
(Although we are accustomed to our decimal number system, which is positional, other ancient number systems, such as the Egyptian number system were not positional, but rather used many additional symbols to represent larger values.)
In a positional number system, the value of each digit is determined by which place it appears in the full number. The lowest place value is the rightmost position, and each successive position to the left has a higher place value.
In our decimal number system, the rightmost position represents the "ones" column, the next position represents the "tens" column, the next position represents "hundreds", etc. Therefore, the number 123 represents 1 hundred and 2 tens and 3 ones, whereas the number 321 represents 3 hundreds and 2 tens and 1 one.
The values of each position correspond to powers of the base of the number system. So for our decimal number system, which uses base 10, the place values correspond to powers of 10:
...  1000  100  10  1 
...  10^3  10^2  10^1  10^0 
1  0  0  1  1  the binary number 
2^4  2^3  2^2  2^1  2^0  place values 
So the binary number 10011 represents the value
(1 * 2^4)  +  (0 * 2^3)  +  (0 * 2^2)  +  (1 * 2^1)  +  (1 * 2^0)  
=  16  +  0  +  0  +  2  +  1 
=  19 
The same principle applies to any number base. For example, the number 2132 base 5 corresponds to
2  1  3  2  number in base 5 
5^3  5^2  5^1  5^0  place values 
So the value of the number is
(2 * 5^3)  +  (1 * 5^2)  +  (3 * 5^1)  +  (2 * 5^0)  
=  (2 * 125)  +  (1 * 25)  +  (3 * 5)  +  (2 * 1) 
=  250  +  25  +  15  +  2 
=  292 
100  =  (1 * 64)  +  (2 * 16)  +  (1 * 4)  +  (0 * 1)  
=  (1 * 4^3)  +  (2 * 4^2)  +  (1 * 4^1)  +  (0 * 4^0)  
Then we use the coefficients of the powers of 4 to form the number as represented in base 4:  
100  =  1 2 1 0  base 4 
One way to do this is to repeatedly divide the decimal number by the base in which it is to be converted, until the quotient becomes zero. As the number is divided, the remainders  in reverse order  form the digits of the number in the other base.
Example: Convert the decimal number 82 to base 6:
82/6  =  13  remainder 4 
13/6  =  2  remainder 1 
2/6  =  0  remainder 2 
The answer is formed by taking the remainders in reverse order: 2 1 4 base 6