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!

—some kind of meme

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 $1.\overline{1}$ is less than one.

[Edited on 2024/08/12] Kind of embarrassing how amazed past-me was by Midy’s theorem, which isn’t even something new. In my defense, it’s not actually super obvious until you derive for yourself the fractions of the balanced form of an obnoxiously large prime-number base.

Halves

$1/2 = 0.888\dots = 1.\overline{888}\dots$

Here, the “$\dots$” 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. $1.2888\dots$ is the same number as $1.3\overline{888}\dots$, and $\overline{5}.888\dots$ is the same number as $\overline{4}.\overline{888}\dots$`

If you’re a p-adic liker, try to figure out what $\dots\overline{888}.0$ and $\dots888.0$ could possibly mean.

Fourths

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

$1/4 = 0.444\dots$
$3/4 = 1.\overline{444}\dots$

Seeing these representations hopefully makes it a little clearer why $1/2$ can be represented as both $0.888\dots$ and $1.\overline{888}\dots$.

Eighths

$1/8=0.222\dots$
$3/8=0.666\dots$
$5/8=1.\overline{666}\dots$
$7/8=1.\overline{222}\dots$

Sixteenths

$1/1\overline{1}=0.111\dots$
$3/1\overline{1}=0.333\dots$
$5/1\overline{1}=0.555\dots$
$7/1\overline{1}=0.777\dots$
$1\overline{8}/1\overline{1}=1.\overline{777}\dots$
$1\overline{6}/1\overline{1}=1.\overline{555}\dots$
$1\overline{4}/1\overline{1}=1.\overline{333}\dots$
$1\overline{2}/1\overline{1}=1.\overline{111}\dots$

Remember, the overlined digits are negative numbers. If you’re not sure what number $1\overline{1}$ is supposed to be, think of it as one times seventeen plus negative one times one.

Thirds

$1/3=0.6\overline{6}6\overline{6}\dots$
$2/3=1.\overline{6}6\overline{6}6\dots$

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

Sixths

$1/6=0.3\overline{3}3\overline{3}\dots$
$5/6=1.\overline{3}3\overline{3}3\dots$

Ninths

$1/1\overline{8}=0.2\overline{2}2\overline{2}\dots$
$2/1\overline{8}=0.4\overline{4}4\overline{4}\dots$
$4/1\overline{8}=0.8\overline{8}8\overline{8}\dots$
$5/1\overline{8}=1.\overline{8}8\overline{8}8\dots$
$7/1\overline{8}=1.\overline{4}4\overline{4}4\dots$
$8/1\overline{8}=1.\overline{2}2\overline{2}2\dots$

Twelfths

$1/1\overline{5}=0.1717\dots$
$5/1\overline{5}=0.7171\dots$
$7/1\overline{5}=1.\overline{7171}\dots$
$1\overline{6}/1\overline{5} = 1.\overline{1717}\dots$

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

Fifths

$1/5=0.37\overline{37}37\overline{37}\dots$
$2/5 = 0.7\overline{37}3\dots$
$3/5=1.\overline{37}37\dots$
$4/5 = 1.\overline{7}37\overline{3}\dots$

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

$1/1\overline{7}=0.2\overline{52}5\dots$
$3/1\overline{7}=0.52\overline{52}\dots$
$7/1\overline{7}=1.\overline{52}52\dots$
$1\overline{8}/1\overline{7}=1.\overline{2}52\overline{5}\dots$

In regular heptadecimal $1/10$ would be written out as $0.1\text{BF}51\text{BF}5\dots$, 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 $0$ and $9_{10}$ less than $10_{10}$; likewise, $9_{10}$ is one less than $10_{10}$ and $9_{10}$ more than $0$; so it only makes sense that $1/10_{10}$ and $9/10_{10}$ would have this kind of symmetry.

I think this makes more sense when you’re deriving the numbers yourself.

