Binary Calculator - Convert Binary, Decimal, Hex & Octal

Free online binary calculator to convert between binary, decimal, hexadecimal, and octal. Perform binary arithmetic operations with step-by-step solutions.

Binary Calculator

First Number

Operation

Convert To

Number Systems

Binary (Base 2)

Uses: 0, 1

Example: 1010 = 10₁₀

Octal (Base 8)

Uses: 0-7

Example: 12 = 10₁₀

Decimal (Base 10)

Uses: 0-9

Example: 10 = 10₁₀

Hexadecimal (Base 16)

Uses: 0-9, A-F

Example: A = 10₁₀

Quick Reference

Dec	Bin	Oct	Hex
0	0000	0	0
1	0001	1	1
2	0010	2	2
3	0011	3	3
4	0100	4	4
5	0101	5	5
6	0110	6	6
7	0111	7	7
8	1000	10	8
9	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
13	1101	15	D
14	1110	16	E
15	1111	17	F

Binary Operations

Addition:1+1=10

Subtraction:10-1=1

Multiplication:11×10=110

Division:100÷10=10

How to Use the Binary Calculator

For Conversion:

1Enter your number in the first input field
2Select the input number system (Binary, Octal, Decimal, or Hex)
3Select "Convert" operation (arrow icon)
4Choose the output number system
5Click "Calculate" to see the result with steps

For Arithmetic Operations:

1Enter the first number and select its number system
2Choose the operation (+, -, ×, ÷)
3Enter the second number (same base as first)
4Select the output number system for the result
5Click "Calculate" to see the result with detailed steps

Understanding Binary Numbers

What is Binary?

Binary is a base-2 number system that uses only two digits: 0 and 1. Unlike the decimal system (base-10) that we use in everyday life, binary is the fundamental language of computers and digital electronics. Every piece of data in a computer—whether it's a number, letter, image, or video—is ultimately represented as a sequence of 1s and 0s.

Why Do Computers Use Binary?

Computers use binary because digital circuits can easily represent two states: on (1) or off (0), high voltage or low voltage. This makes binary extremely reliable and simple to implement in hardware. The two-state system is less prone to errors than systems with more states, and binary arithmetic can be efficiently implemented using logic gates.

Simple and reliable: Only two states to distinguish
Easy to implement in electronic circuits
Less susceptible to noise and errors
Efficient for logical operations

Binary Place Values

In binary, each digit position represents a power of 2, starting from 2⁰ on the right. Here's how the binary number 1011 breaks down:

2³
8

2²
4

2¹
2

2⁰
1

= (1×8) + (0×4) + (1×2) + (1×1) = 8 + 0 + 2 + 1 = 11 in decimal

Common Binary Terms

Bit

The smallest unit of data, representing a single 0 or 1.

Byte

A group of 8 bits, can represent 256 different values (0-255).

Nibble

Half a byte (4 bits), equivalent to one hexadecimal digit.

Word

Typically 16, 32, or 64 bits depending on computer architecture.

Number System Conversions

Binary to Decimal

Multiply each bit by 2 raised to its position and sum the results.

Example: 1101₂ to decimal

= (1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰)

= (1 × 8) + (1 × 4) + (0 × 2) + (1 × 1)

= 8 + 4 + 0 + 1

= 13₁₀

Decimal to Binary

Repeatedly divide by 2 and record remainders, then read backwards.

Example: 13₁₀ to binary

13 ÷ 2 = 6 remainder 1

6 ÷ 2 = 3 remainder 0

3 ÷ 2 = 1 remainder 1

1 ÷ 2 = 0 remainder 1

Reading backwards: 1101₂

Binary to Hexadecimal

Group binary digits in sets of 4 (from right), convert each group to hex.

Example: 11010110₂ to hex

Group: 1101 0110

1101₂ = 13₁₀ = D₁₆

0110₂ = 6₁₀ = 6₁₆

= D6₁₆

Binary to Octal

Group binary digits in sets of 3 (from right), convert each group to octal.

