Filling in the gaps

A rational number is any number that can be written as one integer divided by another, and adding them fills the gaps on the line.

The definition, written carefully

A rational number has the form p over q, where p and q are integers and q is not zero. Rational comes from ratio, not from being reasonable.

Which numbers are already rational

Every integer is rational, because 7 is 7 over 1. Every terminating decimal is, because 0.75 is 75 over 100. The new set does not replace the old ones; it contains them.

One number, many faces

Two thirds, four sixths and 0.666 recurring are one single point on the number line wearing different costumes.

Putting a rational number on the line

The bottom number tells you how many equal pieces to cut each unit into; the top number tells you how many to count.

Comparing without guessing

Give them the same bottom number and compare the tops, or turn both into decimals. Five sevenths is 50 over 70; seven tenths is 49 over 70.

Rational numbers as decimals

Either the division finishes, giving a terminating decimal, or a block of digits repeats for ever. There is no third possibility.

Why it must terminate or repeat

Dividing by 7, the only remainders are 0 to 6. Within seven steps some remainder must repeat, and from then the digits repeat too.

Do not round and call it equal

A calculator shows one third as 0.3333333, which is a rounded copy. Multiply it by 3 and you get 0.9999999 rather than 1.

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Worked Example 1

Show that 0.75, 3/4 and 15/20 are the same rational number.

Reduce each to lowest terms75/100 ÷ 25 = 3/4, and 15/20 ÷ 5 = 3/4

All three reduce to the same thingOne point on the number line written three ways.

Why reducing worksDividing top and bottom by the same number is dividing by 1, which changes nothing.

Worked Example 2

Arrange 2/3, 5/8 and 7/12 in order, smallest first.

Find a common bottomLCM of 3, 8 and 12 is 24

Rewrite2/3 = 16/24, 5/8 = 15/24, 7/12 = 14/24

Now the tops decide it7/12 < 5/8 < 2/3

Check with decimals0.583, 0.625, 0.667. Two independent methods agreeing is worth the extra thirty seconds.

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Activity 14.1

Hunt for the repeating block

What you needPaper and pencil only. A calculator hides the remainders, which are the whole point.

MethodDo the long division, writing down the remainder at every step, until a remainder you have already seen comes back.

Then predictBefore dividing by 13, predict the longest the block could be, then check.

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A rational number has the form p over q, where p and q are integers and q is not zero.

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1What is a rational number, and why must the bottom number not be zero?

2Explain why every integer and every terminating decimal is a rational number.

3Why must the decimal form of a rational number either terminate or recur?

4How can you tell, without dividing, whether a fraction will give a terminating decimal?

5Which is larger, five sevenths or seven tenths, and why is guessing from the digits unsafe?

Question 1 of 10

1A rational number is a number of the form

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Unit 14 · Rational Numbers

Rational Numbers

Mathematics · Grade 7

What You Will Be Able To Do

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Say exactly what a rational number is, and why the bottom cannot be zero

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Place any rational number on the number line

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Compare and order them without guessing from the digits

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Say why every one of them terminates or recurs as a decimal

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Two thirds, four sixths and 0.666 recurring are not three numbers that happen to be close together. They are one single point on the number line wearing three different costumes.

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Presenter notes: Start with the gap left at the end of the last unit. Draw the integer line and ask the class to point at a number between 1 and 2. There is none, and yet half a kilogram of sugar exists and two thirds of a field exists. Ask what they would call the number halfway between 1 and 2, and somebody will say one and a half without hesitating. They already use these numbers daily; today they get a name and a definition.