Base Conversion: Binary, Octal and Hexadecimal

Converts a decimal integer to the given base (2 to 36). Digits above 9 are written A, B, C … . The binary, octal and hexadecimal forms are also shown.

Explanation

This converts a decimal integer into any base from 2 to 36.

In base $n$ every digit carries a weight that is a power of $n$ , just as the decimal system we use every day weights its digits by powers of ten. In binary those weights are powers of two.

255 = 1 \times 2^7 + 1 \times 2^6 + \cdots + 1 \times 2^0

The method is mechanical: divide by the base, note the remainder, divide the quotient again. Keep going until the quotient reaches 0, then read the remainders from the bottom up.

Decimal — the integer to convert, 0 or more
Base — an integer from 2 to 36
Digits above 9 are written $A$ , $B$ , $C$ and so on; in hexadecimal $F$ stands for 15

Example

Convert decimal 255 to binary. Dividing by 2 leaves a remainder of 1 every single time, and after eight steps the quotient hits 0. Reading the remainders from the bottom up gives

255_{(10)} = 11111111_{(2)}

Since $255 = 256 - 1 = 2^8 - 1$ , the answer is exactly eight 1s. The same number is 377 in octal and FF in hexadecimal.

Notes

The base must be an integer from 2 to 36. Ten digits plus twenty-six letters give only 36 distinct symbols, which is where the upper limit comes from.
The input must be a whole number of 0 or more. Negatives and decimals are not handled.
One hexadecimal digit maps to exactly four binary digits. Split FF into 1111 1111 and the correspondence is easy to see, which is why hexadecimal is such a convenient shorthand for binary.
More digits does not mean a bigger number. The smaller the base, the more digits the same value needs.