What Is Binary Arithmetic?
Binary arithmetic is math in base 2, using only the digits 0 and 1 — the native language of every processor, memory chip and digital circuit. Each column represents a power of two (1, 2, 4, 8, 16…), so the binary number 1011 means 8+0+2+1 = 11 in decimal. Addition, subtraction, multiplication and division follow the same column-by-column methods you learned for decimal, just with far simpler rules. This calculator performs all four operations and draws the full working: carry digits in orange, borrow digits in red, one partial-product row per multiplier bit, and quotient-plus-remainder for division.
How to Use This Calculator
1. Enter Number A and Number B — type in the binary box or the decimal box; the two stay in sync automatically (up to 16 bits, 0–65535). 2. Pick an operation with the + − × ÷ buttons. 3. Read the working in the step-by-step panel, exactly as you would write it on paper. 4. Copy the result in whichever base you need: binary, decimal, hexadecimal or octal.
Binary Addition Rules
| Operation | Result | Carry |
|---|---|---|
| 0 + 0 | 0 | — |
| 0 + 1 or 1 + 0 | 1 | — |
| 1 + 1 | 0 | carry 1 |
| 1 + 1 + 1 (incoming carry) | 1 | carry 1 |
Example — 1011 + 110
carries: 1 1 1
1 0 1 1 (11)
+ 1 1 0 (6)
= 1 0 0 0 1 (17)
Binary Subtraction Rules
| Operation | Result | Borrow |
|---|---|---|
| 0 − 0 or 1 − 1 | 0 | — |
| 1 − 0 | 1 | — |
| 0 − 1 | 1 | borrow 1 |
When the top digit is smaller, you borrow 2 from the next column (just as you borrow 10 in decimal). Real processors avoid borrowing entirely: they add the two’s complement of the subtrahend instead, which turns every subtraction into an addition — one reason two’s complement is universal in hardware. When A < B, this calculator shows the negative result and its 8- or 16-bit two’s complement representation.
Binary Multiplication — Shift and Add
Binary long multiplication is decimal’s little sibling: because each multiplier digit is 0 or 1, every partial product is either a shifted copy of A or zero. Multiply 1011 (11) by 110 (6): bit 0 of B is 0 (partial product 0), bit 1 is 1 (write 10110), bit 2 is 1 (write 101100); add them to get 1000010 (66). This shift-and-add method is exactly how simple ALUs multiply — and why multiplying by a power of two is just a left shift.
Binary Division — Quotient and Remainder
Binary long division mirrors the decimal method: test whether the divisor fits into the leading bits, write 1 or 0 in the quotient, subtract, bring down the next bit, repeat. The calculator reports both quotient and remainder and proves the answer with the identity quotient × divisor + remainder = dividend. Division by zero is undefined and flagged clearly.
Binary Multiplication & Division Rules
| Multiplication | Result | Division | Result |
|---|---|---|---|
| 0 × 0 | 0 | 0 ÷ 1 | 0 |
| 0 × 1 or 1 × 0 | 0 | 1 ÷ 1 | 1 |
| 1 × 1 | 1 | divide by 0 | undefined |
Worked Examples for Every Operation
Subtraction — 1101 − 110
borrows: 1
1 1 0 1 (13)
− 1 1 0 (6)
= 1 1 1 (7)
Where a 0 sits above a 1, borrow from the next column (0−1 = 1, borrow 1). Result: 111₂ = 7.
Multiplication — 101 × 11
1 0 1 (5)
× 1 1 (3)
-----------
1 0 1 (bit 0 = 1)
1 0 1 0 (bit 1 = 1, shifted)
-----------
1 1 1 1 (15)
Each 1 bit of the multiplier adds a shifted copy of 101; the partial products sum to 1111₂ = 15 = 5 × 3.
