Binary, Decimal, Hex & Octal Converter

How to use this number base converter

1. Type or paste a number into any box: decimal, hex, binary, octal, or the last box with any base from 2 to 36. All other boxes update as you type. Spaces, underscores and the prefixes 0x, 0b and 0o are allowed, and #FF8800 works in the hex box.
2. Negative numbers: type a minus sign, such as -42. To see how a computer stores it, choose a Bit width (8, 16, 32, 64 or 128): binary, octal and hex then show the two's complement bits. With Signed ticked, a hex or binary value with the top bit set (for example FF in 8 bits) is read as a negative number; untick it to read the bits as unsigned.
3. Fractions: numbers such as 0.625 or 1A.8 work too. When the digits never end, the box shows the first 32 and the message below the boxes shows the repeating part in brackets.
4. Bits: click any bit to flip it. Below the bits you see how many bits the number needs, its bytes in both byte orders, and the integer types it fits in.
5. Show the working explains the conversion step by step, the same way you would do it on paper. It picks the method that fits the box you typed in; choose another one from the list.

There is no size limit for whole numbers: a 300-digit number converts exactly. Nothing you type leaves your browser.

How number bases work

A number base says how many different digits there are and how much each place is worth. In decimal (base 10) the places are worth 1, 10, 100, 1000. In binary (base 2) there are only the digits 0 and 1, and the places are worth 1, 2, 4, 8, 16. In hexadecimal (base 16) the digits are 0–9 and A–F (A = 10 … F = 15), and the places are worth 1, 16, 256, 4096. Octal (base 8) uses 0–7.

Computers work in binary. Programmers write the same bits in hex because one hex digit is exactly 4 bits, so a byte is always two hex digits: 1111 1111 = FF. Octal is used mainly for Unix file permissions such as chmod 755.

How to convert between binary, decimal and hex by hand

Decimal to binary: divide by 2

Divide the number by 2 and write down the remainder. Divide the quotient by 2 again, and repeat until the quotient is 0. The remainders, read from the last to the first, are the binary number. For 156 the remainders are 0, 0, 1, 1, 1, 0, 0, 1, so 156 = 1001 1100. Decimal to hex works the same way with 16: 156 ÷ 16 = 9 remainder 12 (C), so 156 = 9C.

Binary to decimal: add the place values

Write the place values 1, 2, 4, 8, 16 … under the digits from the right, and add the places that have a 1. 1011 0110 = 128 + 32 + 16 + 4 + 2 = 182. For hex, multiply each digit by 1, 16, 256, 4096 …: 2F = 2 × 16 + 15 = 47.

Binary to hex and octal: group the bits

Because 16 = 2⁴, every group of 4 bits (from the right) is one hex digit: 1001 1100 → 9 C. Because 8 = 2³, every group of 3 bits is one octal digit: 10 011 100 → 2 3 4, so 156 = 234 in octal. Going back, replace every hex digit with its 4 bits.

Fractions

For the part after the point, multiply by the new base instead of dividing: 0.625 × 2 = 1.25, 0.25 × 2 = 0.5, 0.5 × 2 = 1.0, so 0.625 = 0.101 in binary. The whole-number parts of the results are the digits, read from the first to the last. Some fractions never end: 0.1 in binary is 0.0(0011), where the digits in brackets repeat forever. That is why 0.1 + 0.2 is 0.30000000000000004 in most programming languages.

Negative numbers and two's complement

Computers store a negative whole number in two's complement: to write -x in N bits, write x in binary, flip every bit and add 1. The top bit then works as the sign (1 = negative). The same bits can be read in two ways, which is why FF is 255 as an unsigned byte and -1 as a signed byte.

Base conversion in code

Java toHexString and toBinaryString print negative numbers as their 32-bit two's complement, while Python bin(-5) prints -0b101. Mask with & 0xFF (or the width you need) to get the bits.

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