Unit 1: The Real Number System

Topic 1: Expressing Repeating Decimals as Fractions

Competencies

· Express repeating decimals as fraction

· Show that repeating decimals are also rational numbers

Suggested ways of teaching this topic: Presentation and Explanation by the Teacher

This is a formal teaching method which involves the teacher presenting and explaining mathematics to the whole class. It can be difficult because you have to ensure that all students understand. This can be a very effective way of:

· Teaching a new piece of mathematics to a large group of students

· Drawing together everyone’s understanding at certain stages of a topic

· Summarizing what has been learnt

As a brain storming activity you may ask students to

· List the types of decimals they know

· Give some examples of repeating decimals

Then, you might start with a question like:

‘How can I express $x=4.1\overline{256}$ as a fraction?’

Then, using examples, you can show them how!

Lesson Notes

Let $x=4.1\overline{256}$

Multiplying the above equation by 1000(where there are 3 repeating decimals), we can have:

$1000x=4125.6\overline{256}$

Subtracting x from 1000x, we get

$$ \begin{align*} 1000x-x&=4121.5 \\ 999x&=4121.5 \\ x&= \frac {4121.5}{999} \\ &= \frac {4121 + \frac {1}{2}}{999} \\ &= \frac {\frac {8243}{2}}{999} \\ &= \frac {8243}{1998} = 4.1\overline{256} \end{align*}$

Example 1:

Find the fraction represented by the repeating decimal .

Let n stand for or 0.77777 … So 10 n stands for or 7.77777 …

10 n and n have the same fractional part, so their difference is an integer.

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You can solve this problem as follows.

So,

Example 2:

Find the fraction represented by the repeating decimal .

Let n stand for or 0.363636 … So 10 n stands for or 3.63636 … and 100 n stands for or 36.3636 …

100 n and n have the same fractional part, so their difference is an integer. (The repeating parts are the same, so they subtract out.)

equation

You can solve this equation as follows:

equation Now simplify to . So

Example 3:

Find the fraction represented by the repeating decimal .

Let n stand for or 0.544444 … So 10 n stands for or 5.444444 … and 100 n stands for or 54.4444 …

Since 100 n and 10 n have the same fractional part, their difference is an integer. (Again, notice how the repeated parts must align to subtract out.)

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