Binary to Decimal Converter

How binary numbers map to decimal

Binary is a base-2 number system. It has only two digits, 0 and 1, and every position in a binary number is worth a power of two: 1, 2, 4, 8, 16 and so on, doubling with each step to the left. Everyday decimal numbers work the same way, except that each position is worth a power of ten.

To read a binary number as a decimal one, find the positions that hold a 1 and add up their values. Positions that hold a 0 add nothing.

decimal = dₙ × 2ⁿ + … + d₂ × 2² + d₁ × 2¹ + d₀ × 2⁰Each digit d is 0 or 1. Position 0 is the rightmost digit.

How to convert binary to decimal

1. Write the binary number and number its digits from the right, starting at 0.
2. Next to each digit, write 2 raised to that digit's position number.
3. Keep only the terms whose digit is 1.
4. Add the kept powers of two.

Example: convert 101101. Counting from the right, the 1s sit at positions 5, 3, 2 and 0, so the value is 32 + 8 + 4 + 1 = 45.

A second method needs no powers. Read the digits from left to right, doubling your running total and adding the next digit. For 101101 that runs 1, then 1 × 2 + 0 = 2, then 2 × 2 + 1 = 5, then 5 × 2 + 1 = 11, then 11 × 2 + 0 = 22, then 22 × 2 + 1 = 45. It lands on the same answer and is easy to do in your head for short numbers.

The converter above follows the same rules. Leading zeros change nothing, so 00101 and 101 both give 5.

What the converter accepts

• Only 0 and 1. An empty box, a digit from 2 to 9, a decimal point, a minus sign or an exponent such as 1e3 makes the answer box show NaN, short for "not a number".
• Whole numbers. Binary fractions such as 101.01 are not read.
• Unsigned values. The byte 11111111 gives 255. The tool does not apply two's complement, so it never reads that byte as −1.
• Up to 53 digits exactly. After any leading zeros, a binary number of up to 53 digits converts exactly (53 ones is 9007199254740991). Longer inputs are rounded to the nearest value a double-precision number can store, so the last digits of the result may be off.

Binary to decimal conversion table

Related conversions

To go the other way, use the decimal to binary converter. Programmers often write long binary values in base 16, which the binary to hex converter does in one step. To add, subtract, multiply or divide binary numbers directly, try the binary calculator.