Fifteenths

$1/1\overline{2}=0.125\overline{8}\dots$
$2/1\overline{2}=0.25\overline{8}1\dots$
$4/1\overline{2}=0.5\overline{8}12\dots$
$7/1\overline{2} = 0.8\overline{125}\dots$
$8/1\overline{2}=1.\overline{8}125\dots$
$1\overline{6}/1\overline{2}=1.\overline{5}8\overline{12}\dots$
$1\overline{4}/1\overline{2}=1.\overline{25}8\overline{1}\dots$
$1\overline{3}/1\overline{2}=1.\overline{125}8\dots$

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

$1/7=0.275\overline{275}\dots$
$2/7=0.5\overline{275}27\dots$
$3/7=0.75\overline{275}2\dots$
$4/7=1.\overline{75}275\overline{2}\dots$
$5/7=1.\overline{5}275\overline{27}\dots$
$6/7=1.\overline{275}275\dots$

Fourteenths

$1/1\overline{3}=0.14\overline{614}6\dots$
$3/1\overline{3}=0.4\overline{614}61\dots$
$5/1\overline{3}=0.614\overline{614}\dots$
$1\overline{8}/1\overline{3}=1.\overline{614}614\dots$
$1\overline{6}/1\overline{3}=1.\overline{4}614\overline{61}\dots$
$1\overline{4}/1\overline{3}=1.\overline{14}614\overline{6}\dots$

Yes, it is confusing that $1\overline{3}_{\text{bal}17}=14_{10}$ and $1\overline{4}_{\text{bal}17}=13_{10}$. Because of how these digits work, though, it was bound to happen somewhere.

Elevenths

$1/1\overline{6}=0.2\overline{8}5\overline{632}8\overline{5}63\dots$
$2/1\overline{6}=0.32\overline{8}5\overline{632}8\overline{5}6\dots$
$3/1\overline{6} = 0.5\overline{632}8\overline{5}632\overline{8}\dots$
$4/1\overline{6} = 0.632\overline{8}5\overline{632}8\overline{5}\dots$
$5/1\overline{6} = 0.8\overline{5}632\overline{8}5\overline{632}\dots$
$6/1\overline{6} = 1.\overline{8}5\overline{632}8\overline{5}632\dots$
$7/1\overline{6} = 1.\overline{632}8\overline{5}632\overline{8}5\dots$
$8/1\overline{6} = 1.\overline{5}632\overline{8}5\overline{632}8\dots$
$1\overline{8}/1\overline{6} = 1.\overline{32}8\overline{5}632\overline{8}5\overline{6}\dots$
$1\overline{7}/1\overline{6} = 1.\overline{2}8\overline{5}632\overline{8}5\overline{62}\dots$

Absolutely beautiful.

I’m not gonna do twenty-seconds.

Thirteenths

$1/1\overline{4} = 0.154\overline{154}\dots$
$2/1\overline{4} = 0.3\overline{7}8\overline{3}7\overline{8}\dots$
$3/1\overline{4} = 0.4\overline{154}15\dots$
$4/1\overline{4} = 0.54\overline{154}1\dots$
$5/1\overline{4} = 0.7\overline{8}3\overline{7}8\overline{3}\dots$
$6/1\overline{4} = 0.8\overline{3}7\overline{8}3\overline{7}\dots$
$7/1\overline{4} = 1.\overline{8}3\overline{7}8\overline{3}7\dots$
$8/1\overline{4} = 1.\overline{7}8\overline{3}7\overline{8}3\dots$
$1\overline{8}/1\overline{4} = 1.\overline{54}154\overline{1}\dots$
$1\overline{7}/1\overline{4} = 1.\overline{4}154\overline{15}\dots$
$1\overline{6}/1\overline{4} = 1.\overline{3}7\overline{8}3\overline{7}8\dots$
$1\overline{5}/1\overline{4} = 1.\overline{154}154\dots$

Another weaving of patterns like we saw with the fifteenths.

Seventeenths

What absolute nonsense awaits us this time?

$1/10 = 0.1$
$2/10 = 0.2$
$3/10 = 0.3$
$4/10 = 0.4$
$5/10 = 0.5$
$6/10 = 0.6$
$7/10 = 0.7$
$8/10 = 0.8$
$1\overline{8}/10 = 1.\overline{8}$
$1\overline{7}/10 = 1.\overline{7}$
$1\overline{6}/10 = 1.\overline{6}$
$1\overline{5}/10 = 1.\overline{5}$
$1\overline{4}/10 = 1.\overline{4}$
$1\overline{3}/10 = 1.\overline{3}$
$1\overline{2}/10 = 1.\overline{2}$
$1\overline{1}/10 = 1.\overline{1}$

Ah. Right.

Just one more:

Eighteenths

$1/11 = 0.1\overline{1}1\overline{1}\dots$

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!