Binary Calculator
Perform arithmetic directly on binary numbers, or convert between binary and decimal — with every step shown.
Binary arithmetic
Convert binary to decimal
Convert decimal to binary
What is the binary number system?
The binary number system is the language computers actually use. It only has two digits — 0 and 1 — and every position represents a power of two rather than a power of ten. This calculator handles three related tasks in one place: full binary arithmetic (add / subtract / multiply / divide), converting a binary value into its decimal equivalent, and converting a decimal value back into binary.
All three tools show the exact working, so you can use this both as a checker for homework and as a learning aid. Numbers of any length are supported — the arithmetic uses arbitrary-precision integers under the hood, so a 200-digit binary value adds and multiplies without losing precision.
Binary calculator, piece by piece
Each card explains one job the calculator does — how it turns binary into decimal and back, and how the four arithmetic operations run column-by-column in base 2.
Binary → decimal — how the converter reads your bits
The rightmost bit is worth 1, then 2, then 4, then 8, doubling each step to the left. The converter multiplies each bit by its place value and sums them. That is why a single "1" moved one place to the left instantly doubles the decimal reading you see below the input.
Decimal → binary — the repeated-division trick
The reverse converter keeps dividing your decimal by 2 and writing down the remainder each time. When the quotient hits zero, the remainders read bottom-up are the binary digits. This is a standard base-conversion algorithm and works for any base by dividing by that base instead of 2.
Binary addition — 1 + 1 carries a 1
Addition works exactly like the decimal method — align, add column by column, carry when you exceed the base. In binary the base is 2, so 1 + 1 makes a carry instead of a 2. The calculator does this with arbitrary-precision integers, so even 200-digit binary sums stay exact.
Binary multiplication & division — shift, add, subtract
Multiplication is the shift-and-add algorithm: for every 1 in the multiplier, copy the multiplicand shifted left by that many places, then add all the rows in binary. Division is the mirror — repeatedly subtract the shifted divisor. Both are what the "×" and "÷" buttons on the calculator run for you.
Features of this calculator
- Add, subtract, multiply, and divide two binary numbers
- Convert between binary, decimal, and hexadecimal
- Supports signed (two's complement) and unsigned integers
- Shows column-by-column working, not just the final answer
- Includes a quick binary/decimal/hex reference table
Frequently asked questions
+Why do computers use binary?
Electronic circuits are cheapest and most reliable when they only have to distinguish two states — on / off, high / low voltage. Two states map naturally to two digits, so every layer above the hardware works in binary too.
+What is the largest number I can enter?
There is no fixed limit. This calculator uses arbitrary- precision integers, so you can add or multiply 200-digit binary numbers and still get exact results.
+Does it handle negative binary numbers?
Yes — put a minus sign in front (e.g. −1011). This tool uses signed arithmetic, not two's complement, so you write the sign explicitly.
+How do I convert to hex or octal instead?
Use the Number Base Converter — it handles binary, octal, decimal and hexadecimal in one step.