Number System Converter
Convert numbers between binary, decimal, hexadecimal, and octal systems.
Convert Between Number Systems
About Number Systems: Different number systems use different bases to represent values. Binary (base-2) uses only 0 and 1, making it fundamental to computing. Decimal (base-10) uses digits 0-9 and is our standard system. Hexadecimal (base-16) uses digits 0-9 and letters A-F, commonly used in computing for memory addresses and color values. Octal (base-8) uses digits 0-7 and was historically important in early computing.
Understanding Number Systems
What Are Number Systems?
Number systems are mathematical frameworks for representing numbers using different bases. Each number system uses a specific set of symbols (digits) and place values based on powers of its base. While humans typically use the decimal (base-10) system, computers and various fields of computer science often utilize other number systems for specific purposes.
The main number systems in computing are:
- Binary (Base-2): Uses only 0 and 1, directly reflecting the on/off states of electronic circuits.
- Decimal (Base-10): Our standard system, using digits 0-9.
- Hexadecimal (Base-16): Uses digits 0-9 and letters A-F (representing values 10-15).
- Octal (Base-8): Uses digits 0-7.
How Number Systems Work
Each position in a number represents a power of the base, increasing from right to left:
Decimal (Base-10) Example: 2359
| Position | Thousands (10³) | Hundreds (10²) | Tens (10¹) | Ones (10⁰) |
|---|---|---|---|---|
| Digit | 2 | 3 | 5 | 9 |
| Value | 2 × 1000 | 3 × 100 | 5 × 10 | 9 × 1 |
| Calculation | 2000 | 300 | 50 | 9 |
Total: 2000 + 300 + 50 + 9 = 2359
Binary (Base-2) Example: 1011
| Position | 2³ | 2² | 2¹ | 2⁰ |
|---|---|---|---|---|
| Digit | 1 | 0 | 1 | 1 |
| Value | 1 × 8 | 0 × 4 | 1 × 2 | 1 × 1 |
| Calculation | 8 | 0 | 2 | 1 |
Total: 8 + 0 + 2 + 1 = 11 (decimal)
Applications of Different Number Systems
Binary (Base-2)
- Fundamental to all computing operations
- Directly represents computer memory states
- Used in digital logic and circuit design
- Foundation of boolean logic operations
Hexadecimal (Base-16)
- Memory addresses in programming and debugging
- Color codes in web development (#RRGGBB format)
- Assembly language and machine code representation
- Network MAC addresses and IPv6 addresses
Octal (Base-8)
- Unix/Linux file permissions
- Used in some older computer systems
- Legacy systems and some embedded applications
- Some scientific and engineering calculations
Decimal (Base-10)
- Everyday human calculations and commerce
- User interfaces and displayed values
- Most mathematical and scientific notation
- Standard for most non-technical communication
Converting Between Number Systems
Converting to Decimal
To convert any number to decimal, multiply each digit by its position value and sum the results, as shown in the examples above.
Converting from Decimal
To convert decimal to another base, repeatedly divide by the target base and record the remainders in reverse order.
Example: Convert 42 (decimal) to binary
- 42 ÷ 2 = 21 remainder 0
- 21 ÷ 2 = 10 remainder 1
- 10 ÷ 2 = 5 remainder 0
- 5 ÷ 2 = 2 remainder 1
- 2 ÷ 2 = 1 remainder 0
- 1 ÷ 2 = 0 remainder 1
Reading the remainders bottom-up: 42 in binary = 101010
Binary-Hexadecimal Shortcut
Since 16 = 2⁴, each hexadecimal digit represents exactly 4 binary digits. This makes conversion between binary and hexadecimal particularly straightforward.
Binary to Hex conversion chart:
Frequently Asked Questions (FAQs)
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