Division — 1011 ÷ 10
1011 (11) ÷ 10 (2) = quotient 101 (5) remainder 1. Check: 101₂ × 10₂ + 1 = 5 × 2 + 1 = 11 = the dividend. Division always returns a quotient and remainder, and dividing by zero is undefined.
Binary / Decimal / Hex Conversion Table
| Decimal | Binary | Hex | Decimal | Binary | Hex |
|---|---|---|---|---|---|
| 0 | 0000 | 0 | 8 | 1000 | 8 |
| 1 | 0001 | 1 | 9 | 1001 | 9 |
| 2 | 0010 | 2 | 10 | 1010 | A |
| 3 | 0011 | 3 | 11 | 1011 | B |
| 4 | 0100 | 4 | 12 | 1100 | C |
| 5 | 0101 | 5 | 13 | 1101 | D |
| 6 | 0110 | 6 | 14 | 1110 | E |
| 7 | 0111 | 7 | 15 | 1111 | F |
Hexadecimal is programmers’ shorthand: each hex digit maps to exactly four binary bits, so 1011 0110₂ is simply B6₁₆. Explore per-bit operations in the Bitwise Calculator (AND, OR, XOR, NOT) and shifting in the Bit Shift Calculator; for how binary meets the analog world, see the ADC Calculator.
Just need to switch bases without arithmetic? The Number System Converter turns any value into binary, decimal, hex and octal at once, with a place-value breakdown.
Curious how the bits are actually computed? The Logic Gate Calculator shows the AND, OR, XOR and other gates behind binary arithmetic, with live symbols and truth tables.
Working with signed (negative) numbers? The Two’s Complement Calculator converts between signed decimal and binary at 8, 16 or 32 bits, showing the invert-and-add-1 steps.
Adding an error-check bit to your data? The Parity Bit Calculator generates even and odd parity and shows the XOR working.
Protecting data against errors? The Hamming Code Calculator adds parity bits that detect and correct single-bit errors, step by step.
Working through logic rather than arithmetic? The Truth Table Generator builds a full truth table from any Boolean expression.
Frequently Asked Questions
How do you add binary numbers?
Column by column from the right: 0+0=0, 0+1=1, 1+1=0 carry 1, and 1+1+1=1 carry 1. Write the sum bit, pass the carry left, repeat. The calculator’s orange row shows every carry in place.
How do you subtract binary numbers?
Use 1−0=1, 1−1=0, and 0−1=1 borrow 1 from the next column. Computers instead add the two’s complement of the subtrahend — same answer, addition-only hardware.
How does binary multiplication work?
Like decimal long multiplication but simpler: each 1 bit in the multiplier contributes a left-shifted copy of the multiplicand; each 0 contributes nothing. Sum the partial products for the result.
How does binary division work?
Binary long division: fit the divisor into the leading bits, write 1 or 0 in the quotient, subtract, bring down the next bit. The calculator verifies quotient × divisor + remainder = dividend.
What is two’s complement?
The standard way computers store negative numbers: invert all bits and add 1. In 8 bits, −5 is 11111011. It lets one adder circuit handle both addition and subtraction, which is why every ALU uses it.
What is 1011 + 1101 in binary?
1011 (11) + 1101 (13) = 11000 in binary, which is 24 in decimal. Several columns generate carries; enter it above to watch each carry appear in orange.
What happens on binary overflow?
Overflow happens when a result needs more bits than the register holds — e.g. two 8-bit numbers whose sum exceeds 255. Hardware sets an overflow flag and the value wraps around. This calculator works to 16 bits and shows the true, non-wrapped result.
How do I convert the result to decimal or hex?
Every result appears at once in binary, decimal, hex and octal. By hand: multiply each bit by its place value (a power of 2) and add for decimal; group bits into fours from the right for hex.
What number bases does this calculator show?
Binary (base 2), decimal (base 10), hexadecimal (base 16) and octal (base 8) simultaneously, plus the bit length — hex being the most useful shorthand since one hex digit equals four bits.