Binary to Decimal Converter (and Decimal to Binary)

About this tool

This converter turns binary numbers into decimal and decimal numbers into binary, with hexadecimal and octal kept in step. All four fields are editable, the arithmetic uses JavaScript BigInt so a 200-digit binary number converts exactly, and a show-work panel writes out the same steps you would do on paper. It converts numbers. If you want to turn letters into their binary or hex byte codes (the word “Hi” into 01001000 01101001), use the Text to Binary converter instead; it encodes text as UTF-8 bytes, which is a different job.

How to use the binary to decimal converter

1. Type or paste a binary number into the Binary field. Spaces and underscores between digit groups are fine, and so is a 0b prefix.
2. Read the result in the Decimal field. Hex and octal update at the same time.
3. To go from decimal to binary, type in the Decimal field instead. A leading minus sign gives a negative result such as -101010.
4. Pick a width (8, 16, 32 or 64 bits) to see the two’s complement pattern, or to read your bits as a signed number.
5. Open the show-work panel to copy the steps into homework or notes.

Positional notation: why binary works

Every digit in a number is worth its own value times a power of the base. In decimal the places are ones, tens, hundreds and so on (10⁰, 10¹, 10²). Binary uses the same idea with base 2, so the places from the right are worth 1, 2, 4, 8, 16, 32, 64, 128 and onward, doubling each time. Because each binary digit (a bit) is only ever 0 or 1, a place is either counted or skipped. To the right of the binary point the places keep halving: 1/2, 1/4, 1/8, which is why 101.101 is 4 + 1 + 0.5 + 0.125 = 5.625.

Worked example: 11010110 to decimal and back

The tool loads with 11010110. Number the places from the right starting at 0. The 1 bits sit in places 7, 6, 4, 2 and 1, so the value is 2⁷ + 2⁶ + 2⁴ + 2² + 2¹ = 128 + 64 + 16 + 4 + 2 = 214. The same number is D6 in hex and 326 in octal.

To convert 214 from decimal to binary, divide by 2 again and again and write down each remainder:

Reading the remainders from the bottom row up gives 11010110, the number we started with.

Two’s complement: how computers store negative numbers

A minus sign is fine on paper, but a register only holds a fixed number of bits. Most hardware stores signed integers in two’s complement: in an n-bit value the top bit is worth −2ⁿ⁻¹ instead of +2ⁿ⁻¹. To write −42 in 8 bits, start from 42 = 00101010, flip every bit to get 11010101, then add 1: 11010110. Notice that this is the same pattern as the default 11010110. Read as unsigned it means 214; read as signed 8-bit it means -42 (that is, −128 + 64 + 16 + 4 + 2). The bits don’t say which reading is meant; the program does. That is why the tool asks for a width, and why it reports “out of range” when 214 is encoded as signed 8-bit, whose largest value is 127.

Common mistakes

• Reading remainders top to bottom. The first remainder is the rightmost bit. Reverse the list.
• Starting the powers at 1. The rightmost place is 2⁰ = 1, not 2.
• Dropping zeros in the middle. 1001 (9) and 11 (3) are different numbers; every 0 holds a place.
• Assuming a leading 1 means negative. Only in a signed format of a known width. 10000000 is 128 unsigned and −128 as signed 8-bit.
• Expecting decimal fractions to end in binary. 0.1 in decimal is 0.0(0011) in binary, repeating forever, which is why floating-point software prints sums like 0.1 + 0.2 slightly off.
• Losing precision in a spreadsheet or plain JavaScript. Ordinary JavaScript numbers are exact only up to 2⁵³ − 1 = 9,007,199,254,740,991, a run of 53 ones in binary. This page uses BigInt, so longer values stay exact.

Use cases

• Checking computer science or digital electronics homework, step by step
• Decoding register values, bit flags and permission masks from a datasheet or log
• Working out IPv4 subnet masks bit by bit, alongside the Subnet Calculator
• Converting hex dumps to decimal; for longer hex work, the Hex to Decimal converter adds byte order and a 16 × 16 chart

The safe-integer limit comes from the MDN reference for Number.MAX_SAFE_INTEGER, and arbitrary-length arithmetic from BigInt.