Decimal to Binary Converter

The standard method for going from base 10 to base 2 is repeated division by 2. Divide the number by 2 and write down the remainder, which is always 0 or 1. Divide the quotient by 2 again and repeat until the quotient reaches 0. The remainders read from the last one to the first are the binary number. For 156: 156 / 2 = 78 r 0, 78 / 2 = 39 r 0, 39 / 2 = 19 r 1, 19 / 2 = 9 r 1, 9 / 2 = 4 r 1, 4 / 2 = 2 r 0, 2 / 2 = 1 r 0, 1 / 2 = 0 r 1. Reading upward gives 10011100.

It works because each remainder is the lowest bit of the number at that stage, and dividing by 2 shifts the remaining bits one place to the right. The first remainder is the ones bit and the last is the highest bit, which is why you read them in reverse.

The other common method subtracts powers of 2. Find the largest power of 2 that fits, subtract it, and repeat with the rest. 156 - 128 = 28, 28 - 16 = 12, 12 - 8 = 4, 4 - 4 = 0. Put a 1 in the 128, 16, 8 and 4 places and 0 everywhere else, and you get 10011100 again. The number of bits you need is one more than the largest power of 2 used: 1,000 needs 10 bits because 512 is the largest power of 2 not above it.

Programs often want a fixed width, so pad with leading zeros: 5 is 101, or 00000101 as a byte. Negative numbers are usually stored in two's complement: write the positive value at the chosen width, invert every bit and add 1, so -5 in 8 bits is 11111011. This converter handles non-negative whole numbers. Fractions are a different story, since many decimals such as 0.1 have no exact binary form and repeat forever (0.000110011...), which is the source of floating-point rounding errors.