Perform basic binary calculations like addition, subtraction, multiplication, and division with ease. Simply enter the binary numbers (using only 0s and 1s), select the operation you want to perform, and get instant results. The final answer will be shown in three formats: binary, decimal, and hexadecimal for better understanding.
The binary number system uses only two digits: 0 and 1. Each digit is called a bit (short for binary digit). Binary numbers are the foundation of modern computing because computers and digital devices work using two states — on and off.
Positional values: Rightmost bit = 2⁰ (1), next = 2¹ (2), next = 2² (4), next = 2³ (8), etc.
Example: 1011₂ = (1×8) + (0×4) + (1×2) + (1×1) = 8 + 0 + 2 + 1 = 11₁₀
Binary addition follows four simple rules:
| A | B | Sum | Carry |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
Example: 1011 + 1101 = 11000 (11 + 13 = 24)
When adding binary numbers with multiple bits, you must include any carry from the previous step — just like in decimal addition. The process is actually simpler because binary uses only two digits: 0 and 1.
Binary subtraction follows four simple rules:
| A | B | Difference | Borrow |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
Example: 1101 − 1011 = 0010 (13 − 11 = 2)
When subtracting, if the top bit is smaller than the bottom bit, you borrow 1 from the next left bit. That borrowed 1 becomes binary 10 (which equals 2 in decimal).
Binary multiplication follows four simple rules:
| A | B | Product |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Example: 101 × 11 = 1111 (5 × 3 = 15)
Binary multiplication works like decimal multiplication. Multiply each digit and shift left for each new row, then add the partial results together.
Binary division is similar to decimal long division and follows simple rules:
| A | B | Result |
|---|---|---|
| 0 | 1 | 0 |
| 1 | 1 | 1 |
| 0 | 0 | Undefined |
| 1 | 0 | Undefined |
Example: 1111 ÷ 11 = 101 (15 ÷ 3 = 5)
Binary division uses repeated subtraction. Compare the divisor with the leftmost bits of the dividend, subtract when possible, and continue step by step like long division.
Binary addition is performed from right to left, carrying over when the sum exceeds 1.
| Step | Operation | Result | Carry |
|---|---|---|---|
| 1 | 1 + 1 | 0 | 1 |
| 2 | 1 + 0 + carry 1 | 0 | 1 |
| 3 | 0 + 1 + carry 1 | 0 | 1 |
| 4 | 1 + carry 1 | 0 | 1 |
Example: 1011 + 1101 = 11000 (11 + 13 = 24)
Binary subtraction is done from right to left, borrowing from the next bit when needed.
| Step | Operation | Result | Borrow |
|---|---|---|---|
| 1 | 1 − 1 | 0 | 0 |
| 2 | 0 − 1 | 1 | 1 |
| 3 | Borrowed 1 → 10 − 0 | 1 | 0 |
| 4 | 1 − 1 | 0 | 0 |
Example: 1101 − 1011 = 0010 (13 − 11 = 2)
Binary multiplication is similar to decimal multiplication using partial products.
| Step | Operation | Partial Result |
|---|---|---|
| 1 | 101 × 1 | 101 |
| 2 | 101 × 1 (shift left) | 1010 |
| 3 | Add partial results | 1111 |
Example: 101 × 11 = 1111 (5 × 3 = 15)
Binary division follows long division by repeatedly subtracting the divisor from the dividend.
| Step | Operation | Quotient Bit | Remainder |
|---|---|---|---|
| 1 | 11 into 11 | 1 | 00 |
| 2 | Bring down next bit (1) | 0 | 01 |
| 3 | 11 into 11 | 1 | 00 |
Example: 1111 ÷ 11 = 101 (15 ÷ 3 = 5)