# Change of base

Students must also be aware of change of base sums. The base that we use is decimal, or 10. Hence a number 1234 written in this base is 1 x 103 +2 x 102 +3 x 101 +4 x 100.In a binary system, the base is 2. There are only two digits that can be used: 1 and 0. Similarly, the base could be any number.

To change base from 10 to a number, we divide the given number by the base and list down the remainders, which when arranged in the opposite order, will give us the number in that base.

**Illustration 6:**

What will 12 in decimal be in binary?

Continue dividing 12 by 2 and list down the successive remainders:

The remainders we get are: 0, 0, 1, 1.

Arrange them in opposite order to get 1, 1, 0, 0.

Hence 12 represented in binary will be 1100. To check whether we are right, we use (1100)2

=1 x 23 +1 x 22 +0 x 21 +0 x 20 =8+4=12.

**Illustration 7:**

What is 121 expressed in base 4 equal to in decimal?

Since the given number 121 is in base 4, we use the above method of converting back to decimal: From the above,(121)4 =1 ́42 +2 ́41 +1 ́40 =16+8+1=25. We can check our answer by working backwards: dividing 25 successively by 4, we get the remainders (25)10 = 1, 2, 1. Hence we get 121

in base 4, which is correct.