Base Converter Calculator
Base Converter Calculator
Convert numbers between binary, decimal, octal, hexadecimal and other bases (2-36) with step-by-step explanations and visualization.
Base Conversion Result
10
Conversion Method:
Step-by-Step Conversion:
Base System Analysis:
Base Representation:
All Base Representations:
Base conversion changes number representation while preserving value.
What is Base Conversion?
Base conversion is the process of changing a number from one base (radix) to another. The base of a number system determines how many digits are used to represent values. Common bases include binary (base 2), octal (base 8), decimal (base 10), and hexadecimal (base 16).
Common Number Systems
Binary (Base 2)
Used in computers
Digital electronics
Octal (Base 8)
Unix permissions
Shorthand for binary
Decimal (Base 10)
Everyday use
Human counting
Hexadecimal (Base 16)
Memory addresses
Color codes
Conversion Methods
1. From Any Base to Decimal
Multiply each digit by base raised to its position:
N = dₙbⁿ + dₙ₋₁bⁿ⁻¹ + ... + d₁b¹ + d₀b⁰
Example: 1010₂ = 1×2³ + 0×2² + 1×2¹ + 0×2⁰ = 8+0+2+0 = 10
2. From Decimal to Any Base
Repeated division by target base:
Divide number by base, record remainder
Continue with quotient until 0
Read remainders in reverse order
Example: 10₁₀ → 1010₂
3. Direct Conversions
Shortcut methods for common bases:
• Binary ↔ Octal: Group bits in 3s
• Binary ↔ Hex: Group bits in 4s
• Octal ↔ Hex: Convert via binary
• Example: 1010 1111₂ → AF₁₆
Digit Systems for Different Bases
| Base | Digits | Max Single Digit | Example Number | Application |
|---|---|---|---|---|
| Binary (2) | 0, 1 | 1 | 1010₂ | Computer circuits |
| Octal (8) | 0-7 | 7 | 752₈ | Unix permissions |
| Decimal (10) | 0-9 | 9 | 255₁₀ | Everyday counting |
| Hexadecimal (16) | 0-9, A-F | F (15) | FF₁₆ | Memory addresses |
| Base 20 | 0-9, A-J | J (19) | 10A₂₀ | Mayan numerals |
| Base 36 | 0-9, A-Z | Z (35) | XYZ₃₆ | URL shorteners |
Real-World Applications
Computer Science & Programming
- Binary representation: How computers store and process data
- Memory addressing: Hexadecimal for memory locations (0x0000 to 0xFFFF)
- Color codes: Hex color values in web design (#FF0000 = red)
- File permissions: Octal notation in Unix/Linux (755, 644)
Electronics & Digital Systems
- Logic gates: Binary operations in digital circuits
- Microprocessor design: Binary arithmetic and addressing
- Network protocols: Hex representation of MAC addresses
- Error detection: Checksums in hexadecimal
Mathematics & Cryptography
- Number theory: Studying properties of different bases
- Cryptography: Base conversions in encryption algorithms
- Data compression: Using different bases for efficient storage
- Encoding schemes: Base64 for binary data in text
Everyday Life
- Timekeeping: Base 60 for seconds/minutes, base 24 for hours
- Measurement: Base 12 for inches in a foot
- Currency: Different bases in historical monetary systems
- Gaming: Hex grids in strategy games
Common Base Conversion Examples
| Binary | Octal | Decimal | Hexadecimal | Application |
|---|---|---|---|---|
| 0000 0000₂ | 0₈ | 0₁₀ | 0x00 | Minimum 8-bit value |
| 0000 1010₂ | 12₈ | 10₁₀ | 0x0A | Binary representation of 10 |
| 0111 1111₂ | 177₈ | 127₁₀ | 0x7F | Maximum 7-bit signed |
| 1111 1111₂ | 377₈ | 255₁₀ | 0xFF | Maximum 8-bit unsigned |
Base Conversion Properties
| Property | Description | Example | Significance |
|---|---|---|---|
| Positional Value | Digit value depends on position | 101₂: 1×4 + 0×2 + 1×1 = 5 | Fundamental to place-value systems |
| Digit Range | Digits from 0 to (base-1) | Base 16: 0-9, A-F (0-15) | Defines valid digits for base |
| Information Density | Higher base = fewer digits for same value | 255₁₀ = 11111111₂ = FF₁₆ | Efficiency in representation |
| Conversion Uniqueness | Each number has unique representation in each base | 10₁₀ = 1010₂ = 12₈ = A₁₆ | Consistent mathematical mapping |
Step-by-Step Conversion Process
Example 1: Binary (1010₂) to Decimal
- Write binary number: 1 0 1 0
- Assign powers of 2: 2³ 2² 2¹ 2⁰ = 8 4 2 1
- Multiply digits by powers: 1×8=8, 0×4=0, 1×2=2, 0×1=0
- Sum results: 8 + 0 + 2 + 0 = 10
- Result: 1010₂ = 10₁₀
Example 2: Decimal (255₁₀) to Hexadecimal
- Start with decimal number: 255
- Divide by 16: 255 ÷ 16 = 15 remainder 15
- Convert remainder to hex digit: 15 = F
- Divide quotient by 16: 15 ÷ 16 = 0 remainder 15 = F
- Read remainders in reverse: FF
- Result: 255₁₀ = FF₁₆
Binary-Hexadecimal-Octal Relationships
| Binary (4-bit) | Decimal | Hexadecimal | Binary (3-bit) | Octal | Pattern |
|---|---|---|---|---|---|
| 0000 | 0 | 0 | 000 | 0 | All zeros |
| 0001 | 1 | 1 | 001 | 1 | First bit |
| 0010 | 2 | 2 | 010 | 2 | Second bit |
| 0100 | 4 | 4 | 100 | 4 | Third bit |
| 1000 | 8 | 8 | - | - | Fourth bit |
| 1111 | 15 | F | 111 | 7 | All ones |
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Frequently Asked Questions (FAQs)
Q: Why do computers use binary?
A: Computers use binary (base 2) because electronic circuits can easily represent two states: on/off, high/low voltage, or magnetic north/south. Binary simplifies circuit design and is more reliable than multi-state systems.
Q: What is the maximum base that can be used?
A: Mathematically, any integer base ≥ 2 can be used. Practically, base 36 is common (0-9, A-Z). Base 64 is also used for encoding. There's no theoretical upper limit, but practicality decreases with very high bases.
Q: How do you handle fractions in different bases?
A: Fractions work similarly in all bases. For example, 0.1 in base 2 represents ½, while 0.1 in base 10 represents 1/10. Conversion involves multiplying fractional parts by the target base repeatedly.
Q: Why is hexadecimal commonly used in programming?
A: Hexadecimal is popular because it's a compact representation of binary (4 bits per hex digit), easy to convert to/from binary, and aligns well with byte boundaries (2 hex digits = 1 byte = 8 bits).
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