Binary Number System Calculator
Perform binary arithmetic and conversions with precision.
Please enter valid binary digits (0 or 1).
Addition (+)
Subtraction (-)
Multiplication (×)
Division (÷)
Please enter valid binary digits (0 or 1).
Binary Result:
Decimal Result:
Hexadecimal:
Understanding the Binary Number System
The binary number system, also known as the base-2 system, is the foundation of all modern computing and digital electronics. Unlike the decimal system (base-10) that humans use daily, which utilizes ten digits (0-9), the binary system uses only two digits: 0 and 1.
How Binary Works
In binary, each position represents a power of 2. Starting from the right (the least significant bit):
- The 1st position is 20 (1)
- The 2nd position is 21 (2)
- The 3rd position is 22 (4)
- The 4th position is 23 (8)
For example, the binary number 1011 is calculated as: (1 × 8) + (0 × 4) + (1 × 2) + (1 × 1) = 11 in decimal.
Binary Arithmetic Rules
Calculating with binary follows specific logic rules, similar to decimal math but with fewer digits to manage:
| Rule | Addition | Multiplication |
|---|---|---|
| 0 & 0 | 0 + 0 = 0 | 0 × 0 = 0 |
| 0 & 1 | 0 + 1 = 1 | 0 × 1 = 0 |
| 1 & 1 | 1 + 1 = 10 (0 carry 1) | 1 × 1 = 1 |
Why Do Computers Use Binary?
Computers use binary because it is physically easy to implement using transistors. A transistor can either be "on" (representing 1) or "off" (representing 0). This binary state allows for high-speed processing with minimal error, as the computer only needs to distinguish between two voltage levels rather than ten.
Common Binary Conversions
To help you understand the scale, here are some common conversions:
- Decimal 5: Binary 101
- Decimal 10: Binary 1010
- Decimal 64: Binary 1000000
- Decimal 255: Binary 11111111