sixth powers of a, if a is an odd throdd number, are: 001, 201, and 401.

The possible endings for sixth powers for the last 1, 2, 3, and 4 digits are: 0, 1, 3, 4; 00, 01, 13, 44; 000, 001, 144, 201, 213, 344, 401, 544; 0000, 0001, 0144, 0201, 0213, 0401, 0544, 1001, 1201, 1344, 1401, 2001, 2144, 2201, 2401, 2544, 3001, 3201, 3213, 3344, 3401, 4001, 4144, 4201, 4401, 4544, 5001, 5201, 5344, 5401.

Now, to summarize the impact of looking at the last digits of numbers in heximal, decimal, and their effectiveness in removing possible perfect powers:

Squares:

– In heximal, looking at the last few digits will remove up to every number except 1/24 (hex) / 1/16 (dec) = 0.0213 (hex) / 0.0625 of all numbers as possible squares.

– In decimal, looking at the last few digits will remove up to every number except 5/200 (hex) / 5/72 (dec) = 0.023 (hex) / 0.06944444… of all numbers as possible squares.

– As you can see heximal removes slightly more numbers as possible squares, the difference is 1/400 (hex) / 1/144 (dec) = 0.0013 (hex) / 0.00694444… (dec), and you need to look at the last 3 digits for the advantage of heximal, otherwise decimal is better, so overall no major advantage for the squares.

Cubes:

– In heximal, looking at the last few digits helps remove up to every number except 20/231 (hex) / 12/91 (dec) = 0.044252235… (hex) / 0.1318681… (dec) of all numbers as possible cubes.

– In decimal, looking at the last few digits helps remove up to every number except 244/1,001 (hex) / 100/217 (dec) = 0.243312244… (hex) / 0.4608295… (dec) of all numbers as possible cubes.

– Like you can see decimal leaves about three and a half times more numbers as possible cubes, that is because decimal can’t exclude any number coprime to ten as possible cubes, but heximal can remove numbers coprime to six as possible cubes, so heximal is much better in terms of cubes.

Sixth Powers:

– In heximal, considering the last digits helps remove up to every number except 13/2,541 (hex) / 9/637 (dec) = 0.003015105… (hex) / 0.0141287… (dec) of all numbers as possible sixth powers.

– In decimal, considering the last digits helps remove up to every number except 44,534/2;345,343 (hex) / 6,250/123,039 (dec) = 0.014545552… (hex) / 0.0507969… (dec) of all numbers as possible sixth powers.

– Conclusion heximal is much better than decimal regarding sixth powers, since decimal leaves over three and a half times more numbers as possible sixth powers compared to heximal.

When we use the same denominator for both heximal and decimal fractions, we can see that heximal generally performs better in removing possible perfect powers compared to decimal.

Comparison of Fraction Representations in Heximal and Decimal

A key aspect of a good positional numbering system is its ability to represent common fractions in a straightforward manner. While decimal does an acceptable job in this regard, heximal excels, only presenting complex representations when the denominator reaches 15 (hex) or 11 (dec). Below is a table showing common fractions in both heximal and decimal representations. Fractions where the numerator is greater than 1 have been excluded, as they tend to have similar representations.

1/x (hex)	Heximal representation	Decimal representation	1/x (dec)
1	1	1	1
2	0.3	0.5	2
3	0.2	0.3	3
4	0.13	0.25	4
5	0.1	0.2	5
10	0.1	0.16	6
11	0.05	0.142857	7
12	0.043	0.125	8
13	0.04	0.1	9
14	0.03	0.1	10
15	0.0313452421	0.09	11
20	0.03	0.083	12
21	0.024340531215	0.076923	13
22	0.023	0.0714285	14
23	0.02	0.06	15
24	0.0213	0.0625	16
25	0.0204122453514331	0.0588235294117647	17
30	0.02	0.05	18
31	0.015211325	0.052631578947368421	19
32	0.014	0.05	20
33	0.014	0.047619	21
34	0.01345242103	0.045	22
35	0.01322030441	0.0434782608695652173913	23
40	0.013	0.0416	24
41	0.01235	0.04	25
42	0.0121502434053	0.0384615	26
43	0.012	0.037	27
44	0.0114	0.03571428	28
45	0.01124045443151	0.0344827586206896551724137931	29
50	0.01	0.03	30
51	0.010545	0.032258064516129	31
52	0.01043	0.03125	32
53	0.01031345242	0.03	33
54	0.01020412245351433	0.02941176470588235	34
55	0.01	0.0285714	35
100	0.01	0.027	36

This table highlights the contrast between the representations of common fractions in heximal and decimal. In heximal, fractions with small denominators are remarkably simple. The first repeating fraction in heximal occurs at 1/5 (hex), while in decimal, it's at 1/3 (dec). Specifically, 1/3 (hex) is represented as 0.2 (hex), while 1/3 (dec) is 0.333333... repeating. This makes working with thirds in heximal far more straightforward, whereas decimal complicates what should be a simpler representation.

Halves and fourths are also simple in heximal: 1/2 (hex) = 0.3 (hex), and 1/4 (hex) = 0.13. The interesting part is that while sevenths have a complex representation in decimal 1/7 (dec) = 0.142857142857..., they have a more manageable representation in heximal 1/11 (hex) = 0.050505050505..., which repeats every 2 digits.

The only advantage decimal has is that fifths have a slightly simpler representation compared to heximal 1/5 (dec) = 0.2 (dec), while 1/5 (hex) = 0.111111111... (hex), however, this advantage comes at the cost of making thirds more complicated, which is generally not worth it.

Additionally, there's an essential detail regarding the frequency of terminating decimals and heximals. It turns out that the frequency of terminating heximals is greater than that of terminating decimals. In the limits, the proportion of these converges to log(5)/log(3), which is approximately 1.244233450... (hex) / 1.4649735... (dec). This means there are approximately one and a half times more numbers with reciprocals as terminating heximals than there are with reciprocals as terminating decimals. This is signifficant because numbers with terminating representations are easy to use precisely in calculations.

Numbers whose reciprocals are terminating heximals are all potential divisors of powers of six, known as 3–smooth numbers. On the other hand, numbers whose reciprocals are terminating decimals don't have a specific name but are all potential divisors of powers of ten, taking the form 2^j x 5^k for some integers j and k.

The length of the period of recurring heximals/decimals is related to the divisors of one less than a power of six or ten. For instance, the reason 1/5 has a period length of 1 in heximal, since 10^1 (hex) – 1 is divisible by 5 (hex). In general, for a reciprocal of a number coprime to six, the length of its period is determined by the smallest exponent of six, such that one less than that power of six is a multiple of that number. An example is 1/11 (hex), which has a period length of 2, since 10^2 (hex) – 1 is divisible by 11 (hex).

Here are some prime powers and their corresponding period lengths for their reciprocals in decimal:

· 5 (hex): Period length = 1

· 11 (hex) / 7 (dec): Period length = 2

· 111 (hex) / 43 (dec): Period length = 3

· 101 (hex) / 37 (dec): Period length = 4

· 5^2 (hex) and 1,235 (hex) / 311 (dec): Period length = 5

· 51 (hex) / 31 (dec): Period length = 10 (hex) / 6 (dec)

· 1;111,111 (hex) / 55,987 (dec): Period length = 11 (hex) / 7 (dec)

· 10,001 (hex) / 1,297 (dec): Period length = 12 (hex) / 8 (dec)

· 31 (hex) / 19 (dec) and 15,231 (hex) / 2,467 (dec): Period length = 13 (hex) / 9 (dec)

· 15 (hex) / 11 (dec) and 245 (hex) / 101 (dec): Period length = 14 (hex) / 10 (dec)

· 35 (hex) / 23 (dec) and 151;341,205 (hex) / 3,154,757 (dec): Period length = 15 (hex) / 11 (dec)

· 21 (hex) / 13 (dec) and 241 (hex) / 97 (dec): Period length = 20 (hex) / 12 (dec)

Keep in mind that the length of the period is always a divisor of p – 1 for any prime p. The maximum length is p – 1, which is the largest divisor of p – 1. This can be proven using Fermat’s Little Theorem.

