Number Converter
Popular Number Converters:
What is a Number Converter?
A number system converter is a tool that converts values between binary (base-2), decimal (base-10), hexadecimal (base-16), and octal (base-8) number systems. It is essential for programming, computer science, digital electronics, and understanding how computers store and process data.
History of Number Measurement
The decimal system originated from Hindu-Arabic mathematicians around the 6th century. Binary was described by Leibniz in 1703 and became the foundation of digital computing. Hexadecimal became dominant in computing because each hex digit represents exactly four binary bits, ideal for memory addresses and color codes.
About This Number Converter
This number converter supports binary (base-2), decimal (base-10), hexadecimal (base-16), and octal (base-8) number systems. It uses base-change algorithms rather than multiplication factors, making it different from standard unit converters.
Understanding Number Systems and Base Conversion
A number system (or numeral system) is a mathematical notation for representing numbers using a set of digits or symbols. The system we use daily — decimal — is just one of many possible bases. Computers rely on binary (base-2), programmers frequently use hexadecimal (base-16), and ancient civilizations developed systems like Roman numerals and sexagesimal (base-60) that still influence modern timekeeping and navigation.
Every positional number system works on the same principle: each digit's value depends on its position. In base-10, the digits represent powers of 10 (ones, tens, hundreds...). In base-2, they represent powers of 2 (1, 2, 4, 8, 16...). Understanding how to convert between these bases is essential for computer science, digital electronics, networking, mathematics, and many engineering fields. This page collects converters for all the number system tools you might need — from binary and hexadecimal to Roman numerals and fractions.
How to Convert Between Number Bases (Step-by-Step)
The universal method for converting between any two number bases involves two stages: first convert the source number to decimal (base-10), then convert from decimal to the target base.
- From any base to decimal: Multiply each digit by its positional value (base raised to the power of its position, counting from 0 on the right) and sum all the products. Example: 1011₂ = 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11₁₀.
- From decimal to any base: Repeatedly divide the decimal number by the target base. Collect the remainders in reverse order — these form the digits of the result. Example: 255₁₀ to hex → 255÷16 = 15 R15(F), 15÷16 = 0 R15(F) → FF₁₆.
- For binary ↔ hexadecimal: Use the shortcut of grouping — every 4 binary digits equal exactly 1 hex digit (0000=0, 1111=F). This avoids going through decimal entirely.
- For binary ↔ octal: Group binary digits in sets of 3 from the right (000=0, 111=7).
- For fractions and decimals: Use the appropriate formula — decimal-to-fraction counts decimal places and simplifies using GCD; fraction-to-decimal divides numerator by denominator.
Essential Number System Formulas
These formulas cover the most common conversions between number systems, fractions, and percentages:
- Base-b to Decimal: N₁₀ = dₙ×bⁿ + dₙ₋₁×bⁿ⁻¹ + ... + d₁×b¹ + d₀×b⁰
- Decimal to Base-b: Repeatedly divide by b; remainders (read bottom-to-top) form the result
- Fraction to Decimal: a/b = a ÷ b
- Decimal to Fraction: Write as digits/10ⁿ (where n = decimal places), then simplify by GCD
- Fraction to Percent: (a/b) × 100%
- Decimal to Percent: d × 100%
- Binary to Hex: Group 4 bits → 1 hex digit (0000=0 through 1111=F)
- Binary to Octal: Group 3 bits → 1 octal digit (000=0 through 111=7)
- Roman Numeral Rule: If a smaller value precedes a larger value, subtract it (IV = 4, IX = 9)
- DMS to Decimal Degrees: DD = degrees + minutes/60 + seconds/3600
Worked Examples — Number System Conversions
Example 1: Convert the binary number 10110101 to decimal, hexadecimal, and octal.
Solution:
Binary to decimal: 1×2⁷ + 0×2⁶ + 1×2⁵ + 1×2⁴ + 0×2³ + 1×2² + 0×2¹ + 1×2⁰ = 128 + 32 + 16 + 4 + 1 = 181.
Binary to hex: Group in fours from right: 1011 0101 → B5₁₆.
Binary to octal: Group in threes from right: 10 110 101 → pad left: 010 110 101 → 265₈.
Answer: 10110101₂ = 181₁₀ = B5₁₆ = 265₈.
Example 2: Convert 0.875 to a fraction in simplest form.
Solution:
Count decimal places: 3 places → denominator = 10³ = 1000.
Write as 875/1000.
Find GCD(875, 1000): 1000 = 1×875 + 125, 875 = 7×125 + 0 → GCD = 125.
