50 Terminating or Repeating?
You’ve seen that when you write a fraction as a decimal, sometimes the decimal terminates, like:
However, some fractions have decimal representations that go on forever in a repeating pattern, like:
It’s not totally obvious, but it is true: Those are the only two things that can happen when you write a fraction as a decimal.
Of course, you can imagine (but never write down) a decimal that goes on forever but doesn’t repeat itself, for example:
But these numbers can never be written as a nice fraction where and are whole numbers. They are called irrational numbers. The reason for this name: Fractions like are also called ratios. Irrational numbers cannot be expressed as a ratio of two whole numbers.
For now, we’ll think about the question: Which fractions have decimal representations that terminate, and which fractions have decimal representations that repeat forever? We’ll focus just on unit fractions.
Definition
A unit fraction is a fraction that has 1 in the numerator. It looks like for some whole number .
Think / Pair / Share
- Which of the following fractions have infinitely long decimal representations and which do not?
- Try some more examples on your own. Do you have a conjecture?
A fraction has an infinitely long decimal expansion if:
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Problem 7
Complete the table below which shows the decimal expansion of unit fractions where the denominator is a power of 2. (You may want to use a calculator to compute the decimal representations. The point is to look for and then explain a pattern, rather than to compute by hand.)
Try even more examples until you can make a conjecture: What is the decimal representation of the unit fraction ?
| Fraction | Denominator | Decimal |
|---|---|---|
Problem 8
Complete the table below which shows the decimal expansion of unit fractions where the denominator is a power of 5. (You may want to use a calculator to compute the decimal representations. The point is to look for and then explain a pattern, rather than to compute by hand.)
Try even more examples until you can make a conjecture: What is the decimal representation of the unit fraction ?
| Fraction | Denominator | Decimal |
|---|---|---|
Marcus noticed a pattern in the table from Problem 7, but was having trouble explaining exactly what he noticed. Here’s what he said to his group:
I remembered that when we wrote fractions as decimals before, we tried to make the denominator into a power of ten. So we can do this:
When we only have 2’s, we can always turn them into 10’s by adding enough 5’s.
Think / Pair / Share
- Write out several more examples of what Marcus discovered.
- If Marcus had the unit fraction , what would be his first step to turn it into a decimal? What would the decimal expansion look like and why?
- Now think about unit fractions with powers of 5 in the denominator. If Marcus had the unit fraction , what would be his first step to turn it into a decimal? What would the decimal ex