In the decimal system we can represent a real number as

.

If we use another base say b then

.

We want to convert a decimal number to binary with the arithmetic calculations done in binary. We can use nested multiplication as in Horner's Algorithm to represent a number:

N = (…(a_n× 10 + a_n - 1)10 +… + a₂)10 + a₁)10 + a₀.

The 10 is converted to b = 1010₂ and each of the a_i must be replaced by (a_i)_b = a _i (here the computer does a table lookup).

N₂ = (…(a _n× b + a _n - 1) b +… + a ₂) b + a ₁) b + a ₀.

Now the binary arithmetic is very easy for the computer.

Example

As binary arithmetic is very tedious for us we would prefer to utilize the division algorithm and perform repeated division by the base recording the remainder each time.
Consider N = (…(a _n× b + a _n - 1) b +… + a ₂) b + a ₁) b + a ₀. Divide both sides of this equation by b , the remainder is then a ₀, repeat and the remainder is a ₁, then a ₂, etc., until the quotient is 0 and we have recovered N_b in the remainders as

a _n…a ₃a ₂a ₁a ₀.

Example Convert 25,431 to binary using the division algorithm.

25,431 = 110001101010111

Converting a fraction to base b is as simple, just use repeated multiplication and recover the integer part each time until the desired accuracy is achieved.

Example Convert 0.1234 to binary.

0.1234 = .0001111110

Now the computer saves a lot of calculation time by using a table lookup. We can do the same thing in our math. Most of us have the binary numbers through 8 memorized. Converting from base 8 to base 2 is just a 3 digit lookup (converting from hexadecimal to binary is just a 4 digit lookup).

Example Covert 25431.1234 to binary by first converting to octal then "looking up" the octal to binary conversion. We will treat the integer and fraction parts separately for now.