An irrational octal base
01/11/2025

Computers rely on binary digits i.e., 0 or 1. As numbers expressed as 0 and 1 are a pain to write or read and take too many characters anyway, hexadecimal is a preferred base. Hence 22_d is rather expressed as 16_h than 10110_b.

All bases are mathematically defined by:

${\sum }_{i=0}^n{a}_i{b}^i$

Where b is the base and a_i the digits that must respect a_i < b.

b is usually an integer but there is at least one irrational that allows the expression of any natural number as a finite and exact representation!

That magical number is noted φ and famously known as the golden number: $\phi =\frac{1+\sqrt{5}}{2}$

Thus 22_d is expressed as 1000101.010001_φ which is actually: ${\phi }^6+{\phi }^2+{\phi }^0+{\phi }^{-2}+{\phi }^{-6}$

Again, numbers expressed in φ base are a pain to write or read and take too many characters anyway. Hence the idea to gather its digits 4 by 4 and express 22_d as 45.44_hφ.

But behold, in φ base, 11_φ = 100_φ and thus B-F hexadecimal digits are never needed, neither 3, 6 or 7!

That allows to represent numbers in octaphi base, using only eight digits: 0, 1, 2, 4, 5, 8, 9 and A.

For example, numerunique exists for more than 2A.04_oφ years. Not bad isn't it?

numerunique of course provides a decimal to octaphi converter (click on the image below to access it):

This proposition is based on Burnol's algorithm presented in "The zeckendorf package" by Jean-François Burnol, 2025 https://ctan.org/pkg/zeckendorf. The octal suggestion was also coined by Jean-François Burnol.

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