Lesson 1, Topic 1
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# Number Bases

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The usual system of counting in recent times is called the decimal or denary system.

This system enables us to write small or large numbers using a combination of digits.

Example:Â  6843, Â  614.35

Each digit in a number has a place value

For example,Â 8321 means 8 thousands,Â  3 hundreds, 2 tens, 1 unit

This can be shown using powers of 10 as follows

This can be written in expanded notation as

8 x 103Â  + 3 x 102 + 2 x 101 + 1 x 100

8 x 1000 + 3 x 100 + 2 x 10 + 1 x 1

8000 + 300 + 20 + 1

Also, the decimal fractionÂ 0.532 means 5 tenths,Â  3 hundredths, 2 thousandthsÂ

i.e. 0.532 = 5 x 10-1 + 3 x 10-2 + 2 x 10-3

Numbers in base 10 are usually written without the subscript ten or 10

431ten is simply written as 431.

Apart from base 10, numbers are counted in other bases such as base two, base five, base seven, base eight etc.

Note: Numbers in other bases are also written in expanded notation.

For example,Â

158 in base five is

158 = 1 x 52 + 5 x 51 + 8 x 50

= 1 x 25 + 5 x 5 + 8 x 1

Any number raised to the power 0 = 1 so in the above example 50 = 1.

### Example

Write the following numbers in expanded form.

a. 5314.23Â
b. 31506eightÂ
c. 232.01fiveÂ
d. 5097ten
e. 163.07ten

Solution

a. 5314.23Â  meansÂ  5314.23 ten

5 x 103 + 3 x 102 + 1 x 101 + 4 x 100 + 2 x 10-1 + 3 x 10-2

b. 31506eightÂ  =Â  3 x 84 + 1 x 83 + 5 x 82 + 0 x 81 + 6 x 80

c. 232.01fiveÂ  = 2 x 52 + 3 x 51 + 2 x 50 + 0 x 5-1 + 1 x 5-2

d. 5097ten = 5 x 103 + 0 x 102 + 9 x 101 + 7 x 100

e. 163.07ten = 1 x 102 + 6 x 101 + 3 x 100 + 0 x 10-1 + 7 x 10-2