Simplify: 875÷125 = 7, 1000÷125 = 8.
Answer: 0.875 = 7/8.
Example 3: Write the year 1999 in Roman numerals.
Solution:
1999 = 1000 + 900 + 90 + 9.
1000 = M, 900 = CM, 90 = XC, 9 = IX.
Combine: M + CM + XC + IX.
Answer: 1999 = MCMXCIX.
Example 4: Convert 3/7 to a percentage.
Solution:
Divide: 3 ÷ 7 = 0.428571...
Multiply by 100: 0.428571... × 100 = 42.8571...%
Answer: 3/7 ≈ 42.86% (rounded to 2 decimal places).
Number System Quick Reference
Common equivalents across decimal, binary, hexadecimal, and octal:
| From | To |
|---|---|
| 0 (decimal) | 0000 (bin) / 0 (hex) / 0 (oct) |
| 10 (decimal) | 1010 (bin) / A (hex) / 12 (oct) |
| 16 (decimal) | 10000 (bin) / 10 (hex) / 20 (oct) |
| 64 (decimal) | 1000000 (bin) / 40 (hex) / 100 (oct) |
| 100 (decimal) | 1100100 (bin) / 64 (hex) / 144 (oct) |
| 127 (decimal) | 1111111 (bin) / 7F (hex) / 177 (oct) |
| 255 (decimal) | 11111111 (bin) / FF (hex) / 377 (oct) |
| 256 (decimal) | 100000000 (bin) / 100 (hex) / 400 (oct) |
| 1000 (decimal) | 1111101000 (bin) / 3E8 (hex) / 1750 (oct) |
| 65535 (decimal) | FFFF (hex) / 177777 (oct) |
| 1/2 | 0.5 = 50% |
| 1/3 | 0.333... = 33.33% |
| 3/4 | 0.75 = 75% |
Number Systems Explained
Binary (Base-2): Uses only digits 0 and 1. Fundamental to all digital computing because electronic circuits have two stable states (on/off). Every piece of data — text, images, video, programs — is ultimately stored as sequences of binary digits (bits). Common groupings: 4 bits = nibble, 8 bits = byte, 16/32/64 bits = word sizes used in modern CPUs.
Octal (Base-8): Uses digits 0–7. Each octal digit maps to exactly 3 binary bits, making it a compact representation of binary. Historically significant in early computing (PDP-8, Unix file permissions: chmod 755 = rwxr-xr-x). Less common today but still used in some embedded systems and Unix/Linux administration.
Decimal (Base-10): Uses digits 0–9. The human-standard system, likely arising because we have 10 fingers. All everyday arithmetic, finance, and measurement uses base-10. Scientific notation (6.022 × 10²³) extends its range for very large or small numbers.
Hexadecimal (Base-16): Uses digits 0–9 and letters A–F. Each hex digit maps to exactly 4 binary bits, making it the preferred human-readable format for binary data. Used for memory addresses, color codes (#FF5733), MAC addresses (00:1A:2B:3C:4D:5E), Unicode code points (U+0041 = "A"), and cryptographic hashes.
Roman Numerals: An additive/subtractive system using letters I(1), V(5), X(10), L(50), C(100), D(500), M(1000). Still used for clock faces, movie copyright dates, Super Bowl numbering, building cornerstones, and formal document outlining. Limited to numbers 1–3999 in standard notation.
Real-World Applications of Number Conversion
Programming & Software Development
Developers constantly convert between binary, hex, and decimal when debugging, working with memory addresses, bitwise operations, color values, and encoding schemes. Understanding hex is essential for reading crash dumps and network packets.
Networking & Cybersecurity
IP addresses (IPv4 uses decimal octets, IPv6 uses hexadecimal), subnet masks (binary manipulation), MAC addresses (hex), and cryptographic hashes (hex representation of binary data) all require base conversion fluency.
Digital Electronics & Hardware
Circuit designers work in binary for logic gates, hex for register values, and octal for some legacy bus architectures. Understanding powers of 2 (1, 2, 4, 8, 16, 32, 64, 128, 256...) is fundamental to hardware design.
Mathematics & Education
Fraction-decimal-percent conversions are core curriculum from elementary school through college. Understanding number bases develops mathematical reasoning about place value and positional notation.
Finance & Commerce
Percentages drive interest rates, tax calculations, discounts, and investment returns. Converting between fractions (1/4 point), decimals (0.25), and percentages (25%) is daily work for financial professionals.