In decimal, different numbers result in various period lengths for their reciprocals:

· 3 (dec) and 3^2 (dec): Period length = 1

· 11 (dec) / 15 (hex): Period length = 2

· 3^3 (dec) and 37 (dec) / 101 (hex): Period length = 3

· 101 (dec) / 245 (hex): Period length = 4

· 41 (dec) / 105 (hex) and 271 (dec) / 1,131 (hex): Period length = 5

· 7 (dec) / 11 (hex) and 13 (dec) / 21 (hex): Period length = 6 (dec) / 10 (hex)

· 239 (dec) / 1,035 (hex) and 4,649 (dec) / 33,305 (hex): Period length = 7 (dec) / 11 (hex)

· 73 (dec) / 201 (hex) and 137 (dec) / 345 (hex): Period length = 8 (dec) / 12 (hex)

· 3^4 (dec) and 333,667 (dec) / 11;052,431 (hex): Period length = 9 (dec) / 13 (hex)

· 9,091 (dec) / 110,031 (hex): Period length = 10 (dec) / 14 (hex)

· 21,649 (dec) / 244,121 (hex) and 513,239 (dec) / 15;000,035 (hex): Period length = 11 (dec) / 15 (hex)

· 9,901 (dec) / 113,501 (hex): Period length = 12 (dec) / 20 (hex)

The length of the period in decimal is determined by the divisors of one less than a power of ten, resulting in varying period lengths for different prime factors.

Perniffages and more substitutes of percentages for heximal and nifftimal compression

In heximal, we have a unique way to represent percentages, perniffages, pertarnexiages, and perunwixionages, which correspond to the decimal concepts of percentages, permillages, and parts per million, respectively.

1. Perniffages (Symbol: ‘%’): Perniffages are equivalent to percentages, but they are based on 100 (hex) instead of 100 (dec). To calculate perniffages, simply move the heximal point two places to the right and add the symbol '%’. Here are some examples:

· 3/2 (hex) = 130% (hex)

· 4/3 (hex) = 120% (hex)

· 4/5 (hex) ≅ 44.444445% (hex)

· 1/2 (hex) = 30% (hex)

Perniffages allow you to express fractions in heximal, making it a convenient way to work with something similar to percentages in this base. For instance, 3/2 (hex) can be represented as 130% (hex).

2. Pertarnexiages (Symbol: ‘‰’): Pertarnexiages are the heximal equivalent of permillages, which are based on 1,000 (hex) instead of 1,000 (dec). To calculate pertarnexiages, move the heximal point three places to the right and add the symbol '‰'. Examples:

· 1/12 (hex) = 43‰ (hex)

· 24/43 (hex) = 332‰ (hex)

· 25/300 (hex) = 54‰ (hex)

· 100/111 (hex) ≅ 500.501…‰ (hex)

· 1 (hex) = 1,000‰ (hex)

Pertarnexiages provide a way to express values with greater precision, especially when dealing with fractions in heximal.

3. Perunwixionages (Symbol: p.u.x): Perunwixionages are the heximal equivalent of parts per million (p.p.m), based on unwixion, and are expressed as p.u.x. To calculate perunwixionages, move the heximal point six places to the right. Examples:

· 43/144 (hex) = 231,043 p.u.x (hex)

· 2,212/3,213 (hex) = 411,412 p.u.x (hex)

· 30/51 (hex) ≅ 325,230.325231… p.u.x (hex)

· 41/101 (hex) ≅ 401,540.154020… p.u.x (hex)

· 1 (hex) = 1;000,000 p.u.x (hex)

· 4/3 (hex) = 1;200,000 p.u.x (hex)

Perunwixionages are useful for expressing extremely small fractions or concentrations, making them ideal for scientific and technical applications in heximal.

These heximal representations of percentages and related concepts provide a convenient way to work with fractional values and concentrations in heximal, similar to how percentages are used in decimal. In addition to these applications, perniffages, pertarnexyages and perunwixionages can be explained with six sided dices. The probability of rolling a 2 or 20 (hex) / 12 (dec) from 2 normal dices is 1% (hex), the probability of rolling a 3 or 30 (hex) / 18 (dec) from 3 normal dices is 1‰ (hex), and the probability of rolling a 10 (hex) / 6 (dec) or 100 (hex) / 36 (dec) from 10 (hex) / 6 (dec) normal dices is 1 p.u.x (hex).

Nifftimal Compression and Base Niff:

Nifftimal compression is a method to reduce the length of numbers expressed in heximal. This compression technique halves the length of the heximal representation by converting it into base niff. Base niff is a base–36 (dec) numbering system, which uses symbols from the Hindu-arabic numerals and Latin alphabet to represent numbers more efficiently. Here's how it works:

Digits in Base Niff (niff):

· 0 (niff) = 00 (hex) / 0 (dec)

· 1 (niff) = 01 (hex) / 1 (dec)

· 2 (niff) = 02 (hex) / 2 (dec)

· 3 (niff) = 03 (hex) / 3 (dec)

· 4 (niff) = 04 (hex) / 4 (dec)

· 5 (niff) = 05 (hex) / 5 (dec)

· 6 (niff) = 10 (hex) / 6 (dec)

· 7 (niff) = 11 (hex) / 7 (dec)

· 8 (niff) = 12 (hex) / 8 (dec)

· 9 (niff) = 13 (hex) / 9 (dec)

· a (niff) = 14 (hex) / 10 (dec)

· b (niff) = 15 (hex) / 11 (dec)

· c (niff) = 20 (hex) / 12 (dec)

· d (niff) = 21 (hex) / 13 (dec)

· e (niff) = 22 (hex) / 14 (dec)

· f (niff) = 23 (hex) / 15 (dec)

· g (niff) = 24 (hex) / 16 (dec)

· h (niff) = 25 (hex) / 17 (dec)

· i (niff) = 30 (hex) / 18 (dec)

· j (niff) = 31 (hex) / 19 (dec)

· k (niff) = 32 (hex) / 20 (dec)

· l (niff) = 33 (hex) / 21 (dec)

· m (niff) = 34 (hex) / 22 (dec)

· n (niff) = 35 (hex) / 23 (dec)

· o (niff) = 40 (hex) / 24 (dec)

· p (niff) = 41 (hex) / 25 (dec)

· q (niff) = 42 (hex) / 26 (dec)

· r (niff) = 43 (hex) / 27 (dec)

· s (niff) = 44 (hex) / 28 (dec)

· t (niff) = 45 (hex) / 29 (dec)

· u (niff) = 50 (hex) / 30 (dec)

· v (niff) = 51 (hex) / 31 (dec)

· w (niff) = 52 (hex) / 32 (dec)

· x (niff) = 53 (hex) / 33 (dec)

· y (niff) = 54 (hex) / 34 (dec)

· z (niff) = 55 (hex) / 35 (dec)

To convert a number from heximal to niff, simply group the heximal digits in pairs and convert each pair into the corresponding niff digit. For example, 12,345 (hex) becomes 1ft (niff), where 01 (hex) converts to 1 (niff), 23 (hex) converts to f (niff), and 45 (hex) converts to t (niff).

Nifftimal compression works for both integers and real numbers in heximal, making it a handy method to represent numbers more efficiently. It's particularly useful for numbers with lengthy heximal representations. Additionally, numbers whose reciprocals terminate in heximal will also terminate in niff, specifically the 3–smooth numbers.

