Chapter 6 of 8

All 8 chapters
  1. 1Laws of powers, readunderstood
  2. 2Zero and negative powers, readunderstood
  3. 3Roots and fractional powers, readunderstood
  4. 4Standard form, readunderstood
  5. 5Calculating in standard form, readunderstood
  6. 6Rational and irrational numbers, readunderstood
  7. 7Surds, readunderstood
  8. 8Accuracy and bounds, readunderstood

M4.1.6 Chapter 6

Rational and irrational numbers

Is a decimal that never ends, such as 0.272727..., really a fraction?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Change a repeating decimal like 0.272727... into a fraction
  • Tell rational numbers from irrational ones like √2 and π

Leila knows that 1/3 = 0.333... The 3s never stop. She asks if every decimal that never stops is a fraction.

A rational number can be written as a fraction of two whole numbers. The whole numbers may be negative, and the bottom number is not 0. Whole numbers are rational, because 5 = 5/1. A decimal that stops is rational, because 0.75 = 3/4.

A decimal that repeats is rational too. A repeating decimal has a block of digits that goes round for ever, such as 0.272727... The block 27 is the repeat.

Now the method. Call the number x, so x = 0.272727... The repeat has 2 digits, so multiply by 10², which is 100. Then the tails line up.

A table with the head Line, Whole and Tail. Rows: 100x, 27, .2727...; x, 0, .2727...; 100x − x, 27, 0.LineWholeTail100x27.2727...x0.2727...100x − x270
The tails are the same, so taking x from 100x removes them.

Draw it first

From a repeating decimal to a fraction

  1. Step 1: the line x = 0.272727.... Step 2: the line 100x = 27.272727.... Step 3: the line 100x − x = 27. Step 4: the line x = 27/99 = 3/11.x = 0.272727...

    1 Call the number x. So x = 0.272727...

  2. Step 1: the line x = 0.272727.... Step 2: the line 100x = 27.272727.... Step 3: the line 100x − x = 27. Step 4: the line x = 27/99 = 3/11.x = 0.272727...100x = 27.272727...

    2 The repeat has 2 digits, so multiply by 100. That gives 100x = 27.272727...

  3. Step 1: the line x = 0.272727.... Step 2: the line 100x = 27.272727.... Step 3: the line 100x − x = 27. Step 4: the line x = 27/99 = 3/11.x = 0.272727...100x = 27.272727...100x − x = 27

    3 Take x away. The tails cancel, so 100x − x = 27. That is 99x = 27.

  4. Step 1: the line x = 0.272727.... Step 2: the line 100x = 27.272727.... Step 3: the line 100x − x = 27. Step 4: the line x = 27/99 = 3/11.x = 0.272727...100x = 27.272727...100x − x = 27x = 27/99 = 3/11

    4 Divide by 99. So x = 27/99, which simplifies to 3/11.

Worked example

Write 0.272727... as a fraction.

  1. Let x = 0.272727... The repeat 27 has 2 digits, so multiply by 100.
  2. 100x = 27.272727... Subtract x: 100x − x = 27.
  3. 99x = 27, so x = 27/99.
  4. Divide the top and bottom by 9: 27/99 = 3/11.

Answer: 3/11

Some repeats are long. 1/7 = 0.142857142857... has the 6 digit repeat 142857. Then you multiply by 10⁶, which is 1 000 000, so that the tails line up.

Worked example

Write 0.142857142857... as a fraction.

  1. Let x = 0.142857142857... The repeat 142 857 has 6 digits, so multiply by 10⁶.
  2. 1 000 000x − x = 142 857.142857... − 0.142857..., which is 142 857.
  3. So 999 999x = 142 857, and x = 142 857/999 999.
  4. Divide the top and bottom by 142 857. The bottom becomes 7, so x = 1/7.

Answer: 1/7

Some numbers cannot be written as a fraction at all. An irrational number cannot be written as a fraction of two whole numbers. The numbers √2 and π are irrational. Their decimals never end and never repeat.

Take a square with sides of 1 cm. Its area is 1 cm². The diagonal cuts it into 2 equal triangles. Now draw a square on the diagonal. The square on the diagonal is made of 4 of those triangles.

On the left, a square with sides of 1 cm, labelled 1 cm, cut by its diagonal into 2 equal triangles, one of them hatched. Under it, the labels area 1 cm² and 2 triangles. On the right, a tilted square drawn on that diagonal, with the side labelled √2 cm. Two cuts across it make 4 of the same triangles, one of them hatched. Under it, the labels area 2 cm² and 4 triangles.1 cm√2 cmarea 1 cm²area 2 cm²2 triangles4 triangles
The square on the diagonal is made of 4 triangles, and the small square of 2 of them. So its area is 2 cm².

