How to Convert Binary to Decimal (and Decimal to Binary and Hex)

To convert binary to decimal, multiply each bit by its place value, a power of 2, and add the results. Reading 101101 from right to left, the place values are 1, 2, 4, 8, 16 and 32, and the bits that are 1 sit under 32, 8, 4 and 1, so 101101₂ = 32 + 8 + 4 + 1 = 45. To go the other way, divide by 2 repeatedly and read the remainders from bottom to top.

Binary place values

Decimal uses powers of 10 (ones, tens, hundreds). Binary uses powers of 2, and each digit, called a bit, is either 0 or 1.

Memorizing the first eight values, 1 to 128, covers a full byte and makes most conversions quick.

Method 1: Add the place values

1. Write the binary number.
2. Under each bit, write its power of 2, starting with 1 on the right.
3. Add the powers of 2 wherever the bit is 1.

Method 2: Double and add (left to right)

This method needs no table. Start at 0, and for each bit from left to right, double the running total and add the bit.

It is fast for long binary strings and is how many computer routines parse numbers.

Decimal to binary: divide by 2

1. Divide the number by 2 and write down the remainder (0 or 1).
2. Divide the quotient by 2 again, recording the remainder.
3. Repeat until the quotient is 0.
4. Read the remainders from last to first.

Reading upward: 156 = 10011100₂. Check: 128 + 16 + 8 + 4 = 156 ✓.

Alternative: subtract powers of 2. Find the largest power of 2 that fits (128), subtract it (156 − 128 = 28), and repeat with the remainder (16, then 8, then 4). Put a 1 in each position you used and 0 elsewhere.

Hexadecimal: binary in groups of four

Hexadecimal (base 16) uses digits 0–9 and A–F. Because 16 = 2⁴, each hex digit corresponds to exactly four bits, which makes conversion a lookup.

Hex to decimal works like binary, with powers of 16: 9C₁₆ = 9 × 16 + 12 = 156.

Web colors are three hex bytes for red, green and blue. #1E90FF is 1E = 30 red, 90 = 144 green and FF = 255 blue. The color converter translates between hex, RGB and other formats.

Octal (base 8) groups bits in threes. It survives mainly in Unix file permissions: 755₈ is 111 101 101, meaning read-write-execute for the owner and read-execute for everyone else.

The number base converter converts between binary, octal, decimal and hex, for any size of number.

Text is stored as numbers too. In ASCII and Unicode, the capital letter A is 65, or 01000001 in binary, and lowercase a is 97, or 01100001; the two differ by a single bit, worth 32. The text to ASCII converter shows the codes for any text, and the bitwise calculator applies AND, OR, XOR and shifts bit by bit.

How many values fit in n bits?

These limits explain many familiar numbers, from the 255 maximum in each color channel to the 2,147,483,647 ceiling of a signed 32-bit integer. The powers of 2 behind them are covered in exponent rules, and how they define kilobytes and gigabytes in data storage units explained.

Negative numbers: two’s complement

Computers store signed integers in two’s complement. The leftmost bit has a negative place value: in 8 bits, it is worth −128 instead of +128.

To negate a number, flip every bit and add 1: 5 = 00000101 → 11111010 → 11111011 = −5. The two’s complement calculator shows each step, and the binary calculator does arithmetic directly in base 2.

Binary fractions

Bits after the binary point are worth ½, ¼, ⅛ and so on. 0.101₂ = ½ + ⅛ = 0.625. Some simple decimals, such as 0.1, never terminate in binary (0.000110011…₂), just as 1/3 never terminates in decimal. That is why many programming languages report 0.1 + 0.2 as 0.30000000000000004 under the IEEE 754 floating-point standard. Financial software avoids the issue by counting in whole cents or using decimal number types.

Common mistakes

• Starting place values from the left. The rightmost bit is always worth 1.
• Reading division remainders top to bottom. Read them from the last remainder to the first.
• Grouping hex digits from the left. Always group four bits starting from the right, padding the left with zeros.
• Forgetting the sign bit when a value is meant to be signed.