Here are some examples of reciprocal fractions in niff:

· 1/1 (niff) = 1 (niff)

· 1/2 (niff) = 0.i (niff)

· 1/3 (niff) = 0.c (niff)

· 1/4 (niff) = 0.9 (niff)

· 1/5 (niff) = 0.777777… (niff)

· 1/6 (niff) = 0.6 (niff)

· 1/7 (niff) = 0.555555… (niff)

· 1/8 (niff) = 0.4i (niff)

· 1/9 (niff) = 0.4 (niff)

· 1/a (niff) = 0.3lllll… (niff)

· 1/b (niff) = 0.39tgd39tgd… (niff)

· 1/c (niff) = 0.3 (niff)

· 1/d (niff) = 0.2rox8b2rox8b… (niff)

· 1/e (niff) = 0.2kkkkk… (niff)

· 1/f (niff) = 0.2eeeee… (niff)

· 1/g (niff) = 0.29 (niff)

· 1/h (niff) = 0.248gxvrj248gxvrj… (niff)

· 1/i (niff) = 0.2 (niff)

· and so on…

The length of periods in base niff compared to heximal is straightforward. In base niff, the length of periods is much shorter than in heximal, specifically half the length when the period in heximal is even, and the same length when the period in heximal is odd. This phenomenon is due to the relationship between the base and the length of the periods in the reciprocals of numbers.

In general, for a base b^n, if the length of the period of the reciprocal of a number m is denoted as 'l,' then the length of the period in base niff is l divided by the greatest common divisor (gcd) of l and n. This means that the exponent n divides the length of the period in such a way that it results in an integer. This property leads to shorter periods in base niff compared to heximal, especially for numbers with even–length periods.

For these reasons, binary (base two), quaternary (base four), and tetradozenal (base dozen–four), are quite similar bases. This is because the only numbers in these bases that result in terminating reciprocals are the powers of 2, which are also known as the 2–smooth numbers. Additionally, the numbers that have short periods in these bases are similar. These similarities make conversion between these bases relatively straightforward.

However, when comparing binary and heximal, the relationship is not as close. Heximal has more numbers that result in terminating reciprocals, and it is not directly related to a power of 2. As a result, the lengths of periods in these bases can differ signifficantly for the same numbers. This explains why some prime numbers in heximal very short periods have, while in binary, the lengths can be longer or shorter, depending on the specific prime.

Extras of heximal

Here is the sequence of powers of 2, 3, 5, 10 (hex), 11 (hex), 14 (hex), 15 (hex), and 20 (hex) in heximal from 0 to 30 (hex):

For 2, we have: 1, 2, 4, 12, 24, 52, 144, 332, 1,104, 2,212, 4,424, 13,252, 30,544, 101,532, 203,504, 411,412, 1;223,224, 2;450,452, 5;341,344.

For 3, we have: 1, 3, 13, 43, 213, 1,043, 3,213, 14,043, 50,213, 231,043, 1;133,213, 3;444,043, 15;220,213, 54;101,043, 250;303,213, 1,231;314,043, 4,134;350,213, 20,451;531,043, 102,235;433,213.

For 5, we have: 1, 5, 41, 325, 2,521, 22,245, 200,201, 1;401,405, 12;212,241, 105;510,125, 545;151,121, 4,502;320,045, 40,120;440,401, 321,043;523,205, 2;445,351;304,441, 22;004,121;335,525, 154;033,052;015,321, 1,342;253,424;133,445, 12,040;240,413;201,001.

For 10 (hex), we have: 1, 10, 100, 1,000, 10,000, 100,000, 1;000,000, 10;000,000, 100;000,000, 1,000;000,000, 10,000;000,000, 100,000;000,000, 1;000,000;000,000, 10;000,000;000,000, 100;000,000;000,000, 1,000;000,000;000,000, 10,000;000;000,000, 100,000;000;000,000, 1;000,000;000;000;000,000.

For 11 (hex), we have: 1, 11, 121, 1,331, 15,041, 205,451, 2;304,401, 25;352,411, 323;320,521, 4,000;530,131, 44,010;231,441, 524,112;550,251, 10;205,242;453,201, 112;302,111;425,211, 1,235;323,230;121,321, 14,032;555,531;334,531, 154,402;555,245;124,241, 2;142,432;552,140;411,051, 24;011,202;513,544;522,001.

For 14 (hex), we have: 1, 14, 244, 4,344, 114,144, 2;050,544, 33;233,344, 554;200,144, 13,531;202,544, 243,121;245,344, 4;332,142;412,144, 113;534,523;014,544, 2,043;221,010;301,344, 33,133;534,145;024,144, 552,403;211,002;430,544, 13;502,453;354,043;313,344, 242;244,120;325,135;240,144, 4,320;350,125;444,430;242,544, 113,330;222,253;555,304;325,344.

For 15 (hex), we have: 1, 15, 321, 10,055, 151,441, 3;241,335, 101;545,401, 1,533;402,215, 33,134;242,121, 1;025,551;000,255, 15;525,425;005,241, 335;023,111;135,535, 10,401;435,112;015,201, 201,231;303,124;324,415, 3;423,354;355,402;203,521, 105;044,133;152,241;510,455, 2,032;234,530;252,552;303,041, 35,012;500,035;202,453;354,135, 1;101,241;101,104;451,014;125,001.

For 20 (hex), we have: 1, 20, 400, 12,000, 240,000, 5;200,000, 144;000,000, 3,320;000,000, 110,400;000,000, 2;212,000;000,000, 44;240,000;000,000, 1,325;200,000;000,000, 30,544;000,000;000,000, 1;015,320;000,000;000,000, 20;350,400;000,000;000,000, 411;412,000;000,000;000,000, 12,232;240,000;000,000;000,000, 245,045;200,000;000,000;000,000, 5;341,344;000,000;000,000;000,000.

As you can see, these are the representations of some perfect powers in heximal.

The next extra is the representation of some mathematical constants in heximal.

Here is the representation of some mathematical constants in heximal. These are the approximations of mathematical constants using 100 (hex) digits after the heximal point:

√2 (hex) ≅ 1.225245314205523321432232430442403340...

√3 (hex) ≅ 1.422042321254545342041213412413330144...

√5 (hex) ≅ 2.122553553151303343124514320340240135...

√10 (hex) ≅ 2.241031220552145325004320404110552053...

√11 (hex) ≅ 2.351252101223212500350350521253304451...

√14 (hex) ≅ 3.055015120532103144313025445120133101...

√15 (hex) ≅ 3.152220240214441344413042502302445532...

Φ, which is the golden ratio = 1/2 x (1+√5) ≅ 1.341254554353431451342235140150120045…

π, which is the fa circle constant ≅ 3.050330051415124105234414053125321103...

τ, which is the true circle constant ≅ 10.141100143234252214513232150255042205...

e, which is Euler’s number ≅ 2.415052053524243123125404352354404354...

γ, which is the Euler–Mascheroni constant ≅ 0.324402323555230430335120432043555545...

log(2) (hex) ≅ 0.405415250322520250433345043122455324...

log(3) (hex) ≅ 1.033144504400030230314515344424100405...

log(5) (hex) ≅ 1.335345534002401505045535310120423150...

log(10) (hex) ≅ 1.443004155122550521152304431551000133...

log(11) (hex) ≅ 1.540152214531523504312232404404343133...

log(14) (hex) ≅ 2.145205224325322155523324353243322514...

log(15) (hex) ≅ 2.221540111333223322024225210145014320...

log₂(3) (hex) ≅ 1.330204002130443040515051424142115320...

log₃(2) (hex) ≅ 0.344140354130455311334034203523054334...

log₁₀(2) (hex) ≅ 0.215321000553230431044203141010101333...

log₁₀(3) (hex) ≅ 0.340234555002325124511352414545454223...

log₁₀(5) (hex) ≅ 0.522004254003012350033154305433331304...

log₁₀(11) (hex) ≅ 1.030325543431050155525451132113410240...

log₁₀(14) (hex) ≅ 1.141325255000243221121401450443433041...

log₁₀(15) (hex) ≅ 1.201023144101534012335155033415344004...

These heximal representations provide a unique perspective on these constants.

Another extra is related to rational powers of six.

Sometimes you might need a sequence that grows exponentially but not as fast as 2^x. In such cases, the rational powers of six can be useful. If you use the sixth root of six as the base of growth, then multiplying it by itself six times gives you six. The sixth root of six is represented as 10^(0.1) (hex) or 10^(1/10) (hex), approximately equal to 1.203100324122051402... (hex) or 1.34800615459728... (dec). The next powers in this sequence are as follows:

· 10^(0.2) (hex) or 10^(1/3) (hex), approximately equal to 1.452255324532505302... (hex) or 1.81712059283214... (dec).

