Number Systems

#### CON

Cursed Octodecinary Numerals is a type of number system where the order of Octodecinary Digits are changed. There are 2 versions of Cursed Octodecinary Numerals. The digit zero would be a big dot (written on paper) or an asterisk (on computer). The digit 9 would be 0 and the digit 17 would be 8. Version 1 contains all the digits from 0 - 9, the letters A - G and a lowercase o.

### SNA

Stick Numerals Association is a type of number system where each digit is represented by sticks. Stick numerals are also known as rod numerals. The most common form of stick numerals is base 28 (7&28).

### WCA

Wingdings Code Association is a type of number system where each digit is represented by symbols except for zero. Each symbol represents a digit such as Cancer being digit 1, Leo being digit 2 and Virgo being digit 3.

## Number 127

The number 127 is a prime number. It is one less than 2^7 (128). It is mainly used to convert between metric and imperial.

### Numeral Systems Where 127 has a repetend less than 15

* Binary
* Quarternary
* Nonadecinary
* Vigesimal
* Duovigesimal

### In Other Numeral Systems

 Duodecinary
 Tridecinary
 Tetradecinary
 Pentadecinary
 Nonadecinary
 Duovigesimal
 Trivigesimal
 Hexavigesimal

## Duodecinary Form

*Note: X is digit 10 and E is digit 11.*

Duodecinary Form is a number system where the base is 12. 12 is a highly composite number so the reciprocals of 2, 3, 4, 6 and 12 have one terminating decimal. Despite five numbers having one terminating decimal, it has multiple drawbacks. Both 5 and 7 have periods larger than 3 and 127 has a reptend of 126. Therefore, duodecinary is very difficult to convert from metric to imperial.

4 and 16 only have 2 as a prime factor. Therefore, in Quarternary and Hexadecimal, only the reciprocals of the powers of 2 terminate.

### Counting in from 1 - 36 in Duodecinary

1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1X, 1E, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2X, 2E, 30

### Fractions in Duodecinary

#### Full Repetend Primes

5 is an example of a full repetend prime number in Duodecinary form.

1/5 = 0.2497

2/5 = 0.4972

3/5 = 0.7249

4/5 = 0.9724

In other words, EEEE/5 = 2497. Therefore, all the repeating digits are 2497 and only the starting digit changes. The order and the digits don't change. 2497 is the Duodecinary form of 4147. 12^4 - 1 = 20735. 20735/5 = 4147.

Other examples of full repetend primes in Duodecinary Form are 7, 17, 31, 41, 43, 53, 67, 101, 103, 113, 127, 137, 139, 149, 151, 163, 173, 197, 223, 257, 269, 281, 283 and 293 (written in decimal).

https://oeis.org/A019340

## Tridecinary Form

Tridecinary Form is a number system where the base is 13. There are several notations of Tridecinary.

Due to 13 being a prime number, only the reciprocals of powers of 13 terminate. However, one less than 13 squared is highly composite which guarantees that commonly used numbers have a repetend of 1 or 2. Therefore, Tridecinary is a useful number system.

### Fractions in Tridecinary

#### Full Repetend Primes

5 is an example of a full repetend prime number in Tridecinary Form.

1/5 = 0.27A5

2/5 = 0.527A

3/5 = 0.7A52

4/5 = 0.A527

In other words, LLLL/5 = 27A5. Therefore, all the repeating digits are 27A5 and only the starting digit changes. The order and the digits don't change. 27A5 is the Tridecinary form of 5712. 13^4 - 1 = 28560. 28560/5 = 5712.

Other examples of full repetend primes in Tridecinary Form are 11, 19, 31, 37, 41, 47, 59, 67, 71, 73, 83, 89, 97, 109, 137, 149, 151, 167, 197, 227, 239, 241, 281 and 293 (written in decimal).

https://oeis.org/A019341

#### Fractional Repetend Primes

17 is an example of a fractional repetend prime number in Tridecinary Form.

1/17 = 0.09L3

2/17 = 0.16K6

3/17 = 0.23A9

4/17 = 0.309L

5/17 = 0.3A92

6/17 = 0.4785

7/17 = 0.5478

8/17 = 0.616K

9/17 = 0.6K61

10/17 = 0.7854

11/17 = 0.8547

12/17 = 0.923A

13/17 = 0.9L30

14/17 = 0.A923

15/17 = 0.K616

16/17 = 0.L309

As shown, since 17 is a quarter repetend number, there are four sets of repeating digits. 09L3 is given to 1/17, 4/17, 13/17 and 16/17. 16K6 is given to 2/17, 8/17, 9/17 and 15/17. 23A9 is given to 3/17, 5/17, 12/17 and 14/17. 4785 is given to 6/17, 7/17, 10/17 and 11/17. In addition, the sets of repeating digits show that 17 has a primitive root with 3, 5, 6, 7, 10, 11, 12 and 14 due to 17 being a prime number one more than a power of 2. 2, 3, 5, 17, 257 and 65537 are primes numbers one more than a power of 2.

#### Reciprocal of 4 and 8

Since the base number is (13) is 1 more than a multiple of 4, but 5 more than a multiple of 8, the repetend of 8 is 2 and the repetend of 4 is 1. 5/8 shares the same set of repeating decimals as 1/8.

1/8 = 0.18

1/4 = 0.3

3/8 = 0.4K

5/8 = 0.81

3/4 = 0.9

7/8 = 0.K4

### List of Notations

Note: Letters are not case sensitive.

 Linear
 X

## Pentadecinary Form

*Note: A is digit 10, K is digit 11, L is digit 12, M is digit 13, N is digit 14.*

Pentadecinary form is a number system where the base is 15.

15 is divisible by both 3 and 5. Therefore 3 and 5 both have terminating decimals. 2 and 7 have a repetend of 1.

### Fractions in Pentadecinary

#### Full Repetend Primes

13 is an example of a full reptend prime number in Pentadecinary Form.

1/13 = 0.124936MLA5K8

2/13 = 0.24936MLA5K81

3/13 = 0.36MLA5K81249

4/13 = 0.4936MLA5K812

5/13 = 0.5K8124936MLA

6/13 = 0.6MLA5K812493

7/13 = 0.8124936MLA5K

8/13 = 0.936MLA5K8124

9/13 = 0.A5K8124936ML

10/13 = 0.K8124936MLA5

11/13 = 0.LA5K8124936M

12/13 = 0.MLA5K8124936

In other words, NNNNNNNNNNNN/M = 124936MLA5K8. Therefore, all the repeating digits are 124936MLA5K8 and only the starting digit changes. The order and the digits don't change. 124936MLA5K8 is the Pentadecinary Form of 9980487530048. 15^12 - 1 = 129746337890624. 129746337890624/13 = 9980487530048.

Other examples of full repetend primes in Pentadecinary Form are 19, 23, 29, 37, 41, 47, 73, 83, 89, 97, 101, 107, 139, 149, 151, 157, 167, 193, 199, 227, 263, 269, 271 and 281 (written in decimal).

https://oeis.org/A019343

#### Fractional Reptend Primes

11 is an example of a fractional reptend prime number in Pentadecinary Form.

1/11 = 0.156L4

2/11 = 0.2AM98

3/11 = 0.4156L

4/11 = 0.56L41

5/11 = 0.6L415

6/11 = 0.82AM9

7/11 = 0.982AM

8/11 = 0.AM982

9/11 = 0.L4156

10/11 = 0.M982A

As shown, since 11 is a half reptend number, there are two sets of repeating digits. 156L4 is given to 1/11, 3/11, 4/11, 5/11 and 9/17. 2AM98 is given to 2/11, 6/11, 7/11, 8/11 and 10/11.

## Octodecinary and Nonary Form

*Note: A is digit 10, K is digit 11, L is digit 12, M is digit 13, N is digit 14, T is digit 15, P is digit 16, H is digit 17.*

Nonary form is a number system where the base is 9. Nonary can be expressed with 2 Ternary Digits. 9 only has 3 as a prime factor. Therefore in Nonary, only the reciprocals of the powers of 3 terminate.

Octodecinary form is a number system where the base is 18. Octodecinary has two versions of the Cursed Octodecinary Numerals (CON). The 3rd, 4th, 6th and 9th powers of 7 are all palindromes, while 7 cubed is a repunit written as 111.

## Nonadecinary Form

*Note: A is digit 10, K is digit 11, L is digit 12, M is digit 13, N is digit 14, T is digit 15, P is digit 16, H is digit 17, R is digit 18.*

Nonadecinary form is a number system where the base is 19.

Due to 19 being a prime number, only the reciprocals of powers of 19 terminate. However, one less than 19 squared is highly composite which guarantees that commonly used numbers have a repetend of 1 or 2. Therefore, Nonadecinary is a useful number system. In addition, the repetend of 127 is 3, making it useful to convert metric to imperial.

## Duovigesimal Form

*Note: A is digit 10, K is digit 11, L is digit 12, M is digit 13, N is digit 14, T is digit 15, P is digit 16, H is digit 17, R is digit 18, B is digit 19, C is digit 20, D is digit 21.*

Duovigesimal form is a number system where the base is 22.

3 and 7 have a repetend of 1. All the prime numbers from 2 - 12 except for 5 have a repetend less than 3 in Duovigesimal. All the numbers from 1 - 24 except 17 and 19 have a repetend less than 5 in Duovigesimal. In addition, the repetend of 127 is 9. All of this makes Duovigesimal a useful number system.

### Fractions in Duovigesimal

#### Full Repetend Primes

5 is an example of a full repetend prime number in Duovigesimal Form.

1/5 = 0.48HM

2/5 = 0.8HM4

3/5 = 0.M48H

4/5 = 0.HM48

In other words, DDDD/5 = 48HM. Therefore, all the repeating digits are 48HM and only the starting digit changes. The order and the digits don't change. 48HM is the Duovigesimal Form of 46851. 22^4 - 1 = 234255. 234255/5 = 46851.

Other examples of full repetend primes in Duovigesimal Form are 17, 19, 31, 37, 41, 47, 53, 71, 83, 107, 131, 139, 191, 193, 199, 211, 223, 227, 233, 269, 281, 283, 307 and 311 (written in decimal).

https://oeis.org/A019349

## Trivigesimal Form

*Note: A is digit 10, K is digit 11, L is digit 12, M is digit 13, N is digit 14, T is digit 15, P is digit 16, H is digit 17, R is digit 18, B is digit 19, C is digit 20, D is digit 21, Y is digit 22.*

Trivigesimal form is a number system where the base is 23. Trivigesimal is the number system with the smallest base where 53 and 79 have a repetend less than 10. Thus, it is useful when you are using the numbers 23, 53 and 79.

## Hexavigesimal Form

*Note: A is digit 10, K is digit 11, L is digit 12, M is digit 13, N is digit 14, T is digit 15, P is digit 16, H is digit 17, R is digit 18, B is digit 19, C is digit 20, D is digit 21, Y is digit 22, W is digit 23, F is digit 24, S is digit 25.*

Hexavigesimal form is a number system where the base is 26. In Hexavigesimal, the prime numbers 2, 3, 5 and 13 have a repetend less than 3.