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:  6843,   614.35

Each digit in a number has a place value

For example  8321 means 8 thousands,  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 103  + 3 x 102 + 2 x 101 + 1 x 10o

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

8000 + 300 + 20 + 1

Also the decimal fraction  0.532 means 5 tenths,  3 hundredths, 2 thousandths

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

431 tens 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.

Example

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. 5314.23  means  5314.23 ten

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

b. 31506eight  =  3 x 84 +1 x 83 +5 x 82 + 0 x 81 + 6 x 8o

c. 232.01 five  = 2 x 52 + 3 x 51 + 2 x 5o + 0 x 5-1 +1 x 5-2

d. 5097 ten = 5 x 103 + 0 x 102 + 9 x 101 + 7 x 10o

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