· 10^(0.3) (hex) or 10^(1/2) (hex), approximately equal to 2.241031220552145325... (hex) or 2.44948974278318... (dec).

· 10^(0.4) (hex) or 10^(2/3) (hex), approximately equal to 3.145114415010052332... (hex) or 3.30192724889463... (dec).

· 10^(0.5) (hex) or 10^(5/10) (hex), approximately equal to 4.241230412503305200... (hex) or 4.45101825354241... (dec).

· 10^1 (hex) or 10^(1/1) (hex), equal to 10 (hex) or 6 (dec).

These are some rational powers of six and the powers of the sixth root of six. You can multiply each term by a power of six and then round the result, creating a sequence like: 121, 145, 224, 315, 424, 1,000, 1,203, 1,452, 2,241, 3,145, 4,241, 10,000, 12,031, 14,523, 22,411, 31,451, 42,413, 100,000... This sequence grows exponentially, but the proportion of each term is determined by the sixth root of six, so it doesn't grow as fast as 2^x but still exhibits the important properties of exponential functions. You can also use a smaller base, such as the fifth root of six, represented as 10^(0.01) (hex) or 10^(1/100) (hex), approximately equal to 1.015004513412251333... (hex) or 1.05103048414282... (dec).

An additional extra is the Benford’s law in heximal. The Benford’s law states that most integers encountered in daily life start with the digit 1, compared with any of the other digits. The values of the frequency of numbers starting with a particular digit in heximal are:

For the digit 1, 0.215321000553230431 (hex) ≅ 21.5321% (hex) of all numbers in heximal start with a 1.

For the digit 2, 0.120513554005054254 (hex) ≅ 12.0514% (hex) of all numbers in heximal start with a 2.

For the digit 3, 0.054403002544132133 (hex) ≅ 5.4403% (hex) of all numbers in heximal start with a 3.

For the digit 4, 0.042522252012111044 (hex) ≅ 4.2522% (hex) of all numbers in heximal start with 4.

For the digit 5, 0.033551301552543210 (hex) ≅ 3.3552% (hex) of all numbers in heximal start with a 5.

In short more than one third of all numbers start with a 1 in average in heximal, and about two ninths start with the digit 2, every other digit is less frequent than the intuitive value of frequency, 0.1111111111… (hex) ≅ 11.1111% (hex), since 1/5 (hex) = 0.11111111111… (hex), and there are 5 possible starting digits, since 0 is impossible.

The next extra is the sequence of the triangular numbers in heximal, which are the sum of the first n integers. The sequence follows as: 1, 3, 10, 14, 23, 33, 44, 100, 113, 131, 150, 210, 231, 253, 320, 344, 413, 443, 514, 550, 1023, 1101, 1140, 1220, 1301, 1343, 1430, 1514, 2003, 2053, 2144, 2240, 2333, 2431, 2530, 3030, 3131, 3233, 3340, 3444, 3553, 4103, 4214, 4330, 4443, 5001, 5120, 5240, 5401, 5523, 10050, 10214, 10343, 10513, 11044, 11220, 11353, 11531, 12110, 12250, 12431, 13013, 13200, 13344, 13533, 14123, 14314, 14510, 15103, 15301, 15500, 20100, 20301, 20503, 21110, 21314, 21523, 22133, 22344, 23000, 23213, 23431, 24050, 24310, 24531, 25153, 25420, 30044, 30313, 30543, 31214, 31450, 32123, 32401, 33040, 33320, 34001, 34243, 34530, 35214, 35503, 40153, 40444, 41140, 41433, 42131, 42430, 43130…

The last extra is the pattern of the last digits of some types of special numbers. Every prime number that ends with 1 in heximal can be represented as a^2+ab+b^2, for some positive integers a and b, this can be proven using the Eisenstein integers, and the fact that every number that ends with 1 in heximal is congruent to 1 mod 3. This is also true for the prime 3. All perfect even numbers end with the digits 44 in heximal, except for 10 (hex), and all perfect even numbers except for 10 (hex) and 44 (hex) end with the digits 144 or 344. As a reference, the list of the Perfect even numbers in heximal follows as: 10, 44, 2144, 101344, 3155033344, 3540210412144, 143045524045344… All Mersenne primes except for 3 (hex) and 11 (hex) end with 31 or 51 in heximal, since any Mersenne number that ends with 11 in heximal is divisible by 11 (hex). As a reference the list of the Mersenne primes in heximal follows as: 3, 11, 51, 331, 101531, 2450451, 15123131, 553032005531, 253040130434434333320331… All trinary repunit primes end with 21 in heximal, and moreover 021 (hex) or 321 (hex). The list of the trinary repunit primes follows as: 21, 5021, 25030321, 11445151222420544105442244513353120450005021…

More Advantages of Heximal Compared to Decimal

If we compare the mathematical properties of six and ten, you will notice that six possesses more mathematical properties. The mathematical properties of ten include: being a semi–prime, as it has only two prime factors: 2 and 5; being a triangular number, since 1 + 2 + 3 + 4 = 10 (dec) or 14 (hex); and being a Pythagorean number, as it can be expressed as a sum of two squares: 1^2 + 3^2 = 10 (dec) or 14 (hex).

On the other hand, the mathematical properties of six are extensive: it is a semi–prime, having 2 prime factors: 2 and 3; it is a triangular number, with 1 + 2 + 3 = 10 (hex) or 6 (dec). Additionally, three and niff are other triangular numbers, which are important for six since 1 + 2 = 3, and 1 + 2 + 3 + 4 + 5 + 10 + 11 + 12 (hex) / 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 (dec) equals 100 (hex) or 36 (dec).

Six is also a superabundant number, as no number smaller than six has an abundancy of 2, which is the abundancy of six. Furthermore, it is a factorial, as 1 x 2 x 3 = 10 (hex) / 6 (dec), and a primorial, since 2 x 3 = 10 (hex) / 6 (dec), representing the product of the first two primes. It is the only number n such that both n and n^2 are triangular numbers. Six is also a perfect number, with an abundancy of 2, since 1/1 + 1/2 + 1/3 + 1/10 (hex) / 1/6 (dec) = 2, while ten is a deficient number, since its abundancy is less than 2, which is 9/5 (dec) / 13/5 (hex) = 1.8 (dec) / 1.445… (hex).

Moreover, six is a highly composite number since no number less than six has 4 divisors. The Euler totient of six is a divisor of six, φ(10) (hex) / φ(6) (dec) = 2, which is a divisor of six. This property is unique to the 3–smooth numbers, excluding the powers of 3, including 1. The reduced totient function of six is a divisor of six, λ(10) (hex) / λ(6) (dec) = 2. This means that any number coprime to six, raised to the sixth power, is 1 more than a multiple of six, and coincidentally also 1 more than a multiple of six squared. This property doesn't hold for all integers, ten included.

Additionally, six is a practical number. For instance, 1 = 1, 2 = 2, 3 = 3, 4 = 1 + 3, 5 = 2 + 3, and 10 (hex) / 6 (dec) = 10 (hex) / 6 (dec). This means that every number between 1 and 10 (hex) / 6 (dec) can be represented as sums of divisors of 10 (hex) / 6 (dec).

As you can see, six possesses more mathematical properties than ten, which is one reason why heximal is superior to decimal. Another reason is the radix economy of a base. The radix economy of a base 'b' is calculated by dividing the number of digits in base 'b' by the logarithm of 'b'. The radix economy of decimal is 10/log(10) (dec) / 14/log(14) (hex), which is approximately equal to 4.34294481903252… (dec) / 4.202024233344143120… (hex).

