Real Numbers — Print

Every number you meet this year is either rational — a fraction — or irrational. Knowing which tells you how it behaves.

What the practice questions cover

These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

Draw a square one unit wide. How long is the diagonal across it?

By Pythagoras it is √2. So write that as a decimal.

a square one unit wide — how long is the diagonal? 1 1 ? it is not 1 it is not 1.4 it is not 1.41 and it never settles 1 1.4 √2 2 the decimals close in on it but never arrive

It is not 1.4, because 1.42 = 1.96. Not 1.41 either — that squares to 1.9881.

Keep going and the decimals get closer but never arrive. There is no fraction that gives it exactly.

And yet the diagonal is right there on the page. You can measure it with string.

So some perfectly real lengths cannot be written as a fraction. Those are the irrational numbers, and together with the fractions they fill the number line completely — with no gaps left over.

Real numbers: nested rational sets beside the irrationals REAL NUMBERS rational integers whole 0, 1, 2, 3, … irrational π √2 never repeats integers add the negatives (−5) · rationals add the fractions (¾, 0.6) every whole number is also an integer, and also rational — the rings nest

The families, nested inside each other

Counting numbers: 1, 2, 3, …

Whole numbers: the counting numbers plus 0.

Integers: whole numbers and their negatives.

Rationals: anything expressible as a fraction — which includes every integer, since 5 = 51.

Reals: the rationals together with the irrationals.

Each family sits inside the next, so -3 is an integer and a rational and a real, all at once.

How to tell rational from irrational

Write it as a decimal and watch what happens.

Stops — like 0.75 — rational.

Repeats a block forever — like 0.3333… or 0.142857142857… — rational.

Goes on forever with no repeating block — like π = 3.14159… or √2 = 1.41421… — irrational.

The root test

The square root of a perfect square is rational: √49 = 7. The square root of any other whole number is irrational: √50 never resolves.

Turning repeating decimals into fractions

0.3 means 0.333… forever. Let x = 0.333…, then 10x = 3.333…

Subtract: 10x - x = 3, so 9x = 3 and x = 13.

The trick is choosing the multiplier so the repeating tails line up and cancel.

Comparing and ordering real numbers

Convert everything to decimals, then compare. 78 = 0.875, √0.81 = 0.9, and 0.87 — so the order is 0.87 < 78 < √0.81.

For irrationals, use a good estimate: π ≈ 3.14 and √10 ≈ 3.16, so π < √10.

Careful with negatives: -34 > -0.8, because -0.75 sits to the right of -0.8.

One form at a time

Never compare a fraction against a decimal against a root in your head. Put them all in the same costume — decimals are usually easiest — and then compare.

Why π is not 227

227 = 3.142857… and repeats, so it is rational. π = 3.14159… and never repeats.

227 is a convenient approximation, close to π but not equal to it. The same goes for 3.14.

Worked examples

Classify

Classify each: -7, 25, √16, √20, 0.6.

  1. -7: integer, and rational since -7 = -71. Every integer is rational.
  2. 25 = 0.4: rational, terminating. Already a fraction.
  3. √16 = 4: integer and rational. 16 is a perfect square, so the root is whole.
  4. √20 ≈ 4.472…: irrational. 20 is not a perfect square.
  5. 0.6 = 23: rational. A repeating decimal is always a fraction in disguise.

All rational except √20, which is irrational.

Repeating to fraction

Write 0.27 as a fraction in lowest terms.

  1. Let x = 0.272727… Name the number so you can operate on it.
  2. The block is two digits, so multiply by 100: 100x = 27.272727… Now the tails match exactly.
  3. Subtract: 100x - x = 27, so 99x = 27 The infinite tails cancel — that is the whole trick.
  4. x = 2799 = 311 Divide top and bottom by 9.

311

Order them

Put in order from least to greatest: 56, 0.8, √0.64, 45.

  1. Convert everything to decimals. One costume makes comparison safe.
  2. 56 ≈ 0.833, 0.8, √0.64 = 0.8, 45 = 0.8 Three of them are exactly equal.
  3. So 0.8 = √0.64 = 45 < 56 Only 56 is larger.

