Converting binary value...
Enter a binary number to see its exact decimal value and place-value breakdown.
How to Convert Binary to Decimal
A Binary to Decimal Converter translates a base-2 number into the base-10 number system used in everyday arithmetic. Binary has only two digits, 0 and 1. Each position represents a power of two, beginning with 2^0 at the rightmost integer digit and increasing as you move left.
To convert `1010`, multiply each digit by its place value: 1 x 2^3, 0 x 2^2, 1 x 2^1, and 0 x 2^0. The set bits contribute 8 and 2, so `1010` in binary equals `10` in decimal. This calculator performs the same process with exact arbitrary-precision arithmetic and shows each contributing bit.
Binary Place Value Chart
Binary place values double from right to left. Memorizing the first eight powers makes byte-sized conversions much faster, while the larger positions are useful in networking, permissions, low-level programming, and computer architecture.
| Position | Power | Decimal Value | Common Meaning |
|---|---|---|---|
| 0 | 2^0 | 1 | Single-bit flag |
| 1 | 2^1 | 2 | Second bit |
| 2 | 2^2 | 4 | Four-value place |
| 3 | 2^3 | 8 | Half-byte high bit |
| 7 | 2^7 | 128 | Unsigned byte high bit |
| 8 | 2^8 | 256 | One above byte maximum |
| 15 | 2^15 | 32,768 | Signed 16-bit high bit |
How to Convert a Binary Fraction to Decimal
Binary fractions use negative powers of two to the right of the radix point. The first fractional position is 2^-1, or 0.5. The next positions are 2^-2, or 0.25, then 2^-3, or 0.125. Each additional position halves the previous place value.
For example, `101.101` combines the integer value 5 with 0.5 and 0.125, producing the exact decimal value `5.625`. Every finite binary fraction has a denominator that is a power of two, which means its decimal representation terminates. The converter therefore shows an exact result rather than a rounded floating-point estimate.
Two's Complement Binary to Negative Decimal
Most computers represent signed integers with two's complement. The interpretation depends on a fixed width such as 8, 16, 32, or 64 bits. When the highest-width bit is zero, the number is non-negative. When it is one, that bit contributes a negative place value.
An 8-bit pattern of `11111011` is 251 when read as unsigned. In signed two's-complement mode, subtract 2^8, or 256, from that unsigned value. The result is `-5`. Width matters: changing or silently padding the bit pattern can change its meaning, so this tool requires the signed input length to exactly match the selected width.
Common Binary Values
| Binary | Decimal | Octal | Hexadecimal |
|---|---|---|---|
| 0000 | 0 | 0 | 0x0 |
| 0001 | 1 | 1 | 0x1 |
| 1010 | 10 | 12 | 0xA |
| 1111 | 15 | 17 | 0xF |
| 01111111 | 127 | 177 | 0x7F |
| 10000000 | 128 | 200 | 0x80 |
| 11111111 | 255 | 377 | 0xFF |
Binary to Decimal FAQ
How do you convert binary to decimal?
Multiply each binary digit by its power-of-two place value, then add the contributions from every set bit. The rightmost integer position begins at 2^0.
What is 1010 in decimal?
Binary `1010` equals decimal `10`. Its set bits contribute 2^3, which is 8, and 2^1, which is 2.
How do you convert a binary fraction to decimal?
Use negative powers of two after the radix point. The first fractional bit contributes 0.5, followed by 0.25, 0.125, and progressively smaller halves.
How do you convert two's complement to decimal?
Read the fixed-width bits as unsigned. When the leading bit is one, subtract 2 raised to the selected bit width to obtain the signed decimal value.
Why does binary use powers of two?
Binary is base 2, so each position has two possible digits and is worth twice the position to its right. Digital hardware maps naturally to two stable states.
Can a binary number be larger than 64 bits?
Yes. This converter uses arbitrary-precision arithmetic and accepts integer values up to 4,096 bits, avoiding the overflow limits of native 64-bit types.