Weird and Wonderful Number Bases

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Number systems are weird

When we think of number systems, we usually imagine the familiar base-10 (decimal) or base-2 (binary). But the idea of a “base” is much more flexible than most people realize. We aren’t restricted to integers ≥ 2. In fact, we can use fractional bases, irrational bases, and even negative or complex ones.

Now, just because we could, does that mean we should?

Short answer: No.
Long answer: Also no.

What is a base, really?

A base $b$ (also called a radix) is just a multiplier for positional notation. A digit string like

$$d_n{d}_{n-1}\dots d_1{d}_0.d_{-1}{d}_{-2}\dots$$

means

$$d_n{b}^n+d_{n-1}{b}^{n-1}+\cdots +d_{1}b+d_0+d_{-1}{b}^{-1}+d_{-2}{b}^{-2}+\dots$$

As long as we agree on a digit set (often $\{0,1,\dots ,\lfloor b\rfloor \}$), the system works, whether $b$ is an integer, a fraction, or even irrational.

A general rule thus emerges: In any base-$b$, the number $b$ is always represented as 10, and $b^n$ is 1 followed by n zeros.

Fractional bases

Take base 1.5. The digits are typically $\{0,1\}$.

• 1000 in base-1.5 means

  $$1\times ({1.5}^3)=3.375$$

• 101 in base-1.5 means

  $$1\times ({1.5}^2)+0\times ({1.5}^1)+1\times ({1.5}^0)=2.25+1=3.25$$

Numbers don’t line up as neatly as in base-10, but the system is perfectly valid.

Irrational bases: base-π

Nothing stops us from picking an irrational base, like $\pi$. Choosing a non-integer base doesn’t stop the positional system from working. It just makes the expansions of other numbers weird.

Now, if you remember from earlier: Any number $b$ in base-$b$ is represented as 10.

• So in base-π, the string 10 represents exactly:

  $$1\times {\pi }^1+0\times {\pi }^0=\pi$$

So if you ever felt the need to express $\pi$ as a rational integer, here you go. It’s 10. _{(in base-π).}

Of course, other numbers get strange expansions. For example, the number 2 in base-π would look like a fraction of 10.

The Golden Ratio base (phinary)

One of the most beautiful non-integer bases is the golden ratio base, also called phinary or the Fibonacci coding system.

The golden ratio is

$$\phi =\frac{1+\sqrt{5}}{2}\approx 1.618$$

In base-$\phi$, we can represent integers using only the digits $\{0,1\}$, with the rule that no two 1s are allowed in a row.

Examples:

• 1φ0 = ${\phi }^2=\phi +1=2.618...$
• 100φ = ${\phi }^3\approx 4.236...$
• Every positive integer has a finite, unique representation in base-$\phi$.

This system is tightly connected to the Fibonacci sequence, since powers of $\phi$ obey Fibonacci-like identities.

Other exotic bases

• Negabinary (base -2): Every integer has a unique representation using only 0 and 1, without needing a minus sign.
• Complex base $i$: The “quater-imaginary” system (base $2i$) can represent every complex integer with digits 0–3.

Why does this matter?

I have no idea. I haven't slep today and just thought it's cool.