An exploration of unconventional number bases

Published: 27/10/2025

We are all familiar with the base-10 number system, where the value of the digits is determined by their positions. For example, the number 1234 will have the value:

$1234_{10} = 1 * 10^3+2*10^2+3*10^1+4*10^0$

Pretty standard, right? Time to change some parts of the system

What's the general form?

It's quite trivial, actually. This system is generally called positional numeral system. Number with digits $x_0 x_1...x_n$ in base $b$ has the value

${x_0x_1...x_n}_b = x_0 * b^{n} + x_1 * b^{n-1} + ... + x_n*b^{0}$

where b is a positive integer, and $x_0,...x_n$ are numbers from 0 to n-1

Negabinary

Let's try to plug $b = -2$ to the equation, and see what happens. As base 2 system uses 0 and 1 as its digits, let's also use them for our hypothetical experiment. A number 1101 in base -2 will have the value

$1101_{-2} = 1*(-2)^3+1*(-2)^2+0*(-2)^1+1*(-2)^0 =-8+4+1=-3$

Unlike traditional positive bases, the multiplier value of negative bases alternates between positive and negative. This makes it somewhat useful for representing negative numbers without having to explicitly represent the negative sign (like in the one's complement). However, it still uses one extra bit anyway, and is much more convoluted to mentally convert, compared to two's complement. This number system is also called negabinary (base -2) or negadecimal (base -10)

Complex base

How about we plug $b = i$ into the equation? The multipliers themselves now have a really interesting property: they loop every 4 digit. The value from right to left goes from i, to -1, to -i, to 1, then back to i. With this, we can create a very naive inefficient number system: we can set every 4th digit (counting from the right) one by one to create a positive number, and every 2nd digit to create any negative number.

$40009000_{i} = 4 + 9 = 13$

$400090_i = -4 -9 = -13$

In this number system, we can have multiple different number representations for the same number. For example, both $40009000_i$ and $50008000_i$ denotes the number 13. Donald Knuth proposed a 2i-based number system called Quater-imaginary base

The Golden Ratio - Phinary

This system is also called base-$\phi$, golden ratio base, or way cooler, "phinary".

This number system stems from the fact that any exponent of $\phi$ can be rewritten as $a+b\phi$. By definition, $\phi^2 = \phi + 1$, so any higher exponent, for example, $\phi^3$, can be rewritten as

$\phi^3 = \phi * \phi^2 = \phi (\phi + 1 ) = \phi^2 + \phi = (\phi+1) + \phi = 1+2\phi$

The same is also true with negative exponents: $\phi^{-2}=1 - \phi$

With this, the number 2 in decimal can be represented as

$10.01_\phi = \phi^1 + \phi^{-2} = (1 + \phi) + (1 - \phi) = 2$

The base doesn't have to stay the same - Factoradic

Another interesting base is called "factoradic" or "factorial base". In this system, the place value from right to left is 0!, 1!, 2!, 3!, and so on. The highest possible digit in each place is also correspondingly determined - the digit of 5! can only be maximal 5, for example. So 341010 in factoradic will be $341010_! = 3*5! + 4*4! + 1*3! + 0*2! + 1*1! + 0*0! = 463_{10}$ Another similar number system use combinatorics instead of factorial for the place value.