History, Art & Design
Roman numerals appear on monuments, tattoos, formal invitations, and film credits. Web designers use hex color codes (#RRGGBB) daily. Understanding these number systems connects mathematics to culture and aesthetics.
Key Takeaways
- All positional number systems work on the same principle — each digit represents a power of the base. Only the base changes.
- To convert between any two bases, go through decimal as an intermediate step: source → decimal → target.
- Binary ↔ hex conversion uses a shortcut: group 4 binary digits = 1 hex digit. Binary ↔ octal: group 3 binary digits = 1 octal digit.
- Fractions become decimals by dividing (a/b = a÷b). Decimals become percentages by multiplying by 100.
- A fraction only terminates in decimal when the denominator (in simplest form) has no prime factors other than 2 and 5.
- Roman numerals use subtractive notation (IV=4, IX=9, XL=40, XC=90, CD=400, CM=900) and max out at 3999.
- Memorizing key equivalents (255=FF=11111111, 1/4=0.25=25%) accelerates mental math and debugging.
Metric Conversion Factor Tables for Number Converter
| Units to convert | Multiply By The Number | Convert as Unit |
|---|---|---|
| Decimal 0 | 0000 | Binary |
| Decimal 1 | 0001 | Binary |
| Decimal 2 | 0010 | Binary |
| Decimal 3 | 0011 | Binary |
| Decimal 4 | 0100 | Binary |
| Decimal 5 | 0101 | Binary |
| Decimal 6 | 0110 | Binary |
| Decimal 7 | 0111 | Binary |
| Decimal 8 | 1000 | Binary |
| Decimal 9 | 1001 | Binary |
| Decimal 10 | A | Hexadecimal |
| Decimal 11 | B | Hexadecimal |
| Decimal 12 | C | Hexadecimal |
| Decimal 13 | D | Hexadecimal |
| Decimal 14 | E | Hexadecimal |
| Decimal 15 | F | Hexadecimal |
Numberconverters & it's abbreviations
| Unit | Abbreviation | Unit | Abbreviation | Unit | Abbreviation |
|---|---|---|---|---|---|
| binary | bin base-2 | decimal | dec base-10 | hexadecimal | hex base-16 |
| octal | oct base-8 |
Frequently Asked Questions
How do I convert decimal to binary?
Repeatedly divide the decimal number by 2 and record the remainders. Read the remainders from bottom to top. For example, 10 ÷ 2 gives remainders 0,1,0,1, so decimal 10 = binary 1010.
How do I convert hex to decimal?
Multiply each hex digit by 16 raised to its position power (starting from 0 on the right). For example, hex FF = 15×16¹ + 15×16⁰ = 240 + 15 = 255.
Why do computers use binary?
Computers use binary because electronic circuits have two stable states (on/off, high/low voltage). This makes binary the most reliable and efficient number system for digital hardware.
What is hexadecimal used for?
Hexadecimal is used in programming for memory addresses, color codes (e.g., #FF0000 for red), MAC addresses, and any context where binary would be too long. Each hex digit represents exactly 4 binary bits.
How do I convert binary to hex?
Group the binary digits into sets of 4 from right to left, then convert each group to its hex equivalent. For example, binary 11111111 = 1111 1111 = F F = hex FF.
Complete list of Number conversion units and its conversion.
- Binary 1010 = Decimal 10
binary to decimal → - Decimal 10 = Binary 1010
decimal to binary → - Hexadecimal A = Decimal 10
hex to decimal →
- Octal 12 = Decimal 10
octal to decimal → - Decimal 10 = Octal 12
decimal to octal → - Binary 1010 = Octal 12
binary to octal →
- Decimal 255 = Binary 11111111
decimal 255 to binary → - Decimal 255 = Hexadecimal FF
decimal 255 to hex → - Decimal 255 = Octal 377
decimal 255 to octal →
- Hexadecimal FF = Binary 11111111
hex FF to binary → - Hexadecimal FF = Decimal 255
hex FF to decimal →
- Decimal 10 = Hexadecimal A
decimal to hex → - Binary 1010 = Hexadecimal A
binary to hex → - Hexadecimal A = Binary 1010
hex to binary →
- Octal 12 = Binary 1010
octal to binary → - Hexadecimal A = Octal 12
hex to octal → - Octal 12 = Hexadecimal A
octal to hex →
- Decimal 100 = Binary 1100100
decimal 100 to binary → - Decimal 100 = Hexadecimal 64
decimal 100 to hex → - Decimal 100 = Octal 144
decimal 100 to octal →