Convert numbers between binary, octal, decimal, hexadecimal and any base up to 62 – with exact BigInt arithmetic, visible calculation steps and fraction support. Everything runs locally in your browser.
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Start from what you are trying to do
- Convert a number between bases and check your workA converted value, and a round-trip back to the original that proves it is right.2 steps, starting with Decimal to Binary Converter
- Represent a negative number in binaryA negative value in two's complement at the bit width you actually need.3 steps, starting with Decimal to Signed Binary Converter
- See how a float is storedThe sign, exponent and mantissa of a floating-point value, and its bytes in memory order.3 steps, starting with IEEE 754 Floating Point Converter
Base Converters
- Number System ConverterConvert any number into binary, octal, decimal, hexadecimal and a custom base of your choice at the same time – with exact BigInt arithmetic, so even very large values never lose precision.
- Any Base ConverterConvert a number between any two bases from 2 to 36 – binary, ternary, octal, decimal, hex, base 32, base 36 and everything in between – exactly, with no size limit.
- Base 2 to 62 ConverterConvert numbers in any base up to 62, where digits are case-sensitive: 0-9, then A-Z for 10-35, then a-z for 36-61.
- Fractional Base ConverterConvert numbers WITH a fractional part between any bases 2-36, using exact fraction arithmetic that detects repeating digits instead of silently rounding them.
- Binary to Decimal ConverterConvert a binary number to decimal by summing the place values – each 1 bit contributes its power of two.
- Decimal to Binary ConverterConvert a decimal number to binary using repeated division by two – the steps panel lists every division and remainder for your exact input, however large.
- Binary Fraction to Decimal ConverterConvert a binary number with a fractional part (such as 101.101) to decimal exactly – digits after the point use negative powers of two: 1/2, 1/4, 1/8 and so on.
- Decimal Fraction to Binary ConverterConvert a decimal fraction to binary by repeated multiplication by two.
- Binary to Hex ConverterConvert binary to hexadecimal by splitting the bits into 4-bit groups (nibbles) – each group maps to exactly one hex digit.
- Hex to Binary ConverterConvert hexadecimal to binary by expanding every hex digit into its 4-bit pattern.
- Binary to Octal ConverterConvert binary to octal by splitting the bits into 3-bit groups – each group is one octal digit 0-7.
- Octal to Binary ConverterConvert octal to binary by expanding every octal digit into its 3-bit pattern.
- Decimal to Hex ConverterConvert a decimal number to hexadecimal by repeated division by 16 – remainders above 9 become the letters A-F.
- Hex to Decimal ConverterConvert hexadecimal to decimal by summing place values – each digit is multiplied by a power of 16, with A-F standing for 10-15.
- Hex Fraction to Decimal ConverterConvert a hexadecimal number with a fractional part (such as A.4) to decimal exactly – digits after the point use negative powers of 16: 1/16, 1/256 and so on.
- Decimal Fraction to Hex ConverterConvert a decimal fraction to hexadecimal by repeated multiplication by 16, with exact detection of repeating hex digits – the float notation used in some language literals.
- Decimal to Octal ConverterConvert a decimal number to octal by repeated division by 8, with every division and remainder shown for your input.
- Octal to Decimal ConverterConvert octal to decimal by summing place values – each digit is multiplied by a power of 8.
- Octal Fraction to Decimal ConverterConvert an octal number with a fractional part (such as 7.4) to decimal exactly – digits after the point use negative powers of 8.
- Decimal Fraction to Octal ConverterConvert a decimal fraction to octal by repeated multiplication by 8, with exact repeating-digit detection instead of silent rounding.
- Hex to Octal ConverterConvert hexadecimal to octal via binary: each hex digit becomes 4 bits, and the bits regroup into 3-bit octal digits.
- Octal to Hex ConverterConvert octal to hexadecimal via binary: 3-bit octal groups merge into 4-bit hex nibbles.
Signed Numbers and Codes
- Signed Binary to Decimal ConverterDecode a binary bit pattern under all four signed conventions at once – unsigned, sign-magnitude, one’s complement and two’s complement – because the SAME bits mean different numbers under each.
- Decimal to Signed Binary ConverterEncode a signed decimal number – positive or negative – into two’s complement, one’s complement or sign-magnitude at 4 to 64 bits, with the encoding steps shown.
- Two's Complement CalculatorCompute the two’s complement of a value or pattern: invert every bit, then add one.
- One's Complement CalculatorCompute the one’s complement – simply invert every bit.
- Sign-Magnitude ConverterConvert between decimal and sign-magnitude form, where the leftmost bit is purely a sign flag and the rest is the ordinary magnitude – intuitive, but with two zeros.
- Binary Bit Width ConverterResize a binary value between 4, 8, 16, 32 and 64 bits.
- Fixed-Point Binary ConverterConvert between decimal values and fixed-point Qm.n binary, where n fraction bits give a resolution of 1/2ⁿ – the format of DSPs and FPGAs that have no floating-point unit.
- IEEE 754 Floating Point ConverterSee exactly how a decimal number is stored as an IEEE 754 float or double – sign, exponent and mantissa bits – and why 0.1 + 0.2 does not equal 0.3.
