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Rational & Irrational numbers

Rational

Changing recurring decimals into fractions

Example 1

What is 0.3636363636363636......  as a fraction? There is a method that you follow to find out!

x = 0.363636363636...    so 100x = 36.3636363636    100x was chosen because the recurring part is two digits long

(100x) - (x) = 36.0000000.........    so 99x = 36                since 99x is the same as (100x) - (x)

(100x) - (x) leaves us with a 'nice' number because the parts after the decimal point match exactly

so x is 36/99

this simplifies to  4/11

 

Example 2

x = 0.2555555555......... After 2 the parts match exactly. The length of the repeating unit is one so find 10x

10x = 2.555555555...........  

(10x) - (x) = 2.55555555....... - 0.255555555...... = 2.5 - 0.2 = 2.3

so 9x = 2.3

so 90x = 23

so x = 23/90

Useful to know...

 

Irrational

 

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