On the other hand, the radix economy of heximal is 10/log(10) (hex) / 6/log(6) (dec), which is approximately equal to 3.203151131212244111… (hex) / 3.34866375930748… (dec). Since the radix economy of heximal is smaller than that of decimal, heximal uses its digits more efficiently. With only 3/5 of the digits of decimal, heximal needs approximately 13/11 (hex) / 9/7 (dec) times more digits to represent large numbers. This can be seen in the comparison of 10^7 (dec) / 14^11 (hex) and 10^13 (hex) / 6^9 (dec), where 13/11 (hex) / 9/7 (dec) is much smaller than 5/3.

This means that a combination lock with 50 (hex) / 30 (dec) digits would be safer if each combination consisted of 5 digits between 0 and 5 compared to 3 digits between 0 and 13 (hex) / 9 (dec). There are 100,000 (hex) / 7,776 (dec) combinations of 5 digits between 0 and 5 compared to the 4,344 (hex) / 1,000 (dec) combinations of 3 digits between 0 and 13 (hex) / 9 (dec).

The base with the lowest radix economy is trinary, base 3, as 3/log(3) is approximately equal to 2.421500210350100010142511350424024351… (hex) / 2.7307176798805121808427204972… (dec). Therefore, it would be even safer for each combination to consist of 14 (hex) / 10 (dec) digits between 0 and 2. There are 1;133,213 (hex) / 59,049 (dec) combinations of 14 digits (hex) / 10 (dec) digits from 0 to 2. The radix economy of binary is slightly greater than that of trinary, as 2/log(2) is approximately equal to 2.515124432030252255313533014322241540… (hex) / 2.8853900817779268147198493620… (dec). Therefore, it is riskier for each combination to consist of 23 (hex) / 15 (dec) digits from 0 to 1 compared to each combination being 14 (hex) / 10 (dec) digits between 0 and 2. There are 411,412 (hex) / 32,768 (dec) combinations of 23 (hex) / 15 (dec) digits between 0 and 1 compared to 1;133,213 (hex) / 59,049 (dec) combinations of 14 (hex) / 10 (dec) digits between 0 and 2.

In summary, trinary is the best base in terms of radix economy, and except for the case of the comparison between base 2 and base 3, and the fact that base 2 and base 4 have the same radix economy, a smaller base will always have a lower radix economy compared to a larger base, but this doesn’t mean that trinary is better than heximal, since the representation of simpler fractions is more complicated, and the divisibility tests are harder.

New Units of Measurement, New Calendar, and New Clock for Heximal

The SI units and the metric system, widely used across the world, rely on decimal conversion for unit conversions. However, using powers of ten in heximal isn't ideal for unit conversions. Therefore, I propose new units and prefixes that use powers of six to simplify calculations in heximal.

There are seven SI base units, with one of them being the unit for "amount," which is essentially just a mathematical constant so there is no need to invent a new unit for amount. Here, I will introduce alternative base units in favour of heximal for four of the six SI base units.

For measuring distance, I propose the "heximal meter," denoted as "mh." It is defined as exactly 78,125/78,732 (dec) or 1;401,405/1;404,300 (hex) of a standard decimal meter, which is approximately 0.99229030127521 (dec) or 0.554200144 (hex). This conversion ensures that the arc length from the equator to the North pole is 1,000;000,000 (hex) heximal meters, equivalent to the arc length from the equator to the North pole, which is 10,000,000 (dec) decimal meters. As you can see the heximal meter is about the same length as the SI meter, so not much would change with this unit, just better for heximal.

For mass, I propose the "heximal gram," denoted as "gh." One heximal gram is defined to be exactly 59,604,644,775,390,625/61,004,779,879,896 (dec) or 2,414;520,255;444,054;001,201/333,425;113,050;213,000 (hex) of a standard decimal kilogram. This value is approximately equal to 977.048763928631325986002… (dec) and exactly equal to 4,305.014311044014251215110353344 (hex).

This definition ensures that 1 heximal meter cube of water has a mass of 1 heximal gram, making it a convenient unit for heximal–based calculations. However, 1 heximal gram is a relatively large amount of mass, almost 1 tonne, so I will introduce prefixes that can be used to represent smaller masses.

It's worth noting that the density of water is 1 heximal gram per heximal meter cube, making the heximal gram a suitable unit for measuring mass.

For time in the heximal system, we simplify calculations by using the "niff" as the principal unit converter. Here's how time is defined in heximal units:

– 1 terrestrial day is defined as 100 (hex) / 36 (decimal) heximal hours. This means that 1 heximal hour is equivalent to 2/3 SI hours, which is equivalent to forty SI minutes.

– 1 heximal hour is defined as 100 (hex) / 36 (decimal) heximal minutes. Therefore, 1 heximal minute is equivalent to 14/13 (hex) / 10/9 (dec) SI minutes, which is equal to 1 SI minute 6 SI seconds and 2/3 of an SI second.

– 1 heximal minute is defined as 100 (hex) / 36 (decimal) heximal seconds. Consequently, 1 heximal second is equivalent to 122/43 (hex) / 50/27 (dec) SI seconds, which is equal to 1 SI second and 23/27 (dec) / 35/43 (hex) of a SI second.

These definitions simplify time calculations in the heximal system, making it more convenient for various applications. A heximal second is denoted as "sh".

For temperature in the heximal system, we use degrees heximal (ºh) as follows:

– The freezing point of water at 1 atmosphere of pressure is defined as 0 degrees heximal.

– The boiling point of water at 1 atmosphere of pressure is defined as 1,000 (hex) degrees heximal.

To convert degrees heximal to degrees Celsius (ºC), you can use the formula nºh = 130/41 (hex) / 54/25 (dec) x nºC, which simplifies to ºh = 2.05432054321… (hex) / 2.16 (dec) x nºC.

To convert degrees heximal to degrees Fahrenheit (ºF), you can use the formula nºh = 6/5 (dec) / 10/5 (hex) x n + –192/5 (dec) / –520/5 (hex) ºF. This means that 520 (hex) / 192 (dec) ºF is equivalent to 520 (hex) / 192 (dec) ºh, which is approximately equal to 88+8/9 (dec) / 224+12/13 (hex) ºC or 88.8888889 (dec) / 224.52 (hex) ºC.

In summary, degrees heximal grows about twice as fast as degrees Celsius and Kelvin, and at a similar rate to degrees Fahrenheit. The meeting point of degrees heximal is positive, unlike the meeting point of degrees Celsius and degrees Fahrenheit, which is –40 (dec) / –104 (hex). The absolute zero of degrees heximal is about –590 (dec) / –2,422 (hex) ºh, equivalent to 0 Kelvin, and 0ºF is approximately –38.4 (dec) / –102.222222222… (hex) ºh.

This temperature scale makes it easier to convert to degrees Celsius compared to Fahrenheit, as it is proportional to Celsius. Although the proportion is a repeating heximal, it's not a repeating decimal. However, conversions from ºC to ºh are easier in the heximal system, since the proportion is a non–repeating heximal, but a repeating decimal.

In the heximal system of units, there is no need for a unit for amount, and units for luminosity intensity and electric current are only relevant in specific contexts. To express other units like speed, pressure, area, volume, acceleration, and more, you can simply use combinations of the base units. Here are some examples:

– The unit of speed is represented as heximal meters per heximal second, denoted as mh/sh.

– The unit of area is represented as square heximal meters, mh^2.

– The unit of pressure can be expressed as heximal grams per square heximal meter per heximal second squared, gh/(mh x (sh)^2), and so on.

This flexible approach allows for the creation of composite units to suit various measurement needs within the heximal system of units.

In the heximal system, the unit of angle measurement is the heximal degree (hº), which replaces the SI degree. Heximal degrees are defined such that 0 heximal degrees is equivalent to 0 radians, and 1,000 (hex) heximal degrees is equal to τ radians, which represents a full turn.

Here are some conversions and examples related to heximal degrees:

– 1hº = τ/1,000 (hex) / τ/216 (dec) rad, approximately 0.010141100143234252 (hex) / 0.02908882086657 (dec) radians.

– 1hº = 5/3 SI degrees, approximately 1.6666667 (dec) / 1.4 (hex) SI degrees.

