Base Converter
Convert numbers between binary, octal, decimal, hexadecimal and any base 2–36.
Enter a number and select its base. Results update instantly in all other bases.
Invalid number for this base
Binary (base 2)
Octal (base 8)
Decimal (base 10)
Hexadecimal (base 16)
Step-by-step decomposition
Each digit multiplied by the base raised to its positional power:
Enter a number above to see the decomposition.
What is a numeral base?
A numeral base (or radix) defines how many distinct symbols a positional number system uses. In base 10 we use ten digits (0–9); in base 2 only 0 and 1.
The value of a number is the sum of each digit multiplied by the base raised to the power of its position, counting from 0 on the right:
where is the base and is the digit at position . For example, 10011 in binary:
Common bases and their uses
| Base | Name | Digits | Typical use |
|---|---|---|---|
| 2 | Binary | 0–1 | Computer hardware, logic gates, data storage |
| 8 | Octal | 0–7 | Unix file permissions (chmod 755), early computing |
| 10 | Decimal | 0–9 | Everyday arithmetic and human notation |
| 16 | Hexadecimal | 0–9, A–F | CSS colors (#FF5733), memory addresses, byte values |
| 32 | Base 32 | 0–9, A–V | Compact binary encoding, TOTP secrets |
| 36 | Base 36 | 0–9, A–Z | Short IDs, URL tokens, case-insensitive encoding |
Frequently Asked Questions
Divide the decimal number by 2 repeatedly, noting the remainder each time, then read the remainders from bottom to top. Example: 19 ÷ 2 = 9 r 1 → 9 ÷ 2 = 4 r 1 → 4 ÷ 2 = 2 r 0 → 2 ÷ 2 = 1 r 0 → 1 ÷ 2 = 0 r 1. Reading remainders bottom to top gives 10011. Check: 1×2⁴ + 1×2¹ + 1×2⁰ = 16 + 2 + 1 = 19 ✓.
Divide the decimal number by 16 repeatedly, noting each remainder. Remainders from 10 to 15 are written A to F. Example: 255 ÷ 16 = 15 r 15 (F) → 15 ÷ 16 = 0 r 15 (F). Reading bottom to top: FF. Check: 15×16¹ + 15×16⁰ = 240 + 15 = 255 ✓.
Multiply each binary digit by 2 raised to its positional power (counting from 0 on the right), then add all results. Example: 10011₂ = 1×2⁴ + 0×2³ + 0×2² + 1×2¹ + 1×2⁰ = 16 + 0 + 0 + 2 + 1 = 19.
Hexadecimal is used because one hex digit represents exactly 4 bits, and two hex digits represent one byte (8 bits). This makes it compact and human-readable for binary data. Common uses: CSS colors (#FF5733), memory addresses (0x7FFF), hash digests (SHA-256 produces 64 hex chars = 32 bytes), and Unicode code points (U+1F600).
The general method is to convert via decimal: binary → decimal → hexadecimal. A direct shortcut between binary and hexadecimal exists: group binary digits in sets of 4 from the right, padding with leading zeros, then replace each group with its hex digit. Example: 10011₂ → 0001 0011 → 1 3 → 13₁₆.
Octal uses eight symbols (0–7), where each digit represents 3 binary bits. It appeared in early computing because 6-bit and 12-bit machines were common. Today it survives primarily in Unix/Linux file permissions: chmod 755 means rwxr-xr-x (7 = 111₂ = rwx, 5 = 101₂ = r-x). Shortcut from binary: group digits in sets of 3. Example: 10011₂ → 010 011 → 23₈.