- Endianness ConverterSwap a value between big-endian and little-endian byte order.
- Binary to Gray Code ConverterConvert weighted binary to reflected Gray code with g = b XOR (b >> 1) – in a Gray sequence, consecutive values differ in exactly one bit, which kills glitches in rotary encoders.
- Gray Code to Binary ConverterConvert reflected Gray code back to weighted binary by cascading XOR from the top bit down – each binary bit is the XOR of the previous binary bit with the current Gray bit.
- Decimal to Gray Code ConverterConvert a decimal number straight to Gray code – first to binary, then one XOR pass.
- Gray Code Table GeneratorGenerate the full decimal-binary-Gray table for 2 to 6 bits.
- BCD to Decimal ConverterDecode binary-coded decimal: each 4-bit group is one decimal digit 0-9.
- Decimal to BCD ConverterEncode a decimal number as BCD – each digit becomes its own 4-bit group, so 429 becomes 0100 0010 1001.
- Excess-3 Code ConverterConvert between decimal and Excess-3 code, where each digit is stored as digit + 3 in four bits – a self-complementing code from the era of decimal arithmetic hardware.
Number Calculators
- Binary CalculatorAdd, subtract, multiply or divide two binary numbers with exact BigInt arithmetic – the addition mode traces every column and carry for your actual operands.
- Binary Addition CalculatorAdd two binary numbers with a full column-by-column carry trace – see exactly where 1+1 writes 0 and carries 1, for your actual inputs.
- Binary Subtraction CalculatorSubtract one binary number from another exactly, including negative results – and see why hardware actually subtracts by adding the two’s complement.
- Binary Multiplication CalculatorMultiply two binary numbers exactly, at any length.
- Binary Division CalculatorDivide two binary numbers exactly, returning the integer quotient and the remainder separately – the form long division and hardware dividers actually produce.
- Hex CalculatorAdd, subtract, multiply or divide hexadecimal numbers exactly – with a decimal cross-check in the result so you can verify at a glance.
- Hex Addition CalculatorAdd two hexadecimal numbers exactly – columns carry at 16, so 8 + 9 writes 1 and carries 1, and the decimal cross-check confirms the result.
- Hex Subtraction CalculatorSubtract hexadecimal numbers exactly, including negative results – with borrows worth 16 rather than 10, which is where hand calculations usually slip.
- Hex Multiplication CalculatorMultiply hexadecimal numbers of any size exactly.
- Hex Division CalculatorDivide hexadecimal numbers exactly, returning quotient and remainder in hex with a decimal cross-check – useful for address arithmetic and alignment calculations.
- Octal CalculatorAdd, subtract, multiply or divide octal numbers exactly – carries and borrows happen at 8, and the decimal equivalents are shown so every step can be checked.
- Base-N CalculatorDo exact arithmetic in ANY base from 2 to 36 – pick the base, pick the operation, and get the result in that base with a decimal cross-check.
- Binary Bit Shift CalculatorShift a value left or right and see the bits move – logical shifts insert zeros, arithmetic right shift copies the sign bit, and every shift is a multiply or divide by two.
- Binary Bitmask CalculatorApply AND, OR, XOR and AND-NOT between a value and a mask, bit by bit – the four operations behind testing, setting, toggling and clearing flags.
Frequently asked questions
Which bases can be converted?
Any base from 2 to 36, using digits 0-9 then A-Z, with dedicated tools for binary, octal, decimal and hexadecimal. Those four have their own pages because bit width, grouping and signedness matter for them in ways they do not for arbitrary bases.
Are conversions exact for very large numbers?
Yes. Arbitrary-precision integer arithmetic is used throughout, so a 300-digit number converts exactly. Converters built on floating point start returning wrong digits somewhere past 2^53 without any warning that they have.
Why is hexadecimal used so widely in computing?
Because one hex digit is exactly four bits, so a byte is always two hex digits and the mapping to binary is direct and readable. Decimal has no such alignment, which is why memory addresses and colour values are written in hex.
How do I convert a fraction?
Use the fraction converters. Fractions are converted digit by digit after the point and can repeat forever in the target base — a tenth is exact in decimal and recurring in binary, which is the root of most floating-point surprises.
What is the difference between a converter and a calculator?
A converter rewrites one number in another base. A calculator does arithmetic while staying in that base, giving a hexadecimal result for hexadecimal inputs with a decimal cross-check beside it.
Why does my binary number not match what I expected?
Usually signedness or width. The same bit pattern means different values as unsigned, two's complement or sign-magnitude, and truncating to 8 bits changes the answer again. The signed-number tools show all three side by side.
Do I need to enter a prefix like 0x?
No. Prefixes are accepted and ignored, since people paste values straight out of code, but the base is taken from the tool or the selector rather than guessed from the text.
Number systems are lenses on the same integers: binary shows the bits hardware stores, hex compresses them for humans, octal survives in permissions, and decimal is where arithmetic intuition lives. The tools below convert between all of them — exactly, with the working shown.
Every converter runs BigInt arithmetic (no 2⁵³ cliff), generates its calculation steps from your actual input, and processes everything locally in your browser. Start with the multi-base converter for orientation, or jump straight to the pair you need.