– Some key angles in heximal degrees: A full turn τ radians is 1,000 (hex) / 216 (dec) hº, half turn τ/2 radians is 300 (hex) / 108 (dec) hº, a third of a turn τ/3 radians is 200 (hex) / 72 (dec) hº, a fourth of a turn τ/4 radians is 130 (hex) / 54 (dec) hº, a sixth of a turn τ/10 (hex) radians is 100 (hex) / 36 (dec) hº, an eighth of a turn τ/12 (hex) radians is 43 (hex) / 27 (dec) hº, a twelfth of a turn τ/20 (hex) radians is 30 (hex) / 18 (dec) hº.

Additionally, SI degrees can also be used in heximal, which can be better, since it is easier to divide angles by 5, but they require considering the last three digits and the sum of the digits when calculating the cosine for angles larger than 1,400 (hex) SI degrees. Heximal degrees have the advantage of only needing the last three digits for cosine calculations. Gradians are not recommended for use in the heximal system as they are less versatile and have limitations in representing certain angles.

In the heximal system, there are prefixes for larger and smaller units based on powers of six, tarnexy, and unwixion. Here are the prefixes and their notations:

Larger Units:

· Hexa (h): Six times larger than the base unit

· Nifffa (n): Niff times larger than the base unit

· Tarna (tr): Tarnexy times larger than the base unit

· Unwina (u): Unwixion times larger than the base unit

· Biwina (b): Biwixion times larger than the base unit

· Triwina (t): Triwixion times larger than the base unit

· Tetrwina (q): Tetrwixion times larger than the base unit

· Pentwina (p): Pentwixion times larger than the base unit

· Unnilwina (un): Unnilwixion times larger than the base unit

· Ununwina (uu): Ununwixion times larger than the base unit

· Umbiwina (ub): Umbiwixion times larger than the base unit

· Untriwina (ut): Untriwixion times larger than the base unit

· And so on...

Smaller Units:

· Hexi (hi): Six times smaller than the base unit

· Nifffi (ni): Niff times smaller than the base unit

· Tarni (tri): Tarnexy times smaller than the base unit

· Unwini (ui): Unwixion times smaller than the base unit

· Biwini (bi): Biwixion times smaller than the base unit

· Triwini (ti): Triwixion times smaller than the base unit

· Tetrwini: (qi): Tetrwixion times smaller than the base unit

· Pentwini: (pi): Pentwixion times smaller than the base unit

· Unnilwini (uni): Unnilwixion times smaller than the base unit

· Ununwini (uui): Ununwixion times smaller than the base unit

· Umbiwini (ubi): Umbiwixion times smaller than the base unit

· Untriwini (uti): Untriwixion times smaller than the base unit

· And so on…

With these prefixes, you can create a wide range of units to suit your needs. For example:

– 1 himh is 1 heximal heximeter, which is one–sixth of a heximal meter (mh).

– 1 uigh is 1 heximal unwinigram, representing one unwixionth of a heximal gram.

– 1 trmh is 1 heximal tarnameter, equivalent to 1 tarnexy heximal meters.

These prefixes can also be applied to units of time (seconds) and heximal degrees, providing flexibility in creating units for various measurements.

After the heximal units is the heximal clock, which is used by some people that support the heximal numbering system, and there is even one on the internet.

The heximal clock is a unique timekeeping system, which offers an interesting alternative to the traditional clock. Its heximal structure allows for a different way of representing time. Here's a breakdown on how it works:

There are six large divisions and fifsy small divisions. The six small divisions have numbers on them from 0 to 5, corresponding to the six’s digit of the number in heximal, and the smaller division correspond to the unit’s digit. Every three small divisions there is one larger small division, that indicates the midpoint between an interval of 2 six heximal hours. That division indicates that the unit’s digit is 3 in heximal. To read an analogue heximal clock just find what is the closest numbered division, which is right to the heximal hour hand, and count how many small divisions left to the current time from that big division to get the unit’s digit of the heximal hour. The heximal minutes and heximal seconds hands read the same way as heximal hours. To read a digital clock is like to read a normal digital clock, it just indicates the time accurately. The time works very simply, there are niff hours per day, so the heximal hour displays a number between 0 (hex) and 55 (hex), there are niff heximal minutes in a heximal hour, so the heximal minute also displays a number between 0 (hex) and 55 (hex), and there are niff heximal seconds in a heximal minute, so the heximal second displays a number between 0 (hex) and 55 (hex). Here is an example: if the time reads 00:00:00 (hex), that means midnight, if the time reads 30:00:00 (hex) that reads midday, and if the day reads 43:00:00 (hex) that reads 18:00:00 (standard) hours, which is close to the sunset. The heximal hour hand moves 1/2 times the speed of the standard hour hand, the heximal minute hand moves 3/2 times the speed of the standard minute hand, and the heximal second hand moves approximately as fast as the standard second hand, but it reverses through less markings. If you want more precision you can use tarniseconds, which are about 1/312 (hex) / 1/116 (dec) standard seconds, so they are the equivalent of a centisecond, and if you need even more precision you can use unwiniseconds, which are about 1/312,350 (hex) / 1/25,194 (dec) seconds, or about 40 (dec) microseconds.

While the heximal clock may not be as widely recognized as the standard base niff forsy clock, it has found a following among those who appreciate its unique approach to timekeeping. It offers an alternative perspective on the passage of time, and its simplicity can be appealing to some.

Now with all of these units I can express some physical constants:

G, which is the Gravitational constant ≅ 0.002144303 (hex) unwini heximal meters cubed per heximal gram per heximal second squared or 0.002144303 (hex) uimh^3gh^(-1)sh^(-2)

c, which is the speed of light ≅ 5,302;115,215.3 (hex) heximal meters per heximal seconds or 0.0053021152153 bimh^(1)sh^(-1)

h, which is the Planck’s constant ≅ 44.225530 (hex) umbiwni heximal grams heximal meters squared per heximal second, or 44.225530 ubigh^(1)mh^(2)sh^(-1)

The next topic is somewhat controversial and relates to the economy. One issue with adopting a heximal system is that virtually all currencies in use today are based on the decimal system. I'll present a hypothetical suggestion for creating a heximal currency, although it's unlikely to be implemented in practice, given the practical challenges involved. This concept envisions a currency similar to the Euro or Dollar, which I'll call the 'heximal euro'.

In this hypothetical heximal currency system, the smallest denomination would be the 'niffi,' equivalent to one–niffth of the base unit. There would be coins for 1 niffi, 2 niffis, 3 niffis, 10 (hex) niffis, 20 (hex) niffis, and 30 (hex) niffis, then for the base unit, the heximal euro, which I will denote as h€, there would be a 1 h€ coin, a 2 h€ coin, a 3 h€ coin, a 10 (hex) h€ bill, a 20 (hex) h€ bill, a 30 (hex) h€ bill, a 100 (hex) h€ bill, a 200 (hex) h€ bill, a 300 (hex) h€ bill, and a 1,000 (hex) h€ bill. Beyond this point, larger bills would be impractical. One advantage of this system is that all transactions smaller than 2,111.10 (hex) h€ could be made without repeating the same coin/bill, since six is a practical number, which is impossible in the current system, since to make a transaction of 4 cents you need to either use at least 2 coins of 1 cent or 2 coins of 2 cents, this is because ten is not a practical number.

The idea here is to convert the prices of products and services to the heximal system and then round the numbers to two heximal places to indicate the number of nifffis. However, it's important to note that transitioning to a new base numbering system for currency would be an enormous undertaking, involving the conversion of existing price structures, wages, and financial systems. Such a transition would require extensive coordination between governments, businesses, and financial institutions.

Additionally, decimal–based currency systems are deeply ingrained in global economies and societies, making any switch to a different base system highly complex and costly. The existing decimal systems have been in use for centuries and are deeply integrated into daily life.

