Converting from Denary to Binary

Understanding Denary

Denary, also known as the decimal system or base-10, is a system based on 10 and comprised of numbers 0 to 9.
It is the number system most commonly used by humans in everyday life.
When we refer to converting from denary, we are referring to converting from this standard decimal system to another, such as binary.

To convert a denary number to binary, start by finding the largest power of 2 that fits into that number. This will be the left-most digit in the binary representation.
If the power of 2 fits, place a 1 under that power. If it does not, place a 0. This becomes your binary digit for that position.
Subtract the value of the power of 2 from the denary number if you placed a 1 for that power.
Move to the next highest power of 2 (proceeding rightward) and repeat the process, carrying on until you reach 2^0.
Your binary number is the sequence of 1’s and 0’s (bits) you have recorded.

For example: to convert the denary number 13 to binary, the process would look like this:
- The largest power of 2 that fits in 13 is 2^3 = 8, so place a 1 under this and subtract 8 from 13 to give 5.
- The largest power of 2 that fits in 5 is 2^2 = 4, so place a 1 under this and subtract 4 from 5 to give 1.
- Now the largest power of 2 that fits in 1 is 2^0 = 1, so place a 1 under this and subtract 1 from 1 to give 0.
- Therefore, the binary equivalent of the denary number 13 is 1101 (reading from the highest power of 2 down).

It can be helpful to double-check your work by reverting your newly converted binary number back to its denary form. This is done by adding up all the 2^n values where n is the position of each 1 in the binary number (starting at position 0 from the right-most bit).
If the converted value matches the original denary value, your binary conversion is correct.