Decimal to Binary Converter

To convert a decimal number to binary, divide by 2 again and again and read the remainders from bottom to top. This converter does it for any size of number and shows every step, handles fractions and negative numbers, and converts back from binary — perfect for homework, computer science classes and programming.

The division method, step by step

Reading the remainders from the bottom up gives 10011100, so 156 in decimal is 10011100 in binary. Check: 128 + 16 + 8 + 4 = 156.

Binary place values

Another way to convert is subtraction: take the largest power of two that fits (128), subtract it (28 left), then the next that fits (16, leaving 12), then 8 (leaving 4), then 4 — and put a 1 in each of those positions.

Fractions

For the part after the decimal point, multiply by 2 repeatedly: the whole-number part of each result (0 or 1) is the next binary digit, read from the top down. 0.625 × 2 = 1.25 (1), 0.25 × 2 = 0.5 (0), 0.5 × 2 = 1.0 (1), so 0.625 = 0.101 in binary. Many decimal fractions, like 0.1, never end in binary — which is why computers store them approximately and 0.1 + 0.2 doesn’t give exactly 0.3 in floating-point arithmetic.

Negative numbers: two’s complement

Computers usually store signed integers in two’s complement: to negate a number, invert every bit and add 1. With 8 bits the range is −128 to 127; with 16 bits, −32,768 to 32,767. The leading bit is 1 for negative numbers.

Worked example

A student converts 42 to binary: 42 ÷ 2 = 21 r0, 21 ÷ 2 = 10 r1, 10 ÷ 2 = 5 r0, 5 ÷ 2 = 2 r1, 2 ÷ 2 = 1 r0, 1 ÷ 2 = 0 r1. Reading up: 101010. For −42 in 8 bits, pad to 00101010, invert to 11010101 and add 1: 11010110. In the binary-to-decimal box with “two’s complement” ticked, 11010110 reads back as −42.

Powers of two to remember

Decimal, binary, octal and hex side by side

Each hex digit stands for exactly four binary digits and each octal digit for three, which is why programmers use hex as a compact way to write binary: 1111 1111 is FF.

Why computers use binary

Digital circuits are most reliable when they only need to tell two states apart — high or low voltage, on or off, magnetised one way or the other. Binary maps perfectly onto those two states, so every number, letter, image and sound in a computer is ultimately stored as a pattern of 0s and 1s. A group of 8 bits is a byte, which can hold 256 different values — enough for one character in basic text encodings or one colour channel in an image.

Common mistakes

• Reading the remainders from top to bottom instead of bottom to top.
• Forgetting the final division that leaves a quotient of 0 and a remainder of 1.
• Dropping leading zeros when a fixed width, such as 8 bits, is required.
• Applying two’s complement to a positive number — it’s only for negatives.
• Treating binary fractions as if every decimal fraction ends: 0.1 repeats forever in binary.

What’s in the template pack

• Decimal-to-binary worksheets with step-by-step answer keys
• Binary-to-decimal worksheets
• Printable binary place value chart
• Powers of two reference table (2⁰ to 2⁶⁴)
• Binary flash cards (0–31)

For computer science teachers and students; the converter itself is free.

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