Introduction
A number system converter transforms values between binary (base 2), octal (base 8), decimal (base 10), and hexadecimal (base 16). These four bases appear throughout computer science curricula, competitive exams, and professional software development. Whether you are verifying homework, debugging memory dumps, or learning positional notation for the first time, instant multi-base output saves time and reduces transcription errors.
Numverto shows every representation simultaneously along with step-by-step working. That transparency helps students understand why 255 decimal equals FF hex and 11111111 binary — not just the final answer. Read our complete number system guide or browse the binary reference table for quick lookups.
Conversion Method & Formula
To convert from any base b to decimal: multiply each digit by b raised to its position index (starting at 0 from the right) and sum. To convert decimal to another base: repeatedly divide by the target base and read remainders bottom-to-top. Hexadecimal groups map cleanly to binary: each hex digit equals four binary bits. Our converter validates digits for the selected input base before applying these rules.
Step-by-Step Examples
Example 1: Binary 1011 to Decimal
Positions (right to left): 2³, 2², 2¹, 2⁰. Calculation: (1×8)+(0×4)+(1×2)+(1×1) = 11 decimal. The tool displays each term in the steps panel.
Example 2: Decimal 255 to Hexadecimal
255 ÷ 16 = 15 remainder 15. Both quotients map to F in hex notation, giving FF₁₆. In binary this is eight consecutive 1-bits — one full byte.
Example 3: Octal 377 to Decimal
3×8² + 7×8¹ + 7×8⁰ = 192 + 56 + 7 = 255 decimal. Octal still appears in Unix file permissions (chmod 755) and legacy systems.
Real-Life Applications
- BCA, BTech, and GATE computer science exam preparation
- Embedded programming and register-level debugging
- Web development colour codes (#RRGGBB hex values)
- Network subnetting and IP address notation
- Digital logic design and FPGA coursework
Advantages of Using This Number System Converter
- Instant conversion across all four major bases with validation
- Step-by-step working suitable for learning and exam checks
- Visual 8-bit binary display for byte-sized values
- No signup — runs entirely in your browser
- Mobile-friendly inputs and copy buttons for each output
Common Mistakes to Avoid
- Reading binary place values left-to-right instead of right-to-left
- Using invalid hex digits (only 0–9 and A–F are allowed)
- Forgetting leading zeros when grouping binary into hex nibbles
- Confusing octal digit 8 or 9 (octal only uses 0–7)
- Applying decimal rules directly without adjusting the base
Learn More
- Binary Arithmetic Calculator
- BCD Converter
- ASCII Converter
- Binary to Hex Converter
- Binary to Decimal Guide
- Binary to Hexadecimal Guide
- Number System Guide
What is a Number System Converter?
A number system converter is a tool that transforms numbers between different positional bases: binary (base 2), octal (base 8), decimal (base 10), and hexadecimal (base 16). Each base uses a different set of digits to represent values, and converting between them is a fundamental skill in computer science, digital electronics, and programming.
This converter handles both integer and fractional numbers (like 1110.111 in binary or 14.875 in decimal), shows step-by-step working for every conversion, and displays results in all four bases simultaneously.
How to Use This Converter
- Enter a number: type any valid number in the input field. Examples:
1011(binary),377(octal),255(decimal),FF(hex), or1110.111(binary with fraction). - Select the input base: choose Binary, Octal, Decimal, or Hexadecimal from the dropdown.
- View all results: conversions to all four bases appear instantly with a visual bit display for binary.
- Learn from step-by-step working: the converter shows each step of the conversion process for educational purposes.
