Binary ↔ Decimal Converter
Convert between binary (base-2) and decimal (base-10) number systems with ease. Essential for computer science, digital electronics, programming, and understanding how computers represent and process numerical data.
Binary ↔ Decimal Converter
Convert between binary (base-2) and decimal (base-10) number systems with detailed explanations
Quick Examples
Click on any example to try the conversion
Binary-Decimal Reference
Quick reference for common binary and decimal equivalents
Small Numbers (0-15)
Powers of 2
Conversion Tips
Quick methods and shortcuts for binary-decimal conversion
Decimal to Binary Shortcuts
- • Powers of 2 are easy: 8 = 1000, 16 = 10000
- • For small numbers, memorize 0-15 in binary
- • Use the division by 2 method for larger numbers
- • Each bit position represents a power of 2
Binary to Decimal Shortcuts
- • Start from the rightmost bit (2^0 = 1)
- • Each position left doubles the value
- • Only add values where the bit is 1
- • 1111 in binary always equals 2^n - 1
How to Use the Binary-Decimal Converter
Master binary and decimal number system conversions
1Choose Conversion Direction
Select the type of conversion you need:
- Decimal to Binary: Convert 25 → 11001
- Binary to Decimal: Convert 1101 → 13
- Auto-detect: Automatically recognize input format
- Batch Mode: Convert multiple numbers at once
2Enter Your Number
Input your number in the appropriate format:
- Decimal: Regular numbers (0-9)
- Binary: Only 0s and 1s (e.g., 101010)
- Validation: Real-time format checking
- Large Numbers: Support for extensive bit patterns
3View Detailed Results
Get comprehensive conversion information:
Binary: 101010
Breakdown: 32 + 8 + 2 = 42
4Learn and Apply
Understand the conversion process:
- • See step-by-step binary arithmetic
- • Learn place value systems
- • Copy results for programming and calculations
Understanding Binary Numbers
Learn how the binary number system works and why it's fundamental to computing
What is Binary?
Binary is a base-2 number system that uses only two digits: 0 and 1. Each position represents a power of 2, making it perfect for digital systems where components can be either "on" (1) or "off" (0).
Place Values in Binary
8
4
2
1
1011₂ = (1×8) + (0×4) + (1×2) + (1×1) = 11₁₀
Why Binary Matters
Binary is the foundation of all digital technology because electronic circuits can easily represent two states: high voltage (1) and low voltage (0).
Digital Electronics
Transistors can be on/off, representing 1/0
Computer Memory
All data stored as sequences of bits (binary digits)
Programming
Bitwise operations and low-level optimization
Network Communication
Data transmission as binary signals
Conversion Methods
Decimal to Binary
Divide by 2, keep track of remainders:
12 ÷ 2 = 6 remainder 0
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Result: 11001
Binary to Decimal
Multiply each digit by its place value:
= 8 + 4 + 0 + 1
= 13₁₀
Common Binary Patterns
Real-World Applications
Discover where binary-decimal conversion is essential in technology and science
Computer Programming
Essential for bitwise operations, memory addressing, binary flags, and understanding how computers store and process data at the lowest level.
if (flags & 0b0010) // Bitwise AND
Digital Electronics
Design and analyze digital circuits, understand logic gates, and work with microcontrollers and embedded systems.
OR Gate: 1 | 0 = 1
XOR Gate: 1 ⊕ 1 = 0
Network Engineering
Calculate subnet masks, understand IP addressing, configure network devices, and analyze network protocols.
Binary: 11000000.10101000.00000001.00000001
Data Analysis
Understand data encoding, file formats, compression algorithms, and binary data structures in databases and file systems.
Binary: 11111111 10000000 01000000
Computer Science Education
Learn fundamental concepts of computer science, algorithms, data structures, and computational thinking.
Bit positions: 0, 1, 2, 3, 4, 5, 6...
Embedded Systems
Program microcontrollers, configure registers, set up interrupts, and control hardware components in IoT and automation projects.
if (PIN & 0b00000001) // Read bit
Mathematical Foundations
Understanding the mathematical principles behind binary number systems
Base Systems Comparison
Decimal (Base-10)
Uses digits 0-9, place values are powers of 10
Binary (Base-2)
Uses digits 0-1, place values are powers of 2
Powers of Two
Understanding powers of 2 is crucial for binary arithmetic:
Binary Arithmetic Examples
Addition
Subtraction
Multiplication
Related Number System Tools
Explore other number base conversion and mathematical tools
Frequently Asked Questions
Why do computers use binary instead of decimal?
Computers use binary because digital circuits can easily represent two states: on (1) and off (0). This corresponds to high and low voltage levels in electronic components. Using more states would require more precise voltage levels, making circuits more complex, slower, and prone to errors.
What's the largest binary number this converter can handle?
Our converter can handle very large numbers, limited mainly by JavaScript's number precision (up to 2⁵³-1 or about 15-16 decimal digits). For most practical applications in programming, electronics, and computer science, this range is more than sufficient.
How do I convert negative numbers to binary?
For negative numbers, computers typically use "two's complement" representation. Our basic converter handles positive integers. For negative numbers in programming, the sign bit and two's complement arithmetic are used, which is a more advanced topic involving fixed-width binary representations.
Can I convert decimal fractions to binary?
Yes, decimal fractions can be converted to binary fractions. For example, 0.5 decimal equals 0.1 binary, and 0.25 equals 0.01 binary. However, many decimal fractions cannot be exactly represented in binary (like 0.1 decimal), which is why floating-point arithmetic can have precision issues.
What's the relationship between binary and hexadecimal?
Hexadecimal (base-16) is closely related to binary because 16 = 2⁴. Each hexadecimal digit represents exactly 4 binary digits (bits). This makes hex a convenient shorthand for binary: 1010₂ = A₁₆, 1111₂ = F₁₆. Programmers often use hex as a more readable way to represent binary data.
How accurate is this converter for educational purposes?
Our converter is completely accurate for educational use and follows standard binary-decimal conversion algorithms. It shows step-by-step breakdowns to help you understand the conversion process, making it perfect for learning computer science fundamentals, digital logic, and mathematical concepts.