So the square on the diagonal has area 2 cm². Its side, which is the diagonal, is the positive number that gives 2 when multiplied by itself. That number is √2 cm.

A square with sides of 1 cm. A dashed diagonal joins two opposite corners and is labelled √2 cm. The sides are labelled 1 cm and 1 cm.1 cm1 cm√2 cm
The diagonal of a square with sides of 1 cm is exactly √2 cm. Mathematicians have proved that √2 is irrational.

Irrational does not mean a long decimal. 1/7 has a long decimal, but it is rational. It means no fraction of whole numbers gives the number exactly. Here are rational and irrational numbers on the number line.

A number line from 0 to 4 with whole number ticks. Points mark 1/2, √2 and π, each at its true place.012341/2√2π
1/2 is rational. √2 and π are irrational. All three sit on the line.

Here is a rule for roots. The square root of a whole number is either a whole number or irrational. Mathematicians have proved this.

Worked example

Say whether each number is rational or irrational: √36, 0.2, √5, π.

  1. 6 × 6 = 36, so √36 = 6. And 6 = 6/1, so √36 is rational.
  2. 0.2 is rational, because 0.2 = 2/10 = 1/5.
  3. 5 is not a square number, so √5 is not a whole number. So √5 is irrational. Its decimal starts like this: √5 = 2.236...
  4. π = 3.14159..., which also never ends or repeats, so π is irrational.

Answer: Rational: √36 and 0.2. Irrational: √5 and π.

A table with the head Number and Kind. Rows: 0.75, rational; 0.333..., rational; √9, rational; √2, irrational; π, irrational.NumberKind0.75rational0.333...rational√9rational√2irrationalπirrational
0.75 = 3/4, 0.333... = 1/3 and √9 = 3 = 3/1. None of these fractions are possible for √2 or π.

See it three ways

With things

Divide 1 by 3 on paper, and divide 1 by 4. See what the digits do.

As a picture

A table with the head Fraction and Decimal. Rows: 1/4, 0.25; 1/8, 0.125; 1/3, 0.333...; 2/11, 0.1818....FractionDecimal1/40.251/80.1251/30.333...2/110.1818...

In symbols

1/4 = 0.25 stops. 1/3 = 0.333... repeats. Every fraction of whole numbers does one or the other.

Try it

Divide 1 by 7 on paper

You need: paper and a pencil.

  1. Divide 1 by 7 with long division. Write each digit and each remainder.
  2. Carry on until the remainder is 1 again.
  3. Look at the digits you wrote.

What you will see: The remainders are 3, 2, 6, 4, 5, 1. When the remainder is 1 again, the digits go round again: 0.142857142857... There are only 6 possible remainders, so a fraction must repeat or stop.

If you cannot try it, this is what happens.
A table with the head Digit and Remainder. Rows: 1, 3; 4, 2; 2, 6; 8, 4; 5, 5; 7, 1.DigitRemainder134226845571

A wrong idea many people have

A number with a very long decimal must be irrational.

Why it feels right: A decimal with many digits looks as if it will never end.

The test that breaks it: 1/7 = 0.142857142857... is long, but it is a fraction. And 0.123456789 stops, so it is 123 456 789/1 000 000 000.

What is true: Irrational means the number cannot be written as a fraction of two whole numbers. Length has nothing to do with it.

A wrong idea many people have

0.999... is a little less than 1, because it never reaches 1.

Why it feels right: Each 9 added gets closer, but the line of 9s never seems to arrive.

The test that breaks it: Use the method. Let x = 0.999... Then 10x = 9.999... and 10x − x = 9, so 9x = 9 and x = 1.

What is true: 0.999... is exactly 1. The method shows that every repeating decimal is a fraction, and 9/9 is 1.

Key words

rational number
a number that can be written as a fraction of two whole numbers, such as 3/4 or −3/4. The bottom number is not 0.
irrational number
a number that cannot be written as a fraction of two whole numbers, such as √2 and π.
repeating decimal
a decimal with a block of digits that goes round for ever, such as 0.272727...

Back to your guess

The answer is A: Yes. A repeating decimal can always be written as a fraction of two whole numbers.

A decimal that repeats can always be written as a fraction of two whole numbers. For 0.272727... it is 3/11. A decimal that never ends and never repeats cannot be, and that is what irrational means.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Which of these numbers is irrational?

    Tap your answer.

    1. Not this one. 49 = 7 × 7, so √49 = 7. That is the fraction 7/1, so it is rational. Go to M4.1.6. Try another.

    2. Yes, that is right. 50 is not a square number, so √50 is not a whole number. The square root of a whole number is a whole number or irrational, so √50 is irrational.

