Why Does It Work?
The process of converting a decimal number to binary using division and recording the remainders works mathematically because of the way binary and decimal numbers are structured. Here is an explanation:
Binary numbers use only two digits - 0
and 1
. Each binary digit represents a power of 2
. Starting from the right, the first digit is 2^0
, the next is 2^1
, then 2^2
, and so on.
Decimal numbers represent sums of powers of 10
. The rightmost digit is 10^0
, the next 10^1
, then 10^2
, etc.
When we divide a decimal number by 2
repeatedly, recording the remainder each time, we are essentially finding out how many 2
s need to be summed to get the original decimal number.
For example, if we start with the decimal 172
, visualize each step below as one digit in the final result.
- 172 / 2 = 86 remainder 0 (2^0 = 1)
- 86 / 2 = 43 remainder 0 (2^1 = 2)
- 43 / 2 = 21 remainder 1 (2^2 = 4)
- 21 / 2 = 10 remainder 1 (2^3 = 8)
- 10 / 2 = 5 remainder 0 (2^4 = 16)
- 5 / 2 = 2 remainder 1 (2^5 = 32)
- 2 / 2 = 1 remainder 0 (2^6 = 64)
- 1 / 2 = 0 remainder 1 (2^7 = 128)
So 172
in binary is 10101100
because:
- 1 x (2^7) = 128
- 0 x (2^6) = 0
- 1 x (2^5) = 32
- 0 x (2^4) = 0
- 1 x (2^3) = 8
- 1 x (2^2) = 4
- 0 x (2^1) = 0
- 0 x (2^0) = 0
Which sums to 172
.
The division algorithm allows us to repeatedly divide out the largest power of 2
possible to recreate the original decimal number as a sum of powers of 2
- which results in the binary representation.