Binary to Decimal Converter
This binary to decimal converter transforms base-2 binary numbers into base-10 decimal values. Enter any binary number to see its decimal equivalent with a step-by-step breakdown showing how each bit's place value contributes to the final result.
Binary to Decimal Converter
Convert between binary and decimal number systems
Frequently Asked Questions
How do you convert binary to decimal?
Multiply each binary digit by 2 raised to the power of its position (starting from 0 on the right). Then add all the results together. For example, binary 1011 = (1 x 2^3) + (0 x 2^2) + (1 x 2^1) + (1 x 2^0) = 8 + 0 + 2 + 1 = 11 in decimal. This positional notation method works for any length binary number.
What is the binary number system?
The binary number system (base 2) uses only two digits: 0 and 1. Each digit position represents a power of 2, just as each position in decimal represents a power of 10. Binary is the fundamental language of computers because digital circuits have two states: on (1) and off (0). All data in computers, from text to images to video, is ultimately stored and processed as binary numbers.
How do you calculate decimal from binary manually?
Write out each bit with its corresponding power of 2, starting from the rightmost bit at position 0. Multiply each bit by its power of 2, then sum the results. For example, 110101: (1x32) + (1x16) + (0x8) + (1x4) + (0x2) + (1x1) = 32 + 16 + 0 + 4 + 0 + 1 = 53. You can also use the doubling method: start from the left, double the running total and add the next bit.
What is 1010 in decimal?
Binary 1010 equals 10 in decimal. The calculation is: (1 x 2^3) + (0 x 2^2) + (1 x 2^1) + (0 x 2^0) = 8 + 0 + 2 + 0 = 10. This is a commonly used example because the binary representation 1010 conveniently converts to the round decimal number 10, making it easy to remember and verify.
What is the largest 8-bit binary number?
The largest 8-bit unsigned binary number is 11111111, which equals 255 in decimal. This is calculated as 2^8 - 1 = 256 - 1 = 255 (the sum of 128+64+32+16+8+4+2+1). An 8-bit number can represent 256 different values (0 to 255). For signed integers using two's complement, an 8-bit number ranges from -128 to 127.
Why do computers use binary?
Computers use binary because digital electronic circuits are most reliably built with two states: high voltage (1) and low voltage (0). This two-state system is extremely resistant to electrical noise and signal degradation, making it highly reliable. While early computers experimented with ternary (base 3) and decimal systems, binary proved the most practical for building fast, accurate, and scalable hardware using transistors.
How do you convert decimal to binary?
Repeatedly divide the decimal number by 2 and record the remainders. Read the remainders from bottom to top to get the binary representation. For example, converting 13: 13/2 = 6 remainder 1, 6/2 = 3 remainder 0, 3/2 = 1 remainder 1, 1/2 = 0 remainder 1. Reading remainders bottom-to-top: 1101. Alternatively, subtract the largest powers of 2 that fit and place a 1 in those positions.
Binary to Decimal Converter
What is Binary to Decimal Conversion?
Binary (base-2) and decimal (base-10) are two of the most important number systems in computing and mathematics. Binary uses only two digits, 0 and 1, while decimal uses ten digits, 0 through 9. Converting between these systems is a fundamental skill in computer science, digital electronics, and programming.
Our converter supports bidirectional conversion: binary to decimal and decimal to binary, with step-by-step breakdowns showing the exact mathematical process used.
How It Works
Each position in a binary number represents a power of 2, just as each position in a decimal number represents a power of 10. The rightmost bit represents 2^0 (which equals 1), the next bit represents 2^1 (which equals 2), then 2^2 (which equals 4), and so on.
Positional Notation:
Binary: b(n) × 2^n + b(n-1) × 2^(n-1) + ... + b(1) × 2^1 + b(0) × 2^0
Binary to Decimal Method
To convert binary to decimal, multiply each bit by its corresponding power of 2 and sum the results:
Example: 10110 (binary) to decimal
- 1 × 2^4 = 16
- 0 × 2^3 = 0
- 1 × 2^2 = 4
- 1 × 2^1 = 2
- 0 × 2^0 = 0
16 + 0 + 4 + 2 + 0 = 22
Decimal to Binary Method
To convert decimal to binary, repeatedly divide the number by 2 and record the remainder. Reading the remainders from bottom to top gives the binary representation:
Example: 22 (decimal) to binary
- 22 ÷ 2 = 11, remainder 0
- 11 ÷ 2 = 5, remainder 1
- 5 ÷ 2 = 2, remainder 1
- 2 ÷ 2 = 1, remainder 0
- 1 ÷ 2 = 0, remainder 1
Reading bottom to top: 10110
Conversion Examples
| Decimal | Binary | Description |
|---|---|---|
| 0 | 0 | Zero |
| 1 | 1 | One |
| 8 | 1000 | Power of 2 |
| 10 | 1010 | Ten |
| 42 | 101010 | Common example |
| 100 | 1100100 | One hundred |
| 255 | 11111111 | Max 8-bit value |
| 1024 | 10000000000 | 1 kilobyte |
Applications
- Computer science: Understanding how data is stored and processed at the hardware level
- Networking: IP address subnetting and subnet mask calculations
- Digital electronics: Logic gate design and circuit analysis
- Programming: Bitwise operations, flags, and permissions
- Data encoding: Understanding character encodings like ASCII and Unicode
- Cryptography: Binary operations form the foundation of many encryption algorithms