0.8 = √0.64 = 45 < 56

Words to know

Rational
Any number writable as ab with integers a and b, b ≠ 0.
Irrational
A real number that cannot be written as a fraction — its decimal never repeats and never ends.
Integer
A whole number, positive, negative or zero.
Terminating / repeating
A decimal that stops, or one that repeats a block forever.

Mistakes to watch for

Calling every square root irrational. √25 is 5 — perfectly rational. A root is irrational only when the number under it is not a perfect square.

Treating 227 or 3.14 as equal to π. They are approximations, and good ones, but π has no exact fraction at all. Writing = where you mean ≈ quietly claims something false.

Mis-ordering negatives. -0.8 is less than -0.75, because it lies further left on the number line. With negatives, bigger digits mean a smaller number.

Assuming a long decimal must be irrational. 17 = 0.142857142857… runs forever but repeats, and anything that repeats is a fraction. Irrational means never repeating.

The Pythagoreans believed every length was a ratio of whole numbers. The diagonal of a square destroyed that belief, and the proof is short enough to fit on a postcard.

Legend says the discoverer, Hippasus, was thrown overboard at sea for revealing it — though the story may have been invented centuries later.

π was proved irrational only in 1761, by Johann Lambert. Nobody had managed it in the previous two thousand years of trying.

Problem 1

Is 0.75 rational?

  1. Yes
  2. No
Worked answer

Yes

0.75 = 34, so yes.

Problem 2

Classify √9.

  1. Rational
  2. Irrational
Worked answer

Rational

√9 = 3, a whole number, so rational.

Problem 3

Classify √7.

  1. Rational
  2. Irrational
Worked answer

Irrational

7 is not a perfect square

√7 = 2.6457… never repeats, so irrational.

Problem 4

Is -5 an integer?

  1. Yes
  2. No
Worked answer

Yes

Integers are the whole numbers and their negatives, so yes.

Problem 5

Write 0.3 as a fraction in lowest terms.

Worked answer

1/3

0.333… = 13

Problem 6

Which of these is irrational?

  1. 0.25
  2. 23
  3. √16
  4. √11
Worked answer

√11

0.25 and 23 are fractions, so rational

√16 = 4, rational

11 is not a perfect square, so √11 is irrational.

Problem 7

Write 0.27 as a fraction in lowest terms.

Worked answer

3/11

Let x = 0.2727…, so 100x = 27.2727…

99x = 27

x = 2799 = 311

Problem 8

Which is largest: 34, 0.7, or √0.64?

  1. 34
  2. 0.7
  3. √0.64
Worked answer

√0.64

34 = 0.75

0.7

√0.64 = 0.8

0.8 is the largest.

Problem 9

Is 227 exactly equal to π?

  1. Yes
  2. No
Worked answer

No

227 = 3.142857… and repeats, so it is rational

π = 3.14159… never repeats

They are close but not equal.

Problem 10

Which describes 0?

  1. Integer and rational
  2. Irrational
  3. Neither
Worked answer

Integer and rational

0 is a whole number and an integer

It is rational because 0 = 01.

Problem 11

Write 0.6 as a fraction in lowest terms.

Worked answer

2/3

10x = 6.666… and x = 0.666…

9x = 6

x = 69 = 23

Problem 12

Which of these lies between 3 and 4?

  1. √5
  2. √10
  3. √17
  4. √20
Worked answer

√10

32 = 9 and 42 = 16

10 lies between 9 and 16

√10 ≈ 3.16

Problem 13

Write 0.45 as a fraction in lowest terms.

Worked answer

5/11

100x - x = 45

99x = 45

x = 4599 = 511

Problem 14

Which is smallest: π, 227, √10, or 3.1?

  1. π
  2. 227
  3. √10
  4. 3.1
Worked answer

3.1

3.1

π ≈ 3.1416

227 ≈ 3.1429

√10 ≈ 3.1623

3.1 is smallest.

Problem 15

√150 lies between two consecutive integers. What is the smaller one?

Worked answer

12

144 < 150 < 169

So 12 < √150 < 13

Problem 16

Is the sum of two irrational numbers always irrational?

  1. Yes, always
  2. No — it can be rational
Worked answer

No — it can be rational

√2 + (-√2) = 0, which is rational

So the sum is not always irrational

It usually is — but "always" is too strong.

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