Binary Converter - Decimal, Hex & Octal Tool
Welcome to the comprehensive Binary Converter designed to help programmers, students, engineers, and computer science enthusiasts convert between Binary, Decimal, Hexadecimal, and Octal number systems instantly with detailed formulas, examples, and educational content.
Multi-Base Number System Converter
Enter a number in any base to see conversions to all other bases
Understanding Number Systems
Binary (Base-2)
Binary is the fundamental numbering system in computing, using only two digits: 0 and 1. Each digit is called a bit (binary digit). Binary directly represents the on/off states of transistors in digital circuits, making it the natural language of computers. All data—text, images, programs—is ultimately stored as binary. Understanding binary is essential for computer science, digital electronics, low-level programming, and comprehending how computers work at the hardware level.
Decimal (Base-10)
Decimal is the standard numbering system humans use daily, with ten digits (0-9). Each position represents a power of 10. Decimal is intuitive for human understanding but requires conversion for computer processing. When programmers write numbers without special notation, they're in decimal. Understanding decimal-to-binary conversion helps bridge human-readable numbers and computer-internal representations. Most programming calculations start in decimal before being processed as binary at the hardware level.
Hexadecimal (Base-16)
Hexadecimal (hex) uses sixteen symbols: 0-9 and A-F (representing 10-15). Each hex digit equals four binary bits (a nibble), making hex a compact representation of binary data. Hex is ubiquitous in computing: memory addresses, color codes (#FF5733), machine code debugging, and data dumps. Two hex digits represent one byte (8 bits), with values 00-FF (0-255 decimal). Hex provides human-readable binary representation—much easier to read than long strings of 0s and 1s.
Octal (Base-8)
Octal uses eight digits (0-7), with each octal digit representing exactly three binary bits. Historically important in computing, octal was commonly used in older systems and Unix file permissions (chmod 755). While less common today than hexadecimal, octal remains relevant in specific applications. Each octal digit maps cleanly to three bits, making octal-to-binary conversion straightforward. Understanding octal helps in legacy system maintenance and Unix/Linux administration.
Conversion Formulas
Binary to Decimal
\[ \text{Decimal} = \sum_{i=0}^{n-1} b_i \times 2^i \]
Where \( b_i \) is the bit at position \( i \) (0 or 1)
Example: Binary 1010 = \( 0×2^0 + 1×2^1 + 0×2^2 + 1×2^3 = 0+2+0+8 = 10 \)
Decimal to Binary
Divide by 2 repeatedly, recording remainders bottom-up
Example: Decimal 10 to binary:
10 ÷ 2 = 5 remainder 0
5 ÷ 2 = 2 remainder 1
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1
Reading bottom-up: 1010
Hexadecimal to Decimal
\[ \text{Decimal} = \sum_{i=0}^{n-1} h_i \times 16^i \]
Where \( h_i \) is the hex digit (0-15, with A=10, B=11, ..., F=15)
Example: Hex A = \( 10×16^0 = 10 \) decimal
Octal to Decimal
\[ \text{Decimal} = \sum_{i=0}^{n-1} o_i \times 8^i \]
Where \( o_i \) is the octal digit (0-7)
Example: Octal 12 = \( 2×8^0 + 1×8^1 = 2+8 = 10 \) decimal
Conversion Examples
| Decimal | Binary | Hexadecimal | Octal |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 8 | 1000 | 8 | 10 |
| 10 | 1010 | A | 12 |
| 15 | 1111 | F | 17 |
| 16 | 10000 | 10 | 20 |
| 255 | 11111111 | FF | 377 |
Practical Applications
Computer Programming
Programmers use different number systems for different purposes. Binary for bitwise operations and understanding memory at the bit level. Hexadecimal for memory addresses, debugging machine code, and representing byte values compactly. Octal for Unix file permissions. Understanding conversions between these systems is essential for low-level programming, systems programming, embedded development, and debugging. Many programming languages support literals in multiple bases: 0b1010 (binary), 10 (decimal), 0xA (hex), 0o12 (octal) all represent the same value.
Digital Electronics and Hardware
Electronics engineers work directly with binary when designing digital circuits. Logic gates operate on binary inputs and outputs. Understanding binary, hex, and octal conversions helps analyze circuit behavior, read datasheets, and debug hardware. Memory chips store data in binary, displayed in hex for readability. Microcontroller programming often requires setting hardware registers using binary or hex values. Number system conversions bridge the gap between human-readable specifications and hardware implementation.
Networking and IP Addresses
Network engineers use binary and hexadecimal extensively. IPv4 addresses are 32-bit binary numbers typically written in decimal (192.168.1.1), but subnet calculations require binary understanding. IPv6 addresses use hexadecimal notation. MAC addresses are written in hex. Understanding binary-to-hex conversion helps with subnetting, CIDR notation, and network troubleshooting. Packet analysis tools display data in hex, requiring conversion skills to interpret network traffic.
