Base Converter

Converting 0xFF to binary or 1011₂ to decimal is quick once you see the method. This base converter handles whole numbers of any length (using exact big-integer arithmetic), fractions, and negative numbers in two’s complement, and shows the repeated-division and place-value steps so you can check homework or understand what your code is doing.

How number bases work

In base b, each position is worth b times the position to its right. In decimal (base 10), 347 means 3 × 100 + 4 × 10 + 7 × 1. In binary (base 2), 1011 means 1 × 8 + 0 × 4 + 1 × 2 + 1 × 1 = 11. Bases above 10 need extra digit symbols, so hexadecimal uses A–F for 10–15, and bases up to 36 use the whole alphabet.

Converting to decimal: place value

Converting from decimal: repeated division

1. Divide the number by the target base.
2. Write down the remainder — it is the next digit, from right to left.
3. Repeat with the quotient until it reaches zero.
4. Read the remainders from last to first.

For fractions, multiply the fractional part by the base instead: the whole-number part of each result is the next digit after the point. The tool shows both procedures.

Binary, octal and hex shortcuts

Because 16 = 2⁴ and 8 = 2³, each hex digit maps to exactly four bits and each octal digit to three. To convert binary to hex, group bits in fours from the right: 1111 0101 = F5. This is why programmers use hex as a compact way to write binary.

Negative numbers: two’s complement

Computers usually store signed integers in two’s complement. To write −x in n bits, subtract x from 2ⁿ — or invert all the bits of x and add 1. In 8 bits, −1 is 1111 1111 (0xFF) and −128 is 1000 0000 (0x80). An n-bit signed value ranges from −2ⁿ⁻¹ to 2ⁿ⁻¹ − 1, so 8 bits hold −128 to 127 and 32 bits hold −2,147,483,648 to 2,147,483,647. Choose the width in the converter to see the pattern your language or processor would store.

Worked example

Convert 255 to binary: 255 ÷ 2 = 127 r1, 127 ÷ 2 = 63 r1, and so on — eight remainders of 1, so 255 = 1111 1111₂, which groups neatly into FF in hex and 377 in octal. Now −42 in 16-bit two’s complement: 42 is 0000 0000 0010 1010; inverting gives 1111 1111 1101 0101 and adding 1 gives 1111 1111 1101 0110, or 0xFFD6. The converter shows the same.

Where you’ll meet different bases

• Binary: bits, flags, network masks and low-level hardware.
• Hexadecimal: colour codes (#FF8800), memory addresses, byte values, hashes and UUIDs.
• Octal: Unix file permissions (chmod 755).
• Base 36: compact IDs and short URLs.
• Base 60 (not in this tool): time and angles — minutes and seconds.

Common values worth knowing

Notation in programming languages

The converter accepts these prefixes when they match the selected base, and ignores spaces, commas and underscores used as digit separators.

Checking your answers

• Convert back the other way — you should get the original number.
• For binary, the last digit shows whether the number is odd (1) or even (0).
• A number with n binary digits lies between 2ⁿ⁻¹ and 2ⁿ − 1.
• Each hex digit is exactly four bits, so an 8-digit hex number is 32 bits.

Precision

Whole numbers are converted exactly with arbitrary-precision integers, so 200-digit numbers are fine. Fractional parts use standard floating-point arithmetic and are rounded to the number of digits you choose; many fractions repeat forever in other bases.

Privacy

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