1. Polynumbers

Box arithmetic begins with the natural numbers — zero and the counting numbers. No negatives, no fractions, no irrationals. Everything built from here stays in that world.

The first extension above the naturals is the polynumber — Wildberger's name for what you already know as a polynomial, reinterpreted as a finite box of natural numbers.

Polynomials as boxes

Start with a polynomial you recognise:

$$p(\alpha) = 2 + 3\alpha + \alpha^2$$

In standard notation this bundles three things together: a constant term, a linear term, and a quadratic term. Strip away the variable and what remains are the coefficients $\{2, 3, 1\}$ together with their degree positions. A polynumber is just that box:

$$p = \langle 2,\; 3,\; 1 \rangle$$

The "variable" $\alpha$ is itself the polynumber $\langle 1 \rangle$ — a box holding the natural number 1. Powers of $\alpha$ are:

$$\alpha^0 = \langle\rangle = 1 \qquad \alpha^1 = \langle 1 \rangle \qquad \alpha^2 = \langle 2 \rangle \qquad \alpha^n = \langle n \rangle$$

So $p = 2\cdot\alpha^0 + 3\cdot\alpha^1 + 1\cdot\alpha^2$ — the box is the coefficient array, nothing more.

Evaluation substitutes a natural number for $\alpha$:

$$p(5) = 2 + 3 \cdot 5 + 5^2 = 42$$

Integers in, integer out. No fractions appear at any step.

The Polynumber type

In boxmath, a single term is a Polynumber: a coefficient paired with a sparse exponent vector that encodes which variables appear and at what power.

Polynumber(coefficient: bigint, exponents: number[])

Field	Meaning
coefficient	Natural number scalar weight
exponents[i]	Power of variable $x_i$

import { Polynumber, Multinumber } from 'boxmath';

new Polynumber(3n, [2]);       // 3x²  — coefficient 3, x²
new Polynumber(1n, [1, 1]);    // xy   — coefficient 1, x¹y¹
new Polynumber(5n, []);        // 5    — constant (no variables)

A Multinumber is the box that holds multiple Polynumber terms together — the full polynomial object:

// p(x) = 2 + 3x + x²
const p = new Multinumber([
  new Polynumber(2n, []),   // 2  (degree 0)
  new Polynumber(3n, [1]),  // 3x (degree 1)
  new Polynumber(1n, [2]),  // x² (degree 2)
]);

p.evaluate([5n]);  // 42n

evaluate hands back a natural number — the box is evaluated term by term and the results summed. This is the whole primitive: no division, no floating point, no scaling anywhere in the chain.