# Converting binary to decimal

## Representing numbers with binary numerals follows the
same rules as representing them with decimal numerals.

- a number is represented as a string of digits
- each digit is at least 0 and less than the base
- we can calculate the value by multiplying each digit by the appropriate power of the base:
- the exponents for these powers begin at zero at the right end
- they increase by one each time we take a step to the left

- adding all these products gives us the value of the number our numeral represents

## As an example, let's do this for 592 base ten.

- our string of digits is 592 (third line)
- each digit is at least 0 and less than 10
- we can calculate the value by multiplying each digit by the appropriate power of 10:
- the exponents, from right to left, are 0, 1, 2 (top line)
- the powers, from right to left, are 1, 10, 100 (second line)
- the products (multiply second line by third line) are then 2, 90, 500 (bottom line)

- adding all these products gives us 592
(converting base 10 to base 10 gives us the numeral we started with; duh)

## Now let's do 1010 base two.

- our string of digits is 1010 (third line)
- each digit is at least 0 and less than 2
- we can calculate the value by multiplying each digit by the appropriate power of 2:
- the exponents, from right to left, are 0, 1, 2, 3 (top line)
- the powers, from right to left, are 1, 2, 4, 8 (second line)
- the products (multiply second line by third line) are then 0, 2, 0, 8 (bottom line)

- adding all these products gives us 10