The base of any number system is determined by the number of digits in the system. For example, we know that binary is a base-2 number system since it uses two digits and decimal is a base-10 system since it uses ten digits. Throughout our lessons, we will distinguish between various number systems by putting a small subscript after the number to indicate the base. For example, a typical decimal number would be written like 462.1510 and a typical binary number would be written like 110.112.

In order to convert between binary and decimal numbers, we need to understand the relationship between the digits of a given number, the position of those digits, and the base of the number system. Let's take another look at our example number in the decimal system.

Hundreds Tens Ones   Tenths Hundredths
102 101 100   10-1 10-2
4 6 2 . 1 5

The first digit of our number is 4, but because this digit is located in the hundreds column, we know it really represents the value 4 * 102 or 400. Similarly, the 6 in the tens column represents the value 6 * 101 or 60. Continuing this pattern, we can express the number 462.1510 as follows:


 4*102   =  4*100  =   400.
 6*101   =  6*10   =    60.
 2*100   =  2*1    =     2.
 1*10-1  =  1*.1   =     0.1
 5*10-2  =  5*.01  =  +  0.05
                       462.15

In general, the relationship between a digit, its position, and the base of the number system is expressed by the following formula:

DIGIT * BASE POSITION #