Even though the idea of a heximal currency sounds intriguing we need to realize that economy is a very complex structure. Therefore, it is unlikely to be implemented in the foreseeable future. Nevertheless, discussing and exploring new ideas in economics and currency systems is always valuable for generating creative solutions to various economic challenges. However, signifficant changes to currency systems should be carefully considered, studied, and planned over an extended period to minimize disruption and ensure a smooth transition.

The last new unit is the hexit, which is the heximal unit of data. 1 hexit = 2.330204002 (hex) / 2.5849625 (dec) bits. The exact conversion is: 1 hexit = log2(10 (hex)) ≅ 2.330 (hex), this is because just like a byte is 12 (hex) bits, a hexit is the exponent of 2 such that 2 raised to it gives 10 (hex).

After that controversial topic is the heximal calendar.

The heximal calendar incorporates some modifications to the Gregorian calendar. In the heximal calendar, there are 20 (hex) / 12 (dec) months, each consisting of 1,405 (hex) / 365 (dec) days, except for leap years. One signifficant change is that heximal weeks are 10 (hex) / 6 (dec) days long, which is one day shorter than Gregorian weeks. This adjustment is made so that 1,000 (hex) heximal hours correspond to exactly 1 heximal week, simplifying time calculations. The months retain their traditional names, but the days of the week have entirely different names. The new names for the heximal week are as follows: First–day, Second–day, Third–day, Fourth–day, Fifth–day, and Sabat. These names draw inspiration from the Portuguese names for the days of the week.

Here is the calendar for a leap year, with tables for the months:

01 January

F	S	T	F	F	S
1	2	3	4	5	10
11	12	13	14	15	20
21	22	23	24	25	30
31	32	33	34	35	40
41	42	43	44	45	50

02 February

F	S	T	F	F	S
1	2	3	4	5	10
11	12	13	14	15	20
21	22	23	24	25	30
31	32	33	34	35	40
41	42	43	44	45	50
51	1	2	3	4	5

03 March

F	S	T	F	F	S
51	1	2	3	4	5
10	11	12	13	14	15
20	21	22	23	24	25
30	31	32	33	34	35
40	41	42	43	44	45
50	1	2	3	4	5

04 April

F	S	T	F	F	S
50	1	2	3	4	5
10	11	12	13	14	15
20	21	22	23	24	25
30	31	32	33	34	35
40	41	42	43	44	45
50	51	1	2	3	4

05 May

F	S	T	F	F	S
50	51	1	2	3	4
5	10	11	12	13	14
15	20	21	22	23	24
25	30	31	32	33	34
35	40	41	42	43	44
45	50	1	2	3	4

10 June

F	S	T	F	F	S
45	50	1	2	3	4
5	10	11	12	13	14
15	20	21	22	23	24
25	30	31	32	33	34
35	40	41	42	43	44
45	50	51	1	2	3

11 July

F	S	T	F	F	S
45	50	51	1	2	3
4	5	10	11	12	13
14	15	20	21	22	23
24	25	30	31	32	33
34	35	40	41	42	43
44	45	50	1	2	3

12 August

F	S	T	F	F	S
44	45	50	1	2	3
4	5	10	11	12	13
14	15	20	21	22	23
24	25	30	31	32	33
34	35	40	41	42	43
44	45	50	51	1	2

13 September

F	S	T	F	F	S
44	45	50	51	1	2
3	4	5	10	11	12
13	14	15	20	21	22
23	24	25	30	31	32
33	34	35	40	41	42
43	44	45	50	1	2

14 October

F	S	T	F	F	S
43	44	45	50	1	2
3	4	5	10	11	12
13	14	15	20	21	22
23	24	25	30	31	32
33	34	35	40	41	42
43	44	45	50	51	1

15 November

F	S	T	F	F	S
43	44	45	50	51	1
2	3	4	5	10	11
12	13	14	15	20	21
22	23	24	25	30	31
32	33	34	35	40	41
42	43	44	45	50	1

20 December

F	S	T	F	F	S
42	43	44	45	50	1
2	3	4	5	10	11
12	13	14	15	20	21
22	23	24	25	30	31
32	33	34	35	40	41
42	43	44	45	50	51

The way to calculate the current day of the week assumes that January 1, 1 BC, in the Gregorian calendar corresponds to January 1, 0, in this calendar, and it is designated as Sabat. To find the day of the week for any date, calculate how many days have passed since January 1, 1 BC, convert this count to heximal, and determine the day of the week based on the last digit. For instance, if the last digit is 0, it corresponds to Sabat, 1 to First–day, 2 to Second–day, 3 to Third–day, 4 to Fourth–day, and 5 to Fifth–day.

Now, let's delve into the heximal calendar. To make it easier to understand, let's begin with the date this blog was published, is 1/14/13,211 (hex) or 1/10/2,023 (dec) in this calendar. The corresponding day of the week is a First-day, which, interestingly, aligns with Sunday 1/14/13,211 (hex) or 1/10/2,023 (dec) in the Gregorian calendar.

The heximal calendar consists of 20 (hex) / 12 (dec) months, with each month having either 51 (hex) / 31 (dec) days (even–indexed months, e.g., April and August) or 50 (hex) / 30 (dec) days (odd–indexed months, e.g., March and September). There is one exception: February, which has 50 (hex) / 30 (dec) days in regular years but gains an extra day, making it 51 (hex) / 31 (dec) days in leap years.

In a leap year, the heximal calendar comprises 1,410 (hex) / 366 (dec) days, resulting in 141 (hex) / 61 (dec) heximal weeks. In non–leap years, there are 1,405 (hex) / 365 (dec) days, equivalent to 140.5 (hex) / 60.833333… (dec) heximal weeks.

A straightforward rule determines leap years: if a year is divisible by 4, it is a leap year, except when it is divisible by 332 (hex) / 128 (dec), in which case it is not. Another way to put it is that a year is a leap year if it is divisible by 4 twice but not divisible by 2 seven or more times. For example, 13,211 (hex) / 2,023 (dec) is not a leap year because it is not divisible by 4, resulting in 1,405 (hex) / 365 (dec) days. In contrast, 13,224 (hex) / 2,032 (dec) is a leap year since it is divisible by 4 but not by 332 (hex) / 128 (dec), resulting in 1,410 (hex) / 366 (dec) days. Lastly, 13,252 (hex) / 2,048 (dec) is not a leap year as it is divisible by 332 (hex) / 128 (dec), leading to 1,405 (hex) / 365 (dec) days.

The calendar loses 1 day to the Gregorian calendar every 22,452 (hex) / 3,200 (dec) years. Since the current year is 13,211 (hex) / 2,023 (dec) in both calendars, it is not surprising that they share the same date. This calendar has 41;014,555 (hex) / 1,168,775 (dec) days, while the Gregorian calendar has 41;015,000 (hex) / 1,168,776 (dec) days, in 3,200 (dec) / 22,452 (hex) years, making them extremely similar in duration. In essence, the difference of 1 day in 22,452 (hex) / 3,200 (dec) years is negligible for most practical purposes.

The only thing left is to recalculate dates, which isn’t that hard, since in most cases you either add 1 day, subtract one day or do nothing to convert days from the Gregorian calendar to this calendar, and in most cases where a year is a leap year in the Gregorian calendar and a leap year in this calendar, or a regular year for both, then after August the days sync, so they are the same between August and December.

Conclusion

In summary, the heximal numbering system presents several advantages over the decimal system, making it a compelling choice for those interested in exploring alternative numeral systems. Heximal simplifies arithmetic with fewer digits and smaller multiplication tables, making calculations more straightforward. It offers advantages in prime factorization, where numbers often have simpler representations or smaller recurring cycles.

Heximal's divisibility tests for prime numbers like 5 and 11 (hex) are straightforward, and since you can discard any number ending with 0, 2, 3 or 4 as a possible prime, it is easier to identify prime numbers in heximal compared to decimal. Moreover, counting with heximal on your fingers allows you to reach higher numbers up to 55 (hex) / 35 (dec) without the need for extra tools or techniques.

