Number base converter (binary, decimal, hex)

How do you convert a number from one base to another?

Each digit is worth its value times a power of the base, counted from 0 at the right. In decimal, 255 is 2 × 100 + 5 × 10 + 5 × 1; in hexadecimal, FF is 15 × 16 + 15 × 1, which is also 255; in binary, 11111111 is 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1. Bases above 10 borrow letters for the extra digits, so A is 10, F is 15 and Z is 35, which is why base 36 is the largest that fits 0–9 and A–Z.

To write a number in another base, divide it by that base again and again and read the remainders from last to first. 255 ÷ 16 is 15 remainder 15, and 15 ÷ 16 is 0 remainder 15, giving F and F: FF. Binary, octal and hex also map straight onto each other, because one hex digit is exactly four bits and one octal digit exactly three.

JavaScript's regular numbers can only hold whole numbers exactly up to 9,007,199,254,740,991 (2^53 − 1); above that, neighbouring integers round to the same value. This converter uses BigInt instead, so every digit of a large number is exact. Negative numbers keep their minus sign in every base, as JavaScript's toString does, and the result also shows the two's complement bit pattern that a fixed-width signed integer stores.

value = dₙ × baseⁿ + … + d₂ × base² + d₁ × base + d₀
FF₁₆ = 15 × 16 + 15 = 255₁₀
255 ÷ 16 = 15 r 15,  15 ÷ 16 = 0 r 15  →  FF