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Lesson 15 · Irrational Numbers

Irrational Numbers

MathematicsSubject
14 minEstimated read
A gap the fractions cannot reach

If you can find a number whose decimals run on for ever without ever repeating, that number cannot be written as a fraction. Such numbers are called irrational.

The definition

An irrational number cannot be written as p over q. Its decimals go on for ever and never fall into a repeating block.

Where you meet them

The square root of 2 is the diagonal of a square with sides of 1 unit, so it is a length you can draw. Pi is another. The roots of 4, 9, 16 and 25 are not irrational, because those are perfect squares.

Pi is not 22 over 7

22/7 agrees with pi to two decimal places, but 22/7 is 3.142857 repeating while pi is 3.14159265 with no repeat anywhere. Using it in a calculation is fine; saying pi equals it is not true.

Why the square root of 2 cannot be a fraction

Suppose it could be, written in lowest terms. Squaring gives top squared equals twice bottom squared, so the top is even.

Substituting back shows the bottom is even too. But we said the fraction was in lowest terms. That is a contradiction, so no such fraction exists.

What kind of argument this is

Assume the opposite, follow the consequences honestly, and arrive somewhere impossible. So the starting point must have been false. This proof is around 2500 years old.

Real numbers

The rationals and irrationals together make the real numbers. The square root of 2 lies between 1.4 and 1.5, then between 1.41 and 1.42, narrowing for ever without landing exactly.

A long decimal is not automatically irrational

One seventeenth has a repeating block sixteen digits long, so on a short display it looks patternless, but it is rational. A calculator screen is far too short to settle the question.

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Worked Example 1

Trap the square root of 2 between two decimals with two places.

Start with whole numbers1² = 1 and 2² = 4, so the root is between 1 and 2
Narrow to one place1.4² = 1.96 and 1.5² = 2.25, so it is between 1.4 and 1.5
Narrow again1.41 < √2 < 1.42
What never happensNo decimal you ever write down squares to exactly 2.
Worked Example 2

Prove that the square root of 2 cannot be written as a fraction.

Assume the opposite√2 = a/b, in lowest terms
Square both sidesa² = 2b², so a² is even, so a is even
Write a as 2k4k² = 2b², so b² = 2k², so b is even too
The contradictionBoth even, so they share a factor of 2. Impossible. √2 is irrational.
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Activity 15.1

Draw a length that no fraction can measure

What you needSquared paper, a ruler, compasses, a sharp pencil. A calculator for the last step only.
MethodDraw a square of side 10 cm, draw a diagonal, measure it, then swing it down onto a number line with compasses.
The pointThe compass lands on an exact point, so the length certainly exists. Then check what the calculator says.
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1

An irrational number cannot be written as p over q with p and q integers.

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1What makes a number irrational, and why is a long decimal not enough on its own?
2Explain the proof that the square root of 2 is irrational.
3Is pi equal to 22 over 7? Explain carefully.
4How can you tell whether a square root is rational without working it out?
5If the square root of 2 cannot be written down exactly, in what sense does it exist?

Question 1 of 10

1An irrational number is one that
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Slide 1 of 4
Unit 15 · Irrational Numbers

Irrational Numbers

Mathematics · Grade 7

What You Will Be Able To Do

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Say what makes a number irrational, and what does NOT

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Decide instantly whether any square root is rational or not

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Follow the 2500-year-old proof that √2 is irrational

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Trap an irrational number between decimals, as closely as you like

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Every fraction terminates or recurs. So a number whose digits run on for ever with no repeating block cannot be a fraction, no matter how hard anyone looks.

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Presenter notes: Open by drawing a square of side one on the board and drawing its diagonal. Ask how long that diagonal is. Somebody will say about 1.4, so ask for the exact value and let the class try. They cannot find one, and today they will learn that nobody can, because no fraction and no finite decimal describes that length. The line is right there in front of them, which is what makes this unsettling and worth an hour.