Binary to Decimal Converter
Convert binary numbers to decimal numbers easily and accurately.
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Table of Contents
Understanding Binary and Decimal Systems
Binary and decimal are two fundamental number systems used in computing and mathematics. Understanding how they work and interact is essential for computer science, programming, and digital electronics.
What is the Decimal System?
The decimal (base-10) system is our everyday number system that uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. It's called "base-10" because each position in a number represents a power of 10:
| Position | Value | Example: 437 |
|---|---|---|
| Hundreds (10²) | 100 | 4 × 100 = 400 |
| Tens (10¹) | 10 | 3 × 10 = 30 |
| Units (10⁰) | 1 | 7 × 1 = 7 |
| Total: | 437 | |
What is the Binary System?
The binary (base-2) system uses only two digits: 0 and 1. It's the foundation of all modern computing systems. In binary, each position represents a power of 2:
| Position | Value | Example: 10110 |
|---|---|---|
| 2⁴ | 16 | 1 × 16 = 16 |
| 2³ | 8 | 0 × 8 = 0 |
| 2² | 4 | 1 × 4 = 4 |
| 2¹ | 2 | 1 × 2 = 2 |
| 2⁰ | 1 | 0 × 1 = 0 |
| Total: | 22 | |
Why Binary is Important in Computing
Binary is fundamental to computing for several reasons:
Physical Implementation
Electronic components can easily represent two states: on/off, high/low voltage, or magnetized/demagnetized, making binary ideal for computers.
Boolean Logic
Binary aligns perfectly with Boolean algebra (TRUE/FALSE operations), which is essential for logical operations in computing.
Data Storage
All data in computers (text, images, videos, programs) is ultimately stored as sequences of binary digits (bits).
Digital Logic Circuits
The building blocks of all computing devices operate using binary signals and logic gates (AND, OR, NOT, etc.).
Conversion Methods
There are two primary methods to convert binary to decimal:
1. Positional Notation Method
This method involves multiplying each binary digit by its corresponding power of 2 based on its position, then adding all results:
Binary: 1011
= (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰)
= 8 + 0 + 2 + 1
= 11
2. Doubling Method
Starting from the leftmost digit, for each bit:
- Double the previous result
- Add the current bit (0 or 1)
Binary: 1011
Start: 0
1: (0 × 2) + 1 = 1
0: (1 × 2) + 0 = 2
1: (2 × 2) + 1 = 5
1: (5 × 2) + 1 = 11
Historical Context
Binary has a rich history in mathematics and computing:
- Ancient China (3rd century BC): The I Ching used binary-like symbols for divination
- 1703: Gottfried Leibniz formalized binary arithmetic in his paper "Explanation of Binary Arithmetic"
- 1930s: Claude Shannon demonstrated how electrical circuits could perform boolean logic
- 1940s: First electronic digital computers used binary for calculations
- Present day: Binary remains the fundamental language of all modern computing systems
Applications of Binary to Decimal Conversion
Understanding binary to decimal conversion is essential in various fields:
Computer Programming
Programmers often need to understand and work with binary data when dealing with low-level operations, bit manipulation, or debugging.
Networking
IP addresses, subnet masks, and network configurations often require conversions between binary and decimal representations.
Digital Electronics
Engineers working with digital circuits, microcontrollers, and embedded systems regularly convert between binary and decimal.
Data Analysis
Understanding binary representations helps when analyzing raw data formats, file structures, or encryption algorithms.
How to Convert Binary to Decimal
Binary (base-2) uses only two digits: 0 and 1. Each position in a binary number represents a power of 2.
Steps to Convert:
-
1Write down the binary number
-
2Starting from the right, multiply each digit by 2 raised to the power of its position (starting from 0)
-
3Add up all the results
11010 = 1×2⁴ + 1×2³ + 0×2² + 1×2¹ + 0×2⁰
= 16 + 8 + 0 + 2 + 0
= 26
Binary Position Values:
2⁰ = 1
2¹ = 2
2² = 4
2³ = 8
2⁴ = 16
2⁵ = 32
2⁶ = 64
2⁷ = 128
Common Examples
Example 1 Basic Numbers
0 = 0
1 = 1
10 = 2
Example 2 Common Values
100 = 4
1000 = 8
10000 = 16
Example 3 Mixed Numbers
101 = 5
110 = 6
111 = 7
Example 4 Larger Numbers
1000 = 8
10000 = 16
100000 = 32