Decimal to Binary Converter
Convert decimal numbers to binary numbers easily and accurately.
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Table of Contents
Complete Guide to Decimal and Binary Number Systems
Understanding Number Systems
Number systems are the foundation of how we represent quantities in mathematics and computing. The two most important systems we'll explore are:
The Decimal System (Base-10)
This is our everyday number system that uses 10 digits (0-9). It's called "base-10" because each position represents a power of 10.
- Example: 423 = 4×10² + 2×10¹ + 3×10⁰ = 400 + 20 + 3
- Each digit's position has a value 10 times greater than the position to its right
The Binary System (Base-2)
The binary system uses only two digits: 0 and 1. It's called "base-2" because each position represents a power of 2.
- Example: 101₂ = 1×2² + 0×2¹ + 1×2⁰ = 4 + 0 + 1 = 5
- Each digit's position has a value 2 times greater than the position to its right
Why Binary Matters in Computing
Binary is fundamental to computing because:
- Physical Implementation: Electronic circuits have two stable states (on/off, high/low voltage), making binary a natural fit.
- Simplicity: With only two states, binary systems are less prone to errors in signal transmission.
- Boolean Logic: Computer operations are based on Boolean algebra, which works with binary values.
- Storage Efficiency: Information can be efficiently encoded using sequences of bits (binary digits).
Binary Place Values
Understanding binary place values is essential for conversion:
| Position | 27 | 26 | 25 | 24 | 23 | 22 | 21 | 20 |
|---|---|---|---|---|---|---|---|---|
| Value | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
Advanced Concepts in Binary Conversion
1. Binary Fractions
Just like decimal numbers can have fractional parts (e.g., 5.25), binary numbers can too:
- The fractional part uses negative powers of 2: 2-1 (0.5), 2-2 (0.25), etc.
- Example: 101.01₂ = 4 + 1 + 0.25 = 5.25₁₀
2. Converting Decimal Fractions to Binary
To convert a decimal fraction to binary:
- Multiply the decimal fraction by 2
- Record the whole part (0 or 1)
- Continue with the fractional part until you get 0 or a repeating pattern
0.625 × 2 = 1.25 (record 1)
0.25 × 2 = 0.5 (record 0)
0.5 × 2 = 1.0 (record 1)
Result: 0.625₁₀ = 0.101₂
3. Special Cases and Patterns
- Powers of 2: Have a single 1 followed by zeros (2=10₂, 4=100₂, 8=1000₂)
- Powers of 2 minus 1: All 1s (3=11₂, 7=111₂, 15=1111₂)
- Some fractions: Have repeating patterns (1/3 = 0.010101...₂)
Applications of Binary in Computing
- Data Storage: All computer data, including text, images, audio, and video, is stored as binary.
- Computer Memory: RAM, ROM, and caches use binary to store information.
- Digital Logic: Computer processors perform calculations using binary logic operations.
- Network Communications: Data transmission over networks uses binary encoding.
- Error Detection/Correction: Binary-based techniques help detect and correct errors in data.
How to Convert Decimal to Binary
To convert decimal to binary, we repeatedly divide the decimal number by 2 and use the remainders to form the binary number.
Steps to Convert:
-
1Divide the decimal number by 2
-
2Write down the remainder (0 or 1)
-
3Repeat with the quotient until it becomes 0
-
4Read the remainders from bottom to top
26 ÷ 2 = 13 remainder 0
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Result: 11010
Decimal to Binary Conversion Table:
0 = 0
1 = 1
2 = 10
3 = 11
4 = 100
5 = 101
6 = 110
7 = 111
8 = 1000
9 = 1001
10 = 1010
11 = 1011
12 = 1100
13 = 1101
14 = 1110
15 = 1111
Common Examples
Example 1 Basic Numbers
0 = 0
1 = 1
2 = 10
Example 2 Common Values
4 = 100
8 = 1000
16 = 10000
Example 3 Mixed Numbers
26 = 11010
42 = 101010
255 = 11111111
Example 4 Larger Numbers
256 = 100000000
512 = 1000000000
1024 = 10000000000