Number Base Comparison Table (0-20)
| Decimal | Binary | Octal | Hex |
|---|---|---|---|
| 0 | 0000 | 0 | 0 |
| 1 | 0001 | 1 | 1 |
| 2 | 0010 | 2 | 2 |
| 3 | 0011 | 3 | 3 |
| 4 | 0100 | 4 | 4 |
| 5 | 0101 | 5 | 5 |
| 6 | 0110 | 6 | 6 |
| 7 | 0111 | 7 | 7 |
| 8 | 1000 | 10 | 8 |
| 9 | 1001 | 11 | 9 |
| 10 | 1010 | 12 | A |
| 11 | 1011 | 13 | B |
| 12 | 1100 | 14 | C |
| 13 | 1101 | 15 | D |
| 14 | 1110 | 16 | E |
| 15 | 1111 | 17 | F |
| 16 | 10000 | 20 | 10 |
| 17 | 10001 | 21 | 11 |
| 18 | 10010 | 22 | 12 |
| 19 | 10011 | 23 | 13 |
| 20 | 10100 | 24 | 14 |
Pattern to notice: Hex F = Decimal 15 = Binary 1111 (all four bits set). This is why 255 = FF = 11111111: it's the maximum value for 8 bits (1 byte).
Why Different Number Bases Exist
Binary (Base 2): The Language of Computers
Computers use binary because digital circuits have two states: on (1) and off (0). Every piece of data (text, images, audio) is ultimately stored as sequences of 1s and 0s. Understanding binary is essential for anyone studying computer science or working with low-level systems.
Hexadecimal (Base 16): Compact Binary Representation
Hex is used by programmers because one hex digit represents exactly 4 binary bits. This makes it much easier to read long binary strings: 11111111 10101010 becomes just FF AA. Common uses include CSS colors (#FF5733), memory addresses (0x7FFF), MAC addresses, and debugging.
Octal (Base 8): Unix File Permissions
Octal maps perfectly to 3 binary bits, making it ideal for Unix/Linux file permissions. The command chmod 755 means: owner=7 (rwx=111), group=5 (r-x=101), others=5 (r-x=101). Octal was also used extensively in older PDP-series computers.
Decimal (Base 10): Human Default
We use decimal because we have 10 fingers. It's the natural system for everyday counting and finance. Computers convert to decimal only for display; internally, everything remains binary.
Conversion Method Cheat Sheet
Binary → Decimal
Multiply each bit by its power of 2 (from right, starting at 20) and sum:
1011 = 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11
Decimal → Binary
Repeatedly divide by 2, collect remainders bottom-to-top:
13 ÷ 2 = 6 R1, 6 ÷ 2 = 3 R0, 3 ÷ 2 = 1 R1, 1 ÷ 2 = 0 R1 → 1101
Binary → Hexadecimal (4-bit grouping)
Group binary digits into sets of 4 from right, replace each group:
1111 0101 → F 5 → F5
Binary → Octal (3-bit grouping)
Group binary digits into sets of 3 from right, replace each group:
011 110 101 → 3 6 5 → 365
Hex → Binary
Replace each hex digit with its 4-bit binary equivalent:
A3 → 1010 0011
Common Exam Questions on Number Systems
Question 1: Convert (255)10 to binary, octal, and hex
Solution:
- Binary: 255 = 128+64+32+16+8+4+2+1 = 11111111
- Octal: Group binary in 3s → 011 111 111 = 377
- Hex: Group binary in 4s → 1111 1111 = FF
Question 2: Convert (1A3)16 to decimal
Solution: 1×16² + A×16¹ + 3×16⁰ = 1×256 + 10×16 + 3×1 = 256 + 160 + 3 = 419
Question 3: How many bits are needed to represent the decimal number 500?
Solution: 2⁸ = 256, 2⁹ = 512. Since 500 < 512, we need 9 bits. (Range of n bits = 0 to 2ⁿ-1)
Question 4: Convert (101.11)2 to decimal
Solution: Integer part: 1×4 + 0×2 + 1×1 = 5. Fraction: 1×0.5 + 1×0.25 = 0.75. Answer: 5.75
Key Formulas to Remember
- Range of n bits: 0 to 2n - 1 (unsigned)
- Number of bits needed for N: ⌈log₂(N+1)⌉
- 1 hex digit = 4 bits (nibble)
- 1 octal digit = 3 bits
- 1 byte = 8 bits = 2 hex digits = max value 255
- Powers of 2: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024...