How the binary calculator works
Type the first and second numbers in binary, pick an operation, and the answer appears in binary with the same sum in decimal and the result in hex. Addition and subtraction are laid out as column sums, with the carries or borrows in small figures above the columns they land in. Multiplication lists one shifted partial product per 1 bit of the multiplier, and division shows each bit brought down, whether the divisor fits, and what is left.
All arithmetic is exact, however long the numbers. To check a single value, the binary to decimal converter shows its place-value working.
The rules of binary arithmetic
Example: 110 × 101 = 110 + 11000 = 11110, which is 6 × 5 = 30.
Worked examples
Example 1 — addition with carries
- Numbers:
- 1011 + 110
- Column 1: 1 + 0: 1
- Column 2: 1 + 1: 0, carry 1
- Column 3: 0 + 1 + 1: 0, carry 1
- Column 4: 1 + carry 1: 0, carry 1
Sum: 10001
Check: 11 + 6 = 17, and 10001 is 16 + 1.
Example 2 — subtraction with borrows
- Numbers:
- 1000 − 11
- Column 1: 0 − 1: borrow → 1
- Column 2: 0 − 1 − borrow: borrow → 0
- Column 3: 0 − borrow: borrow → 1
- Column 4: 1 − borrow: 0
Difference: 101
Check: 8 − 3 = 5.
Every column of a binary addition
| A | B | Carry in | Sum bit | Carry out | Column total (binary) |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 1 |
| 0 | 1 | 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 0 | 1 | 10 |
| 1 | 0 | 0 | 1 | 0 | 1 |
| 1 | 0 | 1 | 0 | 1 | 10 |
| 1 | 1 | 0 | 0 | 1 | 10 |
| 1 | 1 | 1 | 1 | 1 | 11 |
Key facts
- In binary, 1 + 1 = 10: the column gets a 0 and a 1 is carried to the next place on the left.
- Binary multiplication is shift-and-add: each 1 bit of the multiplier adds a copy of the multiplicand shifted left by that bit's position.
- Shifting a binary number left by one place doubles it, and shifting right by one place halves it, dropping any remainder.
- 1011 + 110 = 10001 in binary, which is 11 + 6 = 17 in decimal.
Who needs a binary calculator
Students meet binary addition and shifts early: both are listed in England's GCSE computer science subject content (source: DfE), and exams expect the carries to be shown. Digital electronics students use the same column method to understand adders and ALUs, and programmers use it to check a checksum, an offset or an address calculation by hand. When the numbers live in a fixed-width register and might overflow, the two's complement calculator shows what the hardware would store.
Common binary arithmetic mistakes
- Writing 2 in a column. There is no digit 2 in binary; 1 + 1 is always 0 carry 1.
- Losing the final carry. 1111 + 1 is 10000, five bits; dropping the carry gives 0000.
- Borrowing 10 instead of 2. A borrowed 1 is worth 2 in the column it lands in, not ten.
- Misaligning partial products. Each row shifts one place further left for each higher multiplier bit, including the bits that are 0.
Frequently Asked Questions
What is 1 + 1 in binary?
1 + 1 = 10 in binary, which is two. In a column addition you write 0 and carry 1 to the next column on the left, exactly as 5 + 5 in decimal gives 0 carry 1. And 1 + 1 + 1, with a carry, is 11: write 1, carry 1.
How do you subtract in binary?
Work from the right. 1 − 0 = 1, 1 − 1 = 0 and 0 − 0 = 0. For 0 − 1, borrow from the next column to the left: the borrowed 1 is worth 2 here, so 2 − 1 = 1, and the column you borrowed from goes down by 1. Computers usually subtract by adding the two's complement instead.
How do you multiply binary numbers?
Each bit of the multiplier is 0 or 1, so each row of the long multiplication is either all zeros or the multiplicand shifted left. Write the shifted copies for the 1 bits and add them: 101 × 11 is 101 + 1010 = 1111, which is 5 × 3 = 15.
How does binary long division work?
Bring down one bit of the dividend at a time. If the divisor fits into the running value, write 1 in the quotient and subtract it; if not, write 0. There is no guessing as in decimal, because the only choices are 0 and 1. 1101 ÷ 10 gives 110 remainder 1, or 13 ÷ 2 = 6 r 1.
Can the calculator handle negative binary numbers?
Yes — type a minus sign in front. The working is done on the sizes of the numbers and the sign applied at the end, as it is taught. Division rounds toward zero and the remainder takes the sign of the dividend, so −111 ÷ 10 is −11 remainder −1.
Why is shifting left the same as multiplying by 2?
Every bit moves up one place, and each place is worth twice the one before, so the whole number doubles. 1011 (11) shifted left is 10110 (22). Shifting right halves it and drops any remainder. The bitwise calculator shows shifts at a fixed width.
Related calculators
Notes & Assumptions
- Answers are exact for inputs up to 4,096 binary digits; the step-by-step working is shown for numbers up to 64 bits.
- Negative numbers are entered with a minus sign. Fixed-width overflow and two's complement wrap-around are covered by the twos complement and bitwise calculators.
- Division gives a whole-number quotient and remainder; binary fractions are not produced.
This tool is for education and general calculation. Check arithmetic used in hardware designs or production code against your platform's specification.