Binary to Decimal Conversion Calculator
The binary number system is the foundation of digital computing. It operates on base-2, using only two digits: 0 and 1. This guide explains the binary and decimal number systems, shows how to convert binary to decimal, and includes practical examples, formulas, and conversion tables to help you master the concept.
What Is the Binary System?
The binary system uses base 2. Each digit (bit) represents an increasing power of 2 from right to left.
10110₂= 1×2⁴ + 0×2³ + 1×2² + 1×2¹ + 0×2⁰ = 22₁₀
What Is the Decimal System?
The decimal system uses base 10. Each digit represents a power of 10.
429= 4×10² + 2×10¹ + 9×10⁰ = 429
Binary vs. Decimal
| Feature | Binary (Base 2) | Decimal (Base 10) |
|---|---|---|
| Digits used | 0, 1 | 0–9 |
| Base | 2 | 10 |
| Example | 1100₂ = 12 | 45 = 101101₂ |
Binary to Decimal Conversion Formula
Formula: Decimal = d₀×2⁰ + d₁×2¹ + d₂×2² + ... + dₙ×2ⁿ
100110= 1×2⁵ + 0×2⁴ + 0×2³ + 1×2² + 1×2¹ + 0×2⁰ = 38₁₀
Decimal to Binary Conversion Example
45 ÷ 2 = 22 R1 → 11 R0 → 5 R1 → 2 R1 → 1 R0 → 0 R1
Binary = 101101₂
Binary to Decimal Conversion Table
| Decimal | Binary |
|---|---|
| 0 | 0 |
| 5 | 101 |
| 9 | 1001 |
| 22 | 10110 |
| 29 | 11101 |
| 38 | 100110 |
| 45 | 101101 |
| 60 | 111100 |
Computers use binary because it's easy to distinguish between two electrical states: on (1) and off (0). This allows binary numbers to be processed reliably through logic gates and circuits, making digital computing fast and accurate.
Additional Reasons
- Simplicity of Hardware Design: Binary simplifies the design of digital electronics. With only two states to detect, hardware like transistors, memory cells, and processors can be built with fewer errors and lower cost.
- Error Resistance and Noise Tolerance: Binary signals (high or low voltage) are more resistant to interference or degradation than multi-level signals. This improves data integrity and error detection.
- Efficient Boolean Logic: Computers operate heavily on Boolean logic (AND, OR, NOT), which naturally maps to binary values. This forms the basis of all computational decision-making.
- Storage Efficiency: Binary allows compact and scalable storage systems. Each bit is a minimal unit of information, enabling efficient use of memory and disk space.
- Universality Across Systems: From CPUs to networking protocols and file systems, binary provides a common language that all digital systems can interpret.