How To Convert Number Bases Calculator
A number base is just a choice of how many digits to count to before starting a new column. Converting between bases means regrouping the same quantity into different columns, and the arithmetic is the same place-value arithmetic you already use in decimal.
Case does not matter for hexadecimal. A leading minus sign is allowed, and spaces or underscores are ignored.
Binary
11111111
- Decimal
- 255
- Octal
- 377
- Hexadecimal
- FF
- Bit length
- 8
The formula this calculator uses
value = sum of (digit x base^position)
- base
- How many digits the system uses
- digit
- The symbol in each column, from 0 to base minus 1
- position
- Column index, counting from zero at the right
How to check the result by hand
- 1
Write the digits in place
Line the digits up and number the columns from zero at the right, so the rightmost digit is the base to the power zero.
- 2
Multiply each digit by its place value
In binary the place values are 1, 2, 4, 8, 16 and so on. In hex they are 1, 16, 256, 4096.
- 3
Add the products for decimal
The sum is the value in decimal. For 11111111 that is 128 plus 64 plus 32 plus 16 plus 8 plus 4 plus 2 plus 1, which is 255.
- 4
Group bits for octal or hex
From the right, split the binary digits into groups of three for octal or four for hex, then read each group as a single digit.
- 5
Check the bit length
The number of binary digits tells you how many bits the value needs. Eight bits hold up to 255, sixteen up to 65535, and thirty-two up to about 4.3 billion.
For worked examples, common mistakes and the limits of this formula, read the full How To Convert Number Bases page.