Binary Calculator
Enter two binary numbers and an operation to calculate the result.
Binary Arithmetic Guide
Binary number system
Binary is base-2, using only digits 0 and 1. Each position represents a power of 2: …8, 4, 2, 1.
Addition rules
| A | B | Sum | Carry |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
Related Calculators
How Binary Arithmetic Works
Binary uses only 0 and 1 (base-2). Addition follows the same rules as decimal but carries at 2 instead of 10: 0+0=0, 0+1=1, 1+1=10 (write 0, carry 1), 1+1+1=11 (write 1, carry 1). Subtraction, multiplication, and division work analogously. Computers use binary because electronic circuits have two stable states: on (1) and off (0), making binary the natural language of digital hardware.
Every byte is 8 binary digits. A 32-bit integer can represent 2³² = 4,294,967,296 different values. Understanding binary arithmetic helps with bitwise operations, networking (IP subnets), memory addressing, and debugging low-level code.
Binary Addition Examples
| Binary | Decimal equivalent | Result |
|---|---|---|
| 0101 + 0011 | 5 + 3 | 1000 (= 8) |
| 1010 + 0110 | 10 + 6 | 10000 (= 16) |
| 1111 + 0001 | 15 + 1 | 10000 (= 16) |
| 1100 − 0011 | 12 − 3 | 1001 (= 9) |
Understanding Binary Arithmetic
Binary arithmetic operates on the same principles as decimal arithmetic but uses only two digits: 0 and 1. The binary system is base 2, meaning each digit position represents a power of 2 rather than a power of 10. Addition follows simple rules: 0 plus 0 equals 0, 0 plus 1 equals 1, 1 plus 0 equals 1, and 1 plus 1 equals 0 with a carry of 1 (just as 9 plus 1 in decimal produces 0 with a carry into the tens column). Subtraction uses borrowing similar to decimal subtraction. Multiplication reduces to repeated addition and bit shifting: multiplying by 2 shifts all bits one position left, adding a trailing zero. Division is the inverse of multiplication. Computers use binary arithmetic exclusively because electronic circuits reliably distinguish between two voltage states representing 0 and 1. Understanding binary operations helps programmers work with bitwise operators, flag fields, and low-level memory formats. Digital electronics designers apply binary arithmetic when building adder circuits, multiplexers, and arithmetic logic units inside every modern processor and microcontroller.
Binary Operation Reference Table
| Operation | Binary example | Decimal equivalent |
|---|---|---|
| Addition | 1010 + 0110 | 10 + 6 = 16 (10000) |
| Subtraction | 1010 − 0011 | 10 − 3 = 7 (0111) |
| Multiplication | 1010 × 0011 | 10 × 3 = 30 (11110) |
| Division | 1010 ÷ 0010 | 10 ÷ 2 = 5 (0101) |
| AND | 1100 AND 1010 | 12 AND 10 = 8 (1000) |
| OR | 1100 OR 1010 | 12 OR 10 = 14 (1110) |
Binary Patterns Programmers Should Know
Certain binary patterns appear repeatedly in software and hardware design. A byte containing all 1s (11111111) equals 255 in decimal and is commonly used as a bitmask. Shifting bits left by one position doubles the value; shifting right by one halves it. The most significant bit (leftmost) in signed integer formats indicates the sign of the number under two's complement representation, where negative numbers are encoded so that addition and subtraction circuits require no special cases. Recognizing these patterns helps developers debug low-level code, optimize bitwise operations, and understand hardware register layouts in embedded systems programming and systems architecture work.
