What Is a Number System Converter?
A number system converter translates a value between the four numeral systems used in computing: binary (base 2), octal (base 8), decimal (base 10) and hexadecimal (base 16). They all describe the same quantity — only the digits and grouping differ. Computers store everything in binary, while programmers use hexadecimal and octal as compact shorthands because they map neatly onto groups of bits. This tool converts all four at once and shows a place-value breakdown so you can see exactly how each digit contributes to the total.
How to Use This Converter
1. Type a value in any of the four fields — decimal, binary, hex or octal. 2. The other three update instantly. 3. Read the place-value breakdown to see how each binary bit (highlighted in blue) adds its power-of-2 value (green) to make the decimal total. 4. Use the grouping note below the breakdown to see how the bits cluster into fours for hex and threes for octal. Invalid characters are flagged per field, so hex only accepts 0–9 and A–F, octal 0–7, and binary 0 and 1.
The Four Number Systems at a Glance
| System | Base | Digits | Example (= 2748) |
|---|---|---|---|
| Binary | 2 | 0 1 | 101010111100 |
| Octal | 8 | 0–7 | 5274 |
| Decimal | 10 | 0–9 | 2748 |
| Hexadecimal | 16 | 0–9, A–F | ABC |
How to Convert Decimal to Binary
Divide the decimal number by 2 repeatedly, writing down each remainder, then read the remainders from bottom to top.
Example — 13 to binary
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Read upward: 1101. Check: 8+4+0+1 = 13.
How to Convert Binary to Decimal
Multiply each bit by its place value (a power of 2) and add. For 1101: 1×8 + 1×4 + 0×2 + 1×1 = 13. The breakdown panel above does this visually for whatever number you enter, highlighting each 1-bit and its value.
How to Convert Binary to Hex and Octal
For hexadecimal, group the bits into fours from the right and convert each group to one hex digit. For octal, group into threes. Example: 1010 1011 1100 → A B C (hex); 101 010 111 100 → 5 2 7 4 (octal). This is why hex and octal are so handy — they compress long binary strings without any arithmetic.
Quick Conversion Reference
| Dec | Bin | Hex | Oct | Dec | Bin | Hex | Oct |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 8 | 1000 | 8 | 10 |
| 1 | 1 | 1 | 1 | 10 | 1010 | A | 12 |
| 2 | 10 | 2 | 2 | 15 | 1111 | F | 17 |
| 4 | 100 | 4 | 4 | 16 | 10000 | 10 | 20 |
| 7 | 111 | 7 | 7 | 255 | 11111111 | FF | 377 |
Where Number Base Conversion Is Used
Hex and binary are everywhere in computing: color codes (#FF8800 is three hex bytes of red, green, blue), memory addresses and machine code, MAC and IPv6 addresses, Unicode code points (U+1F600), file permissions in octal (chmod 755), and register values in embedded systems. Once you have the binary, explore per-bit logic in the Bitwise Calculator, do full arithmetic in the Binary Calculator, or shift bits in the Bit Shift Calculator. For how binary meets the analog world, see the ADC Calculator.
Want the logic behind the bits? The Logic Gate Calculator shows AND, OR, NOT, NAND, NOR, XOR and XNOR with live symbols and truth tables.
Need signed (negative) values? The Two’s Complement Calculator encodes negative numbers in 8-, 16- or 32-bit binary with step-by-step working.
Need reflected binary? The Gray Code Converter turns binary into Gray code and back, where only one bit changes between consecutive values.
Frequently Asked Questions
How do I convert decimal to binary?
Divide by 2 repeatedly and record each remainder, then read the remainders bottom to top. 13 → remainders 1,0,1,1 → 1101. Or just type 13 in the decimal field above and read the binary field.
How do I convert binary to decimal?
Multiply each bit by its place value (power of 2) and add: 1101 = 8+4+0+1 = 13. The place-value breakdown shows this for any number you enter.
How do I convert binary to hexadecimal?
Group the bits into fours from the right and convert each group to one hex digit (0–9, A–F). 1011 0110 → B6.
What is the difference between binary, decimal, hex and octal?
They are numeral systems with bases 2, 10, 16 and 8. All represent the same value; computers use binary, while hex and octal are compact shorthands for reading binary.
What is hexadecimal used for?
Writing binary compactly — one hex digit equals four bits. It is used for memory addresses, color codes, MAC addresses and machine code because it is short and less error-prone than long binary.
Can this converter handle large numbers?
Yes, up to 2^53 accurately across all four bases. Enter a value in any field and the other three update instantly with the place-value breakdown.