Base Converter Calculator
Convert numbers between binary, octal, decimal, and hexadecimal bases instantly with our advanced calculator. Get detailed step-by-step solutions for number base conversions.
Number Base Converter Tool
Convert numbers between different bases with detailed step-by-step solution
Master Number Base Conversion with Our Advanced Calculator
Our base converter calculator is an essential tool for students, programmers, and anyone working withcomputer science, programming, and digital electronics. Whether you're solvingmath homework, working with binary arithmetic, studying computer architecture, or exploring digital systems, this tool provides comprehensive solutions with step-by-step explanations.
Number base conversion is fundamental to computer science and digital electronics. Our base converter calculator supports the four most common number systems: binary (base 2), octal (base 8), decimal (base 10), and hexadecimal (base 16). Each base has specific applications: binary for computer memory and logic, octal for Unix file permissions, decimal for everyday arithmetic, and hexadecimal for memory addresses and color codes. This math calculator is particularly useful for programming tasks, computer architecture studies, and digital electronics projects.
Perfect for middle school students learning about different number systems, high school studentsstudying computer science and programming, college students working with computer architecture and digital logic, andprofessionals in software development and electronics. The base converter calculator toolprovides not just the final result, but also the complete mathematical process showing how to convert between different number bases.
Number Base Systems and Applications
| Base | Name | Digits | Applications | Example |
|---|---|---|---|---|
| Base 2 | Binary | 0, 1 | Computer memory, digital logic, binary arithmetic | 1010₂ = 10₁₀ |
| Base 8 | Octal | 0-7 | Unix permissions, legacy systems, compact representation | 17₈ = 15₁₀ |
| Base 10 | Decimal | 0-9 | Everyday arithmetic, human calculations, standard math | 255₁₀ = 255₁₀ |
| Base 16 | Hexadecimal | 0-9, A-F | Memory addresses, color codes, programming | FF₁₆ = 255₁₀ |
Common Mistakes to Avoid
Invalid Digits for Base
Each base has specific valid digits. For example, binary only uses 0 and 1, octal uses 0-7, and hexadecimal uses 0-9 and A-F. Using invalid digits like '8' in binary or 'G' in hexadecimal will cause errors.
Case Sensitivity in Hex
Hexadecimal digits A-F can be written in uppercase or lowercase (A-F or a-f), but mixing cases inconsistently can lead to confusion. It's best to use a consistent format.
Reading Remainders Wrong
When converting from decimal to another base using repeated division, the remainders must be read from bottom to top (last to first) to get the correct result. Reading from top to bottom will give the wrong answer.
How to Convert Between Number Bases
Number base conversion involves two main steps: converting the input number to decimal, then converting the decimal to the target base. Understanding this process is essential for computer science and programming.
Step 1: Convert to decimal using positional notation
Step 2: Convert from decimal using repeated division
The conversion process always goes through decimal as an intermediate step. This two-step approach ensures accuracy and makes the process systematic. For example, to convert from binary to hexadecimal, first convert binary to decimal, then convert decimal to hexadecimal.
Conversion Algorithm
Validate input
Check if digits are valid for the source base
Convert to decimal
Use positional notation formula
Convert to target base
Use repeated division method
Read remainders
Read from bottom to top
Format result
Apply proper formatting for target base
Key Properties of Number Bases
Positional Notation
In base , each digit position represents a power of . The rightmost digit is , the next is , and so on.
Unique Representation
Every positive integer has a unique representation in any base . This is the fundamental theorem of positional notation.
Examples
Decimal to Binary
255 (base 10) → base 2
Process:
- Convert 255 to binary
- Repeated division by 2
- Read remainders bottom to top
Hexadecimal to Decimal
FF (base 16) → base 10
Process:
- Convert FF to decimal
- F = 15, F = 15
- 15 × 16¹ + 15 × 16⁰
Binary to Hexadecimal
1010 (base 2) → base 16
Process:
- Convert 1010 to decimal
- Convert decimal to hex
- Group binary digits
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Solve with AskMathAIFrequently Asked Questions
What is number base conversion?
Number base conversion is the process of changing a number from one base (radix) to another. For example, converting the decimal number 255 to binary (11111111) or hexadecimal (FF). Each base uses a different set of digits and positional notation.
Why do we use different number bases?
Different bases have specific applications: Binary (base 2) is used in computers because digital circuits have two states (on/off). Hexadecimal (base 16) is used for memory addresses and color codes because it's more compact than binary. Octal (base 8) was historically used in computing systems.
How do you convert from decimal to binary?
To convert from decimal to binary, repeatedly divide the decimal number by 2 and record the remainders. Then read the remainders from bottom to top. For example, 13 ÷ 2 = 6 remainder 1, 6 ÷ 2 = 3 remainder 0, 3 ÷ 2 = 1 remainder 1, 1 ÷ 2 = 0 remainder 1. Reading up: 1101₂.
How do you convert from binary to hexadecimal?
To convert from binary to hexadecimal, first group the binary digits into sets of 4 (starting from the right), then convert each group to its hexadecimal equivalent. For example, 1010 1111₂ = AF₁₆, since 1010₂ = A₁₆ and 1111₂ = F₁₆.
What are the valid digits for each base?
Binary uses digits 0-1, octal uses 0-7, decimal uses 0-9, and hexadecimal uses 0-9 and A-F (where A=10, B=11, C=12, D=13, E=14, F=15). The number of valid digits equals the base value.
Why is hexadecimal commonly used in programming?
Hexadecimal is popular in programming because it's more compact than binary (4 binary digits = 1 hex digit), it's easy to convert to/from binary, and it's human-readable. It's used for memory addresses, color codes (RGB), and representing binary data.