Additionally, heximal can be compressed into niff, effectively halving the length of large numbers, a feat not possible in decimal due to its limited digit set. The last advantage is that you can round τ to 10 (hex) if you don’t need too much precision, which makes calculations very easy in heximal, if you round τ to 6 (dec) in decimal calculations won’t be as easy. Dozenal is also a viable alternative to decimal, offering similar length advantages with an easier approach to powers of 2, at the cost of making the primes 5 and 11 (hex) harder to deal with, making it a matter of personal preference between heximal and dozenal.

Throughout this blog, I've provided names for numbers, ordinals, polygons, and various units in the heximal system, as well as introduced a heximal clock and calendar. These resources should equip you with the knowledge and tools needed to explore and use the heximal numbering system in your daily life. I recommend practicing simple arithmetic in heximal until you are comfortable with its use, you can practice it using Wolfram Alpha, since it allows you to compute any expression in any positional numbering system, so solve easy arithmetic problems in heximal verifying the result in Wolfram Alpha, and then try solving harder problems, like linear and quadratic equations, then consider incorporating heximal into your routine, also you don’t need to put a ‘(hex)’ after a heximal number as long as it is possible to understand that the number is in heximal, and reserve decimal for situations where it is necessary.

There are more resources to learn heximal, such as dedicated websites created by people who support the heximal numbering system such as https://www.seximal.net/, and http://shacktoms.org/base–six/base–six.htm, plus there is a heximal clock if you search it on the internet, https://xanthir.com/hex/clock/, which uses the same system I mentioned. There is also a reddit page dedicated to heximal, r/Seximal, so you can hear about opinions of people who use heximal. The last thing I need to say is that some people use powers of 10^4 (hex) in the Reddit page, instead of powers of 10^3 (hex) like I use, so the name of the numbers is different, and they might use spaces differently, but it is still heximal. You can also see more a pages I created in this blog.

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+	0	1	2	3	4	5
0	0	1	2	3	4	5
1	1	2	3	4	5	10
2	2	3	4	5	10	11
3	3	4	5	10	11	12
4	4	5	10	11	12	13
5	5	10	11	12	13	14

x	1	2	3	4	5
0	0	0	0	0	0
1	1	2	3	4	5
2	2	4	10	12	14
3	3	10	13	20	23
4	4	12	20	24	32
5	5	14	23	32	41

F	S	T	F	F	S
1	2	3	4	5	10
11	12	13	14	15	20
21	22	23	24	25	30
31	32	33	34	35	40
41	42	43	44	45	50

F	S	T	F	F	S
1	2	3	4	5	10
11	12	13	14	15	20
21	22	23	24	25	30
31	32	33	34	35	40
41	42	43	44	45	50
51	1	2	3	4	5

F	S	T	F	F	S
51	1	2	3	4	5
10	11	12	13	14	15
20	21	22	23	24	25
30	31	32	33	34	35
40	41	42	43	44	45
50	1	2	3	4	5

F	S	T	F	F	S
50	1	2	3	4	5
10	11	12	13	14	15
20	21	22	23	24	25
30	31	32	33	34	35
40	41	42	43	44	45
50	51	1	2	3	4

F	S	T	F	F	S
50	51	1	2	3	4
5	10	11	12	13	14
15	20	21	22	23	24
25	30	31	32	33	34
35	40	41	42	43	44
45	50	1	2	3	4

F	S	T	F	F	S
45	50	1	2	3	4
5	10	11	12	13	14
15	20	21	22	23	24
25	30	31	32	33	34
35	40	41	42	43	44
45	50	51	1	2	3

F	S	T	F	F	S
45	50	51	1	2	3
4	5	10	11	12	13
14	15	20	21	22	23
24	25	30	31	32	33
34	35	40	41	42	43
44	45	50	1	2	3

F	S	T	F	F	S
44	45	50	1	2	3
4	5	10	11	12	13
14	15	20	21	22	23
24	25	30	31	32	33
34	35	40	41	42	43
44	45	50	51	1	2

+	0	1	2	3	4	5
0	0	1	2	3	4	5
1	1	2	3	4	5	10
2	2	3	4	5	10	11
3	3	4	5	10	11	12
4	4	5	10	11	12	13
5	5	10	11	12	13	14

x	1	2	3	4	5
0	0	0	0	0	0
1	1	2	3	4	5
2	2	4	10	12	14
3	3	10	13	20	23
4	4	12	20	24	32
5	5	14	23	32	41

F	S	T	F	F	S
1	2	3	4	5	10
11	12	13	14	15	20
21	22	23	24	25	30
31	32	33	34	35	40
41	42	43	44	45	50

F	S	T	F	F	S
1	2	3	4	5	10
11	12	13	14	15	20
21	22	23	24	25	30
31	32	33	34	35	40
41	42	43	44	45	50
51	1	2	3	4	5

F	S	T	F	F	S
51	1	2	3	4	5
10	11	12	13	14	15
20	21	22	23	24	25
30	31	32	33	34	35
40	41	42	43	44	45
50	1	2	3	4	5

F	S	T	F	F	S
50	1	2	3	4	5
10	11	12	13	14	15
20	21	22	23	24	25
30	31	32	33	34	35
40	41	42	43	44	45
50	51	1	2	3	4

F	S	T	F	F	S
50	51	1	2	3	4
5	10	11	12	13	14
15	20	21	22	23	24
25	30	31	32	33	34
35	40	41	42	43	44
45	50	1	2	3	4

F	S	T	F	F	S
45	50	1	2	3	4
5	10	11	12	13	14
15	20	21	22	23	24
25	30	31	32	33	34
35	40	41	42	43	44
45	50	51	1	2	3

F	S	T	F	F	S
45	50	51	1	2	3
4	5	10	11	12	13
14	15	20	21	22	23
24	25	30	31	32	33
34	35	40	41	42	43
44	45	50	1	2	3

F	S	T	F	F	S
44	45	50	1	2	3
4	5	10	11	12	13
14	15	20	21	22	23
24	25	30	31	32	33
34	35	40	41	42	43
44	45	50	51	1	2

+	0	1	2	3	4	5
0	0	1	2	3	4	5
1	1	2	3	4	5	10
2	2	3	4	5	10	11
3	3	4	5	10	11	12
4	4	5	10	11	12	13
5	5	10	11	12	13	14

x	1	2	3	4	5
0	0	0	0	0	0
1	1	2	3	4	5
2	2	4	10	12	14
3	3	10	13	20	23
4	4	12	20	24	32
5	5	14	23	32	41

F	S	T	F	F	S
1	2	3	4	5	10
11	12	13	14	15	20
21	22	23	24	25	30
31	32	33	34	35	40
41	42	43	44	45	50

F	S	T	F	F	S
1	2	3	4	5	10
11	12	13	14	15	20
21	22	23	24	25	30
31	32	33	34	35	40
41	42	43	44	45	50
51	1	2	3	4	5

F	S	T	F	F	S
51	1	2	3	4	5
10	11	12	13	14	15
20	21	22	23	24	25
30	31	32	33	34	35
40	41	42	43	44	45
50	1	2	3	4	5

F	S	T	F	F	S
50	1	2	3	4	5
10	11	12	13	14	15
20	21	22	23	24	25
30	31	32	33	34	35
40	41	42	43	44	45
50	51	1	2	3	4

F	S	T	F	F	S
50	51	1	2	3	4
5	10	11	12	13	14
15	20	21	22	23	24
25	30	31	32	33	34
35	40	41	42	43	44
45	50	1	2	3	4

F	S	T	F	F	S
45	50	1	2	3	4
5	10	11	12	13	14
15	20	21	22	23	24
25	30	31	32	33	34
35	40	41	42	43	44
45	50	51	1	2	3

F	S	T	F	F	S
45	50	51	1	2	3
4	5	10	11	12	13
14	15	20	21	22	23
24	25	30	31	32	33
34	35	40	41	42	43
44	45	50	1	2	3

F	S	T	F	F	S
44	45	50	1	2	3
4	5	10	11	12	13
14	15	20	21	22	23
24	25	30	31	32	33
34	35	40	41	42	43
44	45	50	51	1	2