In this case, if a number with a base order than base ten is given to be converted to another number of different base, the given number will be first converted to a number in base ten which will later be converted to a number of the required base.

Example 1.

If for instance we are told to convert 243 base six to base two.

Solution:

First step is to convert the given number in its base to base ten.

243 base six

= 2 x 6² + 4 x 6¹ + 3 x 6°

= 2 x 36 + 4 x 6 + 3 x 1

= 72 + 24 + 3

= 99 base ten

Now, convert it to a number in the required base.

99 base ten = 1100011 base two.

243 base six = 99 base ten = 1100011 base two.

So, 243 base six = 1100011 base two.

Example 2.

If 2354 base nine = “y” base six, find y.

Solution:

First convert to base ten.

2354 base nine

= 2 x 9³ + 3 x 9² + 5 x 9¹ + 4 x 9°

= 1458 + 243 + 45 + 4

= 1750 base ten.

Then convert to the required base.

1750 base ten = 12034 base six.

2354 base nine = 12034 base six, thus;

y = 12034.

## CONVERSION OF DECIMAL NUMBER BASE TO BASE TEN.

The same approach used in the previous conversion is the same that we will use in this very problem, by still making use of our formula **Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³…** at the integral part of the given number. The decimal part is the reverse in power of the integral part (negative powers).

Note before that x raised to power minus 1 is equivalent to 1/x, that is

x–¹ = ¹/x. Also x raised to power minus 2 is equivalent to 1/x² and so on.

Example 1.

Convert 110.101 base two to base ten.

Solution:

110.101 base two

= 1 x 2² + 1 x 2¹ + 0 x 2° + 1 x 2–¹ + 0 x 2–² + 1 x 2–³

= 1 x 4 + 1 x 2 + 0 x 1 + 1 x ½ + 0 x ¼ + 1 x ⅛

= 4 + 2 + 0 + ½ + 0 + ⅛

= 6 + ½ + ⅛

= 6 ⅝

110.101 base two = 6.625 base ten.

Example 2.

Express 131.32 base six in base ten.

Solution:

131.32 base six

= 1 x 6² + 3 x 6¹ + 1 x 6° + 3 x 6–¹ + 2 x 6–²

= 1 x 36 + 3 x 6 + 1 x 1 + 3 x ¹/6 + 2 x ¹/36

= 36 + 18 + 1 + ³/6 + ²/36

= 55 + ½ + ¹/18

= 55 + (10/18)

= 55 + 5/9

131.32 base six = 55.556 base ten.

Example 3.

Express 0.135 base seven to base ten.

Solution:

0.135 base seven

= 0 x 7° + 1 x 7–¹ + 3 x 7–² + 5 x 7–³

= 0 x 1 + 1 x ¹/7 + 3 x ¹/7² + 5 x ¹/7³

= 0 + ¹/7 + ³/49 + 5/343

= 0 + (75/343)

= 0.219 base ten.

**QUESTIONS AND SOLUTIONS.**

Q1. Convert 432 base five to a number in

(a). Base three. (b) Base two.

(a). Base three

**Solution:**

Step one: convert to base ten

432 base five = 4 x 5² + 3 x 5¹ + 2 x 5°

= 100 + 15 + 2 = 117 base ten.

Step two: Now convert 117 base ten to the required base three.

432 base five = 11100 base three

(b). Base two

**Solution:**

Step one: convert to base ten, we have done that in Q1a.

Step two: convert 117 base ten to the required base two.

432 base five = 1110101 base two.

Q2. Convert:

(a). 11.11 base two. (b). 123.12 base three to base ten.

Solution: (a).

11.11 base two = 1 x 2¹ + 1 x 2° + 1 x 2–¹ + 1 x 2–²

= 1 x 2 + 1 x 1 + 1 x ½ + 1 x ¼

= 2 + 1 + ½ + ¼

= 3 + ¾

Therefore 11.11 base two = 3.75 base ten.

(b). 123.12 base three to base ten.

Solution:

= 1 x 3² + 2 x 3¹ + 3 x 3° + 1 x 3–¹ + 2 x 3–²

= 1 x 9 + 2 x 3 + 3 x 1 + 1 x ⅓ + 2 x ¹/9

= 9 + 6 + 3 + ⅓ + ²/9

= 18 + 5/9

= 18.556 base ten.