Example: 11010110₂ to octal

Group: 011 010 110

011₂ = 3₈

010₂ = 2₈

110₂ = 6₈

= 326₈

Binary Arithmetic Operations

Binary Addition

Rules:

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 10 (0, carry 1)

1 + 1 + 1 = 11 (1, carry 1)

Example:

1010

+ 0111

10001

Binary Subtraction

Rules:

0 - 0 = 0

1 - 0 = 1

1 - 1 = 0

0 - 1 = 1 (borrow 1)

Example:

1010

- 0011

0111

Binary Multiplication

Rules:

0 × 0 = 0

0 × 1 = 0

1 × 0 = 0

1 × 1 = 1

Example:

101

× 11

101

+ 1010

1111

Binary Division

Process:

Similar to long division

Divide step by step

Subtract and bring down

Example:

1100 ÷ 11 = 100

(12 ÷ 3 = 4 in decimal)

Real-World Applications of Binary

Computer Programming

All programming languages ultimately compile to binary machine code. Understanding binary helps in low-level programming, debugging, and optimization.

Cryptography

Binary operations like XOR are fundamental to encryption algorithms. Bitwise operations on binary data form the basis of secure communication.

Networking

IP addresses, subnet masks, and network protocols use binary. Understanding binary is essential for network configuration and troubleshooting.

Game Development

Binary flags and bitmasking are used for efficient state management, permissions, and game logic in video game programming.

Mobile Apps

Mobile applications use binary for data storage, image processing, and efficient memory management on resource-constrained devices.

Artificial Intelligence

Neural networks and machine learning algorithms process data in binary format, making binary understanding crucial for AI development.

Related Math Calculators

Hex Calculator

Convert hexadecimal numbers

Scientific Calculator

Advanced math operations

Percentage Calculator

Calculate percentages easily

Fraction Calculator

Add, subtract fractions

About This Binary Calculator

This binary calculator was developed by CalcFinex, a team dedicated to creating accurate, educational mathematical tools. Our calculator implements industry-standard algorithms for number system conversions and binary arithmetic operations, ensuring precise results for students, developers, and professionals.

100%

Accurate Results

Verified algorithms for all conversions

Number Systems

Binary, Decimal, Hex, Octal support

∞

Free Usage

No limits, no registration required

Our step-by-step solutions help users understand the conversion process, making this tool ideal for learning binary concepts. The calculator follows standard computer science conventions and is regularly updated to maintain accuracy and performance.

Frequently Asked Questions (FAQs)

Learn More About Binary

Educational Resources

📚
Khan Academy - Computer Science
Free courses on binary and computer science fundamentals
📖
Math is Fun - Binary System
Interactive tutorials on binary numbers
🎓
Coursera - Digital Systems
University-level courses on digital systems and binary

Technical References

🔧
MDN - Bitwise Operators
JavaScript bitwise operations documentation
💡
GeeksforGeeks - Binary System
Comprehensive programming tutorials on binary
📝
Wikipedia - Binary Number
Detailed encyclopedia article on binary numbers

Why Our Binary Calculator is Better

✓

Multiple Number Systems

Convert between Binary, Decimal, Hexadecimal, and Octal - all in one place. Most calculators only support 2-3 systems.

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Step-by-Step Solutions

See exactly how conversions and calculations are performed. Perfect for learning and verification.

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Arithmetic Operations

Perform addition, subtraction, multiplication, and division in any number system - not just conversions.

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Educational Content

Comprehensive guides, examples, and FAQs to help you understand binary concepts thoroughly.

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Quick Reference Tables

Built-in conversion tables and operation rules for instant reference while working.

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Mobile Responsive

Works perfectly on all devices - desktop, tablet, and mobile. Calculate on the go!

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No Ads or Popups

Clean, distraction-free interface. No annoying ads or popups interrupting your work.

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100% Free Forever

No premium features, no subscriptions, no hidden costs. Everything is completely free.

Binary Calculator - Convert Binary, Decimal, Hex & Octal

Free online binary calculator to convert between binary, decimal, hexadecimal, and octal. Perform binary arithmetic operations with step-by-step solutions.