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Lesson 2, Topic 2
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Division of Number Bases

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Example: Division of Number Bases

(i) $$\scriptsize 1111.01_2 \div 1101_2$$

(ii) $$\scriptsize 242_5 \div 14_5$$

(iii) $$\scriptsize 83A6{16} \: \div \: 6_{16}$$

Solution

In Binary Division, you first convert to base 10, do the division and then convert the answer to the given base.

(i) $$\scriptsize 1111.01_2 \div 1101_2$$

convert to base 10

i.e. $$\scriptsize 1111.01_{2} \\ \scriptsize = 1 \: \times \: 2^3 \:+\: 1 \:\times\: 2^2 \:+ \:1 \: \times \: 2^1 \: +\: 1 \: \times \: 2^0 \\ \scriptsize \: + \: 0 \: \times \: 2^{-1}\: + \: 1 \: \times\: 2^{-2}$$

:> $$\scriptsize 8 \: + \: 4 \: + \: 2 \: + \: 1 \: + \: 0 \: + \: 0.25 = 15.25$$

$$\scriptsize 1101_{2} \\ \scriptsize = 1 \:\times \: 2^3 \: + \: 1 \: \times\: 2^2 \: + \: 1 \: \times \: 2^1 \: + \: 1 \: \times \: 2^0$$

:>$$\scriptsize 8 \: + \: 4 \: + \: 2 \: + \: 1 = 13$$

i.e $$\scriptsize 1111.01_2 \div 1101_2 \equiv 15.25 \div 13$$

= $$\scriptsize 1.173$$

Convert 1.173 to base 2

i.e. 0.173 = 0.0012

Therefore, $$\scriptsize 1.173 = 1.002_2$$

Solution

(ii) $$\scriptsize 242_5 \div 14_5$$

First convert the given numbers to base 10.

$$\scriptsize 2\: \times \: 5^2 \: +\: 4 \: \times \: 5^1 \: + \: 2 \: \times \: 5^0 = 72_{10}$$

$$\scriptsize 1 \:\times \:5^1 \:+ \:4 \:\times\: 5^0 = 9_{10}$$

$$\therefore \frac {72_{10}}{9_{10}} \scriptsize = 8_{10}$$

We then convert 810 back to base 5

Therefore, $$\scriptsize 8_{10} = 13_{5}$$

Solution

(iii) $$\scriptsize 83A6_{16} \: \div \: 8_{16}$$

Convert both numbers to base 10

$$\scriptsize 83A6_{16} \: \div \: 8_{16} \\ \scriptsize = 8 \: \times \: 16^3 \: + \: 3 \: \times \: 16^2 \: + \: 10 \: \times \: 16^1 \: + \: 6 \: \times \: 16^0 \\ \scriptsize = 32768 \: + \: 768 \: + \: 160 \: + \: 6 \\ \scriptsize = 33,702$$

$$\scriptsize 8_{16} \\ \scriptsize = 8 \: \times \: 16^0 \\ \scriptsize = 8 \: \times \: 1 \\ \scriptsize = 8$$

i.e $$\scriptsize 83A6_{16} \: \div \: 8_{16} \equiv 33,702 \: \div \: 8 \\ \scriptsize = 4212.75$$

Convert back to base 16

Whole number part

Decimal part

Whole number part = $$\scriptsize 1074$$

Decimal part = $$\scriptsize C$$

$$\scriptsize \therefore \: Answer = 1074.C_{16}$$

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