    3. Not this one. This is the repeating decimal for 1/3. It is a fraction of whole numbers, so it is rational. Go to M4.1.6. Try another.

    See the answer

    √50

    50 is not a square number, so √50 is not a whole number. The square root of a whole number is a whole number or irrational, so √50 is irrational.

    • If you chose A: 49 = 7 × 7, so √49 = 7. That is the fraction 7/1, so it is rational. Go to M4.1.6.
    • If you chose C: This is the repeating decimal for 1/3. It is a fraction of whole numbers, so it is rational. Go to M4.1.6.

  2. Write 0.545454... as a fraction in its simplest form.

    Tap your answer.

    1. Not this one. This is 0.54, which stops. The repeat needs the bottom number 99, which is one less than 100. Go to M4.1.6. Try another.

    2. Not this one. This is equal to the answer, but it is not in simplest form. Divide the top and bottom by 9. Go to M4.1.6. Try another.

    3. Yes, that is right. The repeat 54 has 2 digits, so 99x = 54. Then x = 54/99 and dividing the top and bottom by 9 gives 6/11.

    See the answer

    6/11

    The repeat 54 has 2 digits, so 99x = 54. Then x = 54/99 and dividing the top and bottom by 9 gives 6/11.

    • If you chose A: This is 0.54, which stops. The repeat needs the bottom number 99, which is one less than 100. Go to M4.1.6.
    • If you chose B: This is equal to the answer, but it is not in simplest form. Divide the top and bottom by 9. Go to M4.1.6.

  3. What does it mean to say a number is irrational?

    Tap your answer.

    1. Yes, that is right. That is the whole definition. The decimal of an irrational number never ends and never repeats, but the fraction test is what counts.

    2. Not this one. 1/7 has a long decimal and is rational. Irrational is about fractions, not about length. Go to M4.1.6. Try another.

    3. Not this one. A decimal that never stops can repeat, such as 0.333..., and that is rational. An irrational number has a decimal that never ends and never repeats. Go to M4.1.6. Try another.

    See the answer

    It cannot be written as a fraction of two whole numbers.

    That is the whole definition. The decimal of an irrational number never ends and never repeats, but the fraction test is what counts.

    • If you chose B: 1/7 has a long decimal and is rational. Irrational is about fractions, not about length. Go to M4.1.6.
    • If you chose C: A decimal that never stops can repeat, such as 0.333..., and that is rational. An irrational number has a decimal that never ends and never repeats. Go to M4.1.6.

  4. The picture shows a square with sides of 1 cm. Which statement about its diagonal is right?

    A square with sides of 1 cm. A dashed diagonal joins two opposite corners and is labelled √2 cm. The sides are labelled 1 cm and 1 cm.1 cm1 cm√2 cm
    Tap your answer.

    1. Not this one. 1.4 × 1.4 = 1.96, not 2. A decimal like 1.4 is only close to √2. The exact value is not a fraction of whole numbers. Go to M4.1.6. Try another.

    2. Not this one. The 2 is the area of the square built on the diagonal. The length of the diagonal is the side of that square, √2 cm. Go to M4.1.6. Try another.

    3. Yes, that is right. The square on the diagonal has area 2 cm², so the diagonal is √2 cm. √2 is irrational.

    See the answer

    The diagonal is √2 cm, which cannot be written exactly as a fraction of whole numbers.

    The square on the diagonal has area 2 cm², so the diagonal is √2 cm. √2 is irrational.

    • If you chose A: 1.4 × 1.4 = 1.96, not 2. A decimal like 1.4 is only close to √2. The exact value is not a fraction of whole numbers. Go to M4.1.6.
    • If you chose B: The 2 is the area of the square built on the diagonal. The length of the diagonal is the side of that square, √2 cm. Go to M4.1.6.

  5. x = 0.272727... The repeat 27 has 2 digits. Why do we multiply by 100 and not by 10?

    Tap your answer.

    1. Not this one. 10x = 2.7272... and x = 0.2727... have tails that are one digit out of step. Subtracting does not cancel them. Go to M4.1.6. Try another.

    2. Yes, that is right. A shift of 2 places moves the repeat exactly one block along, so 100x and x have the same tail.

    3. Not this one. Multiplying by 2 gives 2x = 0.5454..., which does not move the repeat along. The point must shift 2 places, and that means multiplying by 10², which is 100. Go to M4.1.6. Try another.

    See the answer

    The repeat has 2 digits, so shifting the point 2 places lines up the repeating tails.

    A shift of 2 places moves the repeat exactly one block along, so 100x and x have the