Decimal to Binary Converter

How decimals become binary

Decimal is base 10 and binary is base 2. Converting between them means rewriting the same quantity using only the digits 0 and 1, where each position is worth a power of two (1, 2, 4, 8, 16 and so on) instead of a power of ten.

Every whole number has exactly one binary form. For example, 13 is 8 + 4 + 1, so it is written 1101: a 1 in the 8s, 4s and 1s positions and a 0 in the 2s position.

How to convert decimal to binary

The standard method for whole numbers is repeated division by 2. Divide the number by 2, note the remainder (always 0 or 1), then divide the quotient by 2 again. Stop when the quotient reaches 0. The remainders, read from last to first, are the binary digits.

Example: convert 45. 45 ÷ 2 = 22 remainder 1; 22 ÷ 2 = 11 remainder 0; 11 ÷ 2 = 5 remainder 1; 5 ÷ 2 = 2 remainder 1; 2 ÷ 2 = 1 remainder 0; 1 ÷ 2 = 0 remainder 1. Reading the remainders from the bottom up gives 101101.

Another example: 156 gives remainders 0, 0, 1, 1, 1, 0, 0, 1 from first to last, so reading backwards the result is 10011100.

For the fractional part of a number, multiply by 2 instead and collect the whole-number parts. Take 0.625: 0.625 × 2 = 1.25 gives a 1; 0.25 × 2 = 0.5 gives a 0; 0.5 × 2 = 1.0 gives a 1 and leaves nothing. So 0.625 is 0.101 in binary.

What the converter shows

• Negative numbers get a minus sign in front of the binary form of their magnitude, so −5 becomes −101. This is not two's complement.
• Fractions are expanded digit by digit. 0.5 is 0.1 and 3.75 is 11.11, because those values are sums of powers of two.
• Most other decimals never end in binary. 0.1 repeats forever in base 2, so the converter prints the 55-digit binary value of the double-precision number closest to 0.1 rather than 0.1 exactly.
• Large numbers show the bits of the nearest double-precision value. Integers above 9007199254740991 (2^53 − 1) can lose their lowest bits, which then appear as trailing zeros.
• An empty box counts as 0.

Decimal to binary conversion table

Related conversions

Check your work in the opposite direction with the binary to decimal converter. For a more compact notation, convert to base 16 with the decimal to hex converter, or do arithmetic on binary numbers with the binary calculator.