4.6 repeating decimals

• Wiskunde
• Middelbare school
• havo, vwo
• Leerjaar 2

Today

- I understand the differences between natural, whole, rational, irrational and real numbers.

- I understand what a repeating decimal is and know the line notation involved.

I'll show you how this works.

√​a​​​⋅√​b​​​=√​a⋅b​​​

​√​b​​​​​√​a​​​​​=√​​b​​a​​​​​

A repeating decimal

We can write this as

​3​​1​​=0,333...

Two other examples

All numbers with repeating decimals can be written as a fraction.

We can always convert a repeating decimal to a fraction using the following fun algorithm:

Consider 0,99999...

Let a = 0,999...

Consider 0,99999...

Let a = 0,999...

then 100a = 99,999...

Consider 0,99999...

Let a = 0,999...

then 100a = 99,999...

100a - a = 99a

Consider 0,99999...

Let a = 0,999...

then 100a = 99,999...

100a - a = 99a

and

99,999... - 0,999... = 99

Consider 0,99999...

Let a = 0,999...

then 100a = 99,999...

100a - a = 99a

and

99,999... - 0,999... = 99

so 99a = 99 and this means 

a=​99​​99​​=1

Another example

Practice!