Online Binary | Decimal | Hex | Octal Converter

Binary (Base 2)

Accepts: 0, 1

Decimal (Base 10)

Accepts: 0-9

Hexadecimal (Base 16)

Accepts: 0-9, A-F

Octal (Base 8)

Accepts: 0-7

Step-by-Step Calculation

Number System Conversion Table

Quick reference table showing equivalent values across all number systems (0-20):

Decimal	Binary	Octal	Hexadecimal

Mathematical Decimal Types

Terminating Decimals

Numbers that have a finite number of digits after the decimal point. These decimals end after a certain number of places.

Examples:

• 3.14 - Two decimal places
• 0.5 - One decimal place (equivalent to 1/2)
• 2.75 - Two decimal places (equivalent to 11/4)
• 0.125 - Three decimal places (equivalent to 1/8)

Recurring (Repeating) Decimals

Non-terminating decimals where a sequence of digits repeats infinitely. These are rational numbers that cannot be expressed as terminating decimals.

Examples:

• 1/3 = 0.333... or 0.3̄ - The digit 3 repeats infinitely
• 1/6 = 0.1666... or 0.16̄ - The digit 6 repeats infinitely
• 2/11 = 0.181818... or 0.1̄8̄ - The sequence 18 repeats
• 1/7 = 0.142857142857... - The sequence 142857 repeats

Note: The bar notation (vinculum) above digits indicates the repeating sequence.

Non-Recurring Decimals (Irrational Numbers)

Non-terminating decimals where the digits do not repeat in any pattern. These represent irrational numbers that cannot be expressed as fractions.

Examples:

• π (Pi) = 3.14159265358979... - The ratio of a circle's circumference to its diameter
• e (Euler's number) = 2.71828182845904... - The base of natural logarithms
• √2 = 1.41421356237309... - The square root of 2
• φ (Golden ratio) = 1.61803398874989... - The golden ratio

Note: These numbers have infinite, non-repeating decimal expansions and cannot be expressed as simple fractions.

Key Distinction:

Terminating and recurring decimals are rational numbers (can be expressed as fractions), while non-recurring decimals are irrational numbers (cannot be expressed as fractions).

User Guide

How to Use This Tool

• Enter a Value: Type a number in any of the four input fields (Binary, Decimal, Hexadecimal, or Octal)
• Auto-Convert: The tool automatically converts your input to all other number systems in real-time
• View Calculation: A detailed step-by-step calculation guide appears below showing how the conversion was performed
• Copy Results: Click the "Copy Results" button to copy all converted values to your clipboard
• Clear Individual Fields: Hover over any input field and click the × button to clear it
• Reset All: Click the "Reset All" button to clear all fields and start fresh
• Quick Examples: Use the quick convert buttons to see example conversions instantly

Input Validation

• Binary: Only accepts 0 and 1
• Decimal: Only accepts digits 0-9
• Hexadecimal: Accepts 0-9 and A-F (case insensitive)
• Octal: Only accepts digits 0-7
• Invalid characters are automatically rejected as you type

Advanced Features

• Real-time Conversion: Instant conversion as you type
• Step-by-Step Guide: Detailed mathematical breakdown of each conversion
• Highlighted Results: The active input field is highlighted for easy identification
• Responsive Design: Works seamlessly on desktop, tablet, and mobile devices
• Conversion Table: Reference table for quick lookups
• Clipboard Support: One-click copy of all results

Interesting Facts

Did You Know?

Binary is the foundation of all modern computing. Every piece of data in your computer is ultimately represented as a series of 1s and 0s!

Historical Facts

• The binary system was formally described by Gottfried Leibniz in 1679, though earlier mathematicians had explored similar concepts
• The decimal system (base 10) likely originated because humans have 10 fingers
• Hexadecimal became popular in computing because one hex digit represents exactly four binary digits (bits)
• The octal system was widely used in early computing before hexadecimal became more common
• Ancient civilizations used various number bases: Babylonians used base 60, Mayans used base 20

Computing Facts

• A single hex digit can represent values from 0 to 15 (16 different values)
• 8 bits (binary digits) = 1 byte = 2 hexadecimal digits
• IPv6 addresses are written in hexadecimal notation
• Color codes in web design use hexadecimal (e.g., #FFFFFF for white)
• File permissions in Unix/Linux systems are often expressed in octal notation (e.g., 755)

Mathematical Facts

• You can convert to any base, not just 2, 8, 10, and 16 - these are simply the most commonly used
• In base n, the largest single digit is n-1
• Each position in a number represents a power of the base (units, base¹, base², base³, etc.)
• Binary arithmetic is actually simpler than decimal - there are only 4 basic addition facts (0+0, 0+1, 1+0, 1+1)

Additional Tips & Tricks

Conversion Shortcuts

• Binary to Hex: Group binary digits in sets of 4 from right to left, then convert each group
• Binary to Octal: Group binary digits in sets of 3 from right to left, then convert each group
• Powers of 2: Memorize common powers (2, 4, 8, 16, 32, 64, 128, 256, 512, 1024) for faster mental math
• Hex Letters: Remember A=10, B=11, C=12, D=13, E=14, F=15
• Quick Check: The last digit in binary tells you if the decimal number is odd (1) or even (0)

Best Practices

• Always double-check your conversions, especially for critical applications
• Use the step-by-step guide to understand the conversion process, not just the result
• Start with small numbers when learning to build confidence
• Practice mental conversion for common values (10, 16, 32, 64, 100, 255)
• Use the conversion table as a reference when working with small numbers

Professional Tips

• Programmers: Learn to recognize hex patterns in memory addresses and color codes
• Network Engineers: Practice converting subnet masks between decimal and binary
• Students: Use the step-by-step guide to learn the conversion algorithms
• Debugging: Binary representation helps understand bitwise operations and flags
• Data Analysis: Understanding hexadecimal helps in analyzing binary file formats

Practice Makes Perfect!

The more you practice converting between number systems, the faster and more intuitive it becomes. Try converting everyday numbers you encounter!

Use Cases & Applications

Programming & Software Development

• Bitwise Operations: Understanding binary is essential for bit manipulation and logical operations
• Memory Addresses: Hexadecimal is used to represent memory locations in debugging
• Color Coding: Web colors are specified in hexadecimal (e.g., #FF5733)
• Character Encoding: ASCII and Unicode values are often expressed in hex
• Binary Flags: Using binary to set and check multiple boolean flags efficiently

Networking & System Administration

• IP Addressing: Converting between dotted decimal and binary for subnet calculations
• Subnet Masks: Understanding CIDR notation requires binary conversion
• MAC Addresses: Typically displayed in hexadecimal format
• File Permissions: Unix/Linux permissions are often set using octal notation (e.g., chmod 755)
• Port Numbers: Converting between decimal and hexadecimal for network analysis

Education & Learning

• Computer Science: Fundamental topic in CS curriculum and programming courses
• Digital Electronics: Understanding binary logic gates and circuits
• Mathematics: Learning different number systems and positional notation
• Information Theory: Understanding data representation and encoding
• Cryptography: Many encryption algorithms work with binary and hexadecimal

Hardware & Electronics

• Microcontrollers: Programming embedded systems requires understanding binary and hex
• Assembly Language: Machine code instructions are represented in hex
• Hardware Registers: Configuring hardware by setting binary flags in registers
• Digital Signal Processing: Working with binary data in signal processing applications
• FPGA Programming: Designing digital circuits using binary logic

Security & Forensics

• Malware Analysis: Examining binary executables in hexadecimal editors
• Hash Values: Cryptographic hashes are displayed in hexadecimal
• Packet Inspection: Analyzing network traffic at the binary level
• File Signatures: Identifying file types by their hex signatures (magic numbers)
• Memory Forensics: Examining RAM dumps in hexadecimal format

Web Design Example

Color #FF5733 in hex = RGB(255, 87, 51)

FF = 255 (Red), 57 = 87 (Green), 33 = 51 (Blue)

IP Subnet Example

IP: 192.168.1.0

Binary: 11000000.10101000.00000001.00000000

File Permissions

chmod 755 in octal

= 111 101 101 in binary

= rwxr-xr-x

Practical Examples

Common Conversions

Example 1: Converting 42

Decimal: 42

Binary: 101010

Octal: 52

Hexadecimal: 2A

Example 2: Converting 255

Decimal: 255

Binary: 11111111

Octal: 377

Hexadecimal: FF

Example 3: Converting 1000

Decimal: 1000

Binary: 1111101000

Octal: 1750

Hexadecimal: 3E8

Example 4: Powers of 2

Decimal: 1024

Binary: 10000000000

Octal: 2000

Hexadecimal: 400

Real-World Examples

Example: HTTP Status Code

The HTTP status code "404 Not Found":

• Decimal: 404
• Binary: 110010100
• Hexadecimal: 194
• Octal: 624

Example: RGB Color - Pure Red

The color code for pure red:

• Hexadecimal: #FF0000
• Red Channel (FF): Decimal 255, Binary 11111111
• Green Channel (00): Decimal 0, Binary 00000000
• Blue Channel (00): Decimal 0, Binary 00000000

Example: ASCII Character 'A'

The letter 'A' in different number systems:

• Decimal: 65
• Binary: 01000001
• Hexadecimal: 41
• Octal: 101

Step-by-Step Conversion Example

Converting Binary 1011 to Decimal:

1011₂ = (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰)
     = (1 × 8) + (0 × 4) + (1 × 2) + (1 × 1)
     = 8 + 0 + 2 + 1
     = 11₁₀

Converting Decimal 156 to Hexadecimal:

156 ÷ 16 = 9 remainder 12 (C in hex)
9 ÷ 16 = 0 remainder 9

Reading remainders from bottom to top: 9C₁₆

About This Tool

This Number System Converter is a professional-grade, free online tool designed to help programmers, students, engineers, and anyone working with different number bases. It provides instant, accurate conversions between Binary (Base 2), Decimal (Base 10), Hexadecimal (Base 16), and Octal (Base 8) number systems.

Key Features

• Real-Time Conversion: Instant conversion as you type, no submit button needed
• Comprehensive Guide: Step-by-step explanation of how each conversion is calculated
• Input Validation: Automatic validation ensures only valid characters for each base
• Responsive Design: Works perfectly on all devices - desktop, tablet, and mobile
• Professional UI: Clean, modern interface with intuitive controls
• Educational: Perfect learning tool with detailed explanations and reference tables
• Free to Use: No registration, no ads, completely free tool

Who Is This For?

• Programmers & Developers: Quick conversions for debugging, memory addresses, and bitwise operations
• Computer Science Students: Learning tool for understanding number systems and conversions
• Network Engineers: Converting IP addresses, subnet masks, and MAC addresses
• Embedded Systems Developers: Working with registers and binary configurations
• Digital Electronics Engineers: Analyzing and designing digital circuits
• Security Professionals: Analyzing binary data and hexadecimal dumps
• Anyone Learning: Perfect for homework, self-study, or quick reference

Technical Details

• Technology: Built with pure HTML, CSS, and JavaScript - no external dependencies
• Performance: Client-side processing ensures instant results with no server delays
• Accuracy: Supports large numbers with precise conversion algorithms
• Privacy: All conversions happen in your browser - no data is sent to any server
• Accessibility: Designed with semantic HTML and ARIA attributes for screen readers
• SEO Optimized: Comprehensive metadata for better search engine visibility

Our Mission

To provide the most user-friendly, accurate, and educational number system converter available online. We believe in making complex concepts accessible to everyone, whether you're a beginner learning about number systems or a professional needing quick conversions.

Why Different Number Systems?

Different number systems serve different purposes in computing:

• Binary (Base 2): The language of computers - all digital data is ultimately binary
• Decimal (Base 10): The number system humans naturally use for everyday counting
• Hexadecimal (Base 16): Compact representation of binary data, widely used in programming
• Octal (Base 8): Used in Unix file permissions and some legacy systems

Understanding these number systems and being able to convert between them is a fundamental skill in computer science, programming, and digital electronics. This tool makes that process effortless and educational.

Feature	Details
Price	Free
Rendering	Client-Side Rendering
Language	JavaScript
Paywall	No

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About This Tool

• Real-time conversion as you type
  Input validation for each number system
  Step-by-step calculation guide showing the math behind conversions
  Result highlighting to distinguish inputs from outputs
  Individual clear buttons (× icon) on each field
  Reset All button to clear everything
  Copy Results button for clipboard functionality
  Quick Convert buttons with example values
  Comprehensive conversion table (0-20)

How It Works?