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Heptadecimal representation of fractions

It’s tempting to call numbers like 0.5 or 1.25 or 6.28 “decimals”. But decimals only exist in base 10. What we have here are heptadecimals.

For the record, the base-agnostic way of referring to that little dot between the integer part and the fractional part of a number is to call it a radix point.

weka li weka!

Remember: Negative digits are negative. If the next significant digit after the radix point is negative, then the integer part is actually one less than it appears to be. The number 1.❶ is less than one.

Halves

1/2 = 0.888... = 1.❽❽❽...

Here, the “...” means that the pattern continues forever to the right. Which means...

That’s right, there are two ways to write the number one-half; a zero followed by infinitely many eights after the radix point, AND a 1 followed by infinitely many negative-eights after the radix point. They’re the same number!

In fact, any time you see a heptadecimal that ends in infinitely many eights, you can replace those eights with negative-eights and add one to the position immediately to the left of where the eights started, and you’ll get the same number. 1.2888... is the same number as 1.3❽❽❽..., and ❺.888... is the same number as ❹.❽❽❽...

If you’re a p-adic liker, try to figure out what ...❽❽❽.0 and ...888.0 could possibly mean.

Fourths

This is a sub-heading under “halves” because fourths are just halves of halves.

1/4 = 0.444...
3/4 = 1.❹❹❹...

Seeing these representations hopefully makes it a little clearer why 1/2 can be represented as both 0.888... and 1.❽❽❽....

Eighths

1/8 = 0.222...
3/8 = 0.666...
5/8 = 1.❻❻❻...
7/8 = 1.❷❷❷...

Sixteenths

1/1❶ = 0.111...
3/1❶ = 0.333...
5/1❶ = 0.555...
7/1❶ = 0.777...
1❽/1❶ = 1.❼❼❼...
1❻/1❶ = 1.❺❺❺...
1❹/1❶ = 1.❸❸❸...
1❷/1❶ = 1.❶❶❶...

Remember, the circled digits are negative numbers. If you’re not sure what number 1❶ is supposed to be, think of it as one times seventeen plus negative one times one.

Thirds

1/3 = 0.6❻6❻...
2/3 = 1.❻6❻6...

Here we start to see an intriguing pattern: Repeating digits in the cycle of A, -A, A, -A, and so forth.

Sixths

1/6 = 0.3❸3❸...
5/6 = 1.❸3❸3...

Ninths

1/1❽ = 0.2❷2❷...
2/1❽ = 0.4❹4❹...
4/1❽ = 0.8❽8❽...
5/1❽ = 1.❽8❽8...
7/1❽ = 1.❹4❹4...
8/1❽ = 1.❷2❷2...

Twelfths

1/1❺ = 0.1717...
5/1❺ = 0.7171...
7/1❺ = 1.❼❶❼❶
1❻/1❺ = 1.❶❼❶❼...

Remember, all the other twelfths are covered under either “thirds” or “fourths”.

Fifths

1/5 = 0.37❸❼37❸❼...
2/5 = 0.7❸❼3...
3/5 = 1.❸❼37...
4/5 = 1.❼37❸...

This is where you really start to see the fascinating patterns that you get from the balanced heptadecimal representations of fractions. You’d probably get something similar from other balanced bases, but I don’t feel like checking tbh.

Tenths

1/1❼ = 0.2❺❷5...
3/1❼ = 0.52❺❷...
7/1❼ = 1.❺❷52...
1❽/1❼ = 1.❷52❺...

In regular heptadecimal 1/10 would be written out as 0.1BF51BF5..., which looks different but really isn’t. It’s just that when you bring negative digits into the mix, you can really see the symmetry of the fractions. 1 is one more than zero and 9(DEC) less than 10(DEC); likewise, 9(DEC) is one less than 10(DEC) and 9(DEC) more than zero; so it only makes sense that 1/10(DEC) and 9/10(DEC) would have this kind of symmetry.

I think this makes more sense when you’re deriving the numbers yourself. I can’t really figure out how to adequately word what I mean.

Fifteenths

1/1❷ = 0.125❽...
2/1❷ = 0.25❽1...
4/1❷ = 0.5❽12...
7/1❷ = 0.8❶❷❺...
8/1❷ = 1.❽125...
1❻/1❷ = 1.❺8❶❷...
1❹/1❷ = 1.❷❺8❶...
1❸/1❷ = 1.❶❷❺8...

Look at this! This is beautiful! This is two separate patterns of positive and negative digits interweaving with each other, like a pair of ropes braiding around each other.

Sevenths

1/7 = 0.275❷❼❺...
2/7 = 0.5❷❼❺27...
3/7 = 0.75❷❼❺2...
4/7 = 1.❼❺275❷...
5/7 = 1.❺275❷❼...
6/7 = 1.❷❼❺275...

Fourteenths

1/1❸ = 0.14❻❶❹6...
3/1❸ = 0.4❻❶❹61...
5/1❸ = 0.614❻❶❹...
1❽/1❸ = 1.❻❶❹614...
1❻/1❸ = 1.❹614❻❶...
1❹/1❸ = 1.❶❹614❻...

Yes, it is confusing that BS 1❸ = DEC 14 and BS 1❹ = DEC 13. Because of how these digits work, though, it was bound to happen somewhere.

Elevenths

1/1❻ = 0.2❽5❻❸❷8❺63...
2/1❻ = 0.32❽5❻❸❷8❺6...
3/1❻ = 0.5❻❸❷8❺632❽...
4/1❻ = 0.632❽5❻❸❷8❺...
5/1❻ = 0.8❺632❽5❻❸❷...
6/1❻ = 1.❽5❻❸❷8❺632...
7/1❻ = 1.❻❸❷8❺632❽5...
8/1❻ = 1.❺632❽5❻❸❷8...
1❽/1❻ = 1.❸❷8❺632❽5❻...
1❼/1❻ = 1.❷8❺632❽5❻❸...

Absolutely beautiful.

I’m not gonna do twenty-seconds.

Thirteenths

1/1❹ = 0.154❶❺❹...
2/1❹ = 0.3❼8❸7❽...
3/1❹ = 0.4❶❺❹15...
4/1❹ = 0.54❶❺❹1...
5/1❹ = 0.7❽3❼8❸...
6/1❹ = 0.8❸7❽3❼...
7/1❹ = 1.❽3❼8❸7...
8/1❹ = 1.❼8❸7❽3...
1❽/1❹ = 1.❺❹154❶...
1❼/1❹ = 1.❹154❶❺...
1❻/1❹ = 1.❸7❽3❼8...
1❺/1❹ = 1.❶❺❹154...

Another weaving of patterns like we saw with the fifteenths.

Seventeenths

What absolute nonsense awaits us this time?

1/10 = 0.1
2/10 = 0.2
3/10 = 0.3
4/10 = 0.4
5/10 = 0.5
6/10 = 0.6
7/10 = 0.7
8/10 = 0.8
1❽/10 = 1.❽
1❼/10 = 1.❼
1❻/10 = 1.❻
1❺/10 = 1.❺
1❹/10 = 1.❹
1❸/10 = 1.❸
1❷/10 = 1.❷
1❶/10 = 1.❶

Ah. Right.

Just one more:

Eighteenths

1/11 = 0.1❶1❶...

I’m stopping here, lest I keep going forever. If you’re curious what lies beyond, I encourage you to do some math of your own!