__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**