Data Representation and File Formats
File formats and data encoding use multiple number systems. File headers (magic numbers) are specified in hex. Color codes in graphics use hex (#RRGGBB). Binary file editors display data in hexadecimal. Understanding these conversions helps developers parse file formats, implement custom protocols, debug data encoding issues, and work with binary file formats. Image processing, audio encoding, and video compression all involve understanding data at the binary level, often viewed in hex.
Common Questions
Why do computers use binary instead of decimal?
Computers use binary because digital circuits have two stable states: on/off, high voltage/low voltage, or charged/uncharged. Transistors—the building blocks of computers—work as switches with these two states. Binary (0 and 1) naturally represents these states. While theoretically possible to build ternary or decimal computers, binary is simpler, more reliable, and requires less power. Every transistor decisively represents either 0 or 1, making binary the most practical choice for digital electronics and computer architecture.
Why is hexadecimal so popular in computing?
Hexadecimal provides compact, human-readable representation of binary data. One hex digit represents exactly 4 bits (a nibble), and two hex digits represent one byte (8 bits). This clean mapping makes hex perfect for representing binary data—FF is much easier to read and remember than 11111111. Memory addresses, machine code, color codes, and data dumps all use hex because it balances compactness with readability. Hex bridges the gap between raw binary (hard to read) and decimal (doesn't align with byte boundaries).
What's the relationship between binary and octal/hex?
Binary, octal, and hex have clean mathematical relationships. Each octal digit represents exactly 3 binary bits (000-111 maps to 0-7). Each hex digit represents exactly 4 binary bits (0000-1111 maps to 0-F). This makes conversion between binary and octal/hex straightforward—just group bits. For example, binary 101010 groups as 101|010 = octal 52, or groups as 10|1010 = hex 2A. This bit-grouping relationship makes octal and hex excellent shortcuts for reading and writing binary values without conversion calculations.
How do you convert between hex and octal directly?
There's no direct hex-to-octal formula—you must go through decimal or binary as an intermediate. The easiest method: convert hex to binary (4 bits per hex digit), then group the binary into 3-bit chunks for octal. For example, hex A (1010 binary) becomes octal 12 (001|010 binary). Alternatively, convert hex to decimal, then decimal to octal. Direct hex-octal conversion isn't common in practice since each serves different purposes—hex for general computing, octal mainly for Unix permissions.
What are the common mistakes in number system conversion?
Common errors include: (1) Forgetting that A-F in hex represent 10-15, not letters, (2) Reading binary remainders top-down instead of bottom-up when converting from decimal, (3) Using invalid digits (8-9 in octal, G-Z in hex), (4) Confusing the base in calculations (treating hex A as decimal 10 in formulas), (5) Forgetting leading zeros in binary-to-hex/octal grouping. Always validate inputs for the correct digit range: binary (0-1), octal (0-7), decimal (0-9), hex (0-F).
Quick Reference Guide
Conversion Shortcuts
- Binary to Hex: Group 4 bits from right (0110|1010 = 6A)
- Binary to Octal: Group 3 bits from right (110|101|010 = 652)
- Hex to Binary: Each hex digit = 4 bits (F = 1111, A = 1010)
- Octal to Binary: Each octal digit = 3 bits (7 = 111, 5 = 101)
- Powers of 2: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024...
Common Values
- 255 decimal: FF hex, 377 octal, 11111111 binary (1 byte max)
- 256 decimal: 100 hex, 400 octal, 100000000 binary
- 1024 decimal: 400 hex, 2000 octal (1 kilobyte)
- Powers of 16: 16, 256, 4096, 65536...
- Powers of 8: 8, 64, 512, 4096...
Why Choose RevisionTown Resources?
RevisionTown is committed to providing accurate, user-friendly tools and educational resources across diverse topics. While we specialize in mathematics education for curricula like IB, AP, GCSE, and IGCSE, we also create practical tools for technical applications like this Binary Converter.
Our converter combines precision with instant calculations and comprehensive explanations to help students, programmers, engineers, and computer science enthusiasts understand and apply number system conversions effectively in programming, digital electronics, networking, and computer science education.
Note: This Binary Converter handles conversions between Binary (Base-2), Decimal (Base-10), Hexadecimal (Base-16), and Octal (Base-8) number systems. Enter a value in any base, and the converter automatically displays equivalents in all other bases. Binary uses digits 0-1, Octal uses 0-7, Decimal uses 0-9, and Hexadecimal uses 0-9 and A-F. This tool is essential for computer science education, programming, digital electronics, networking, and understanding how computers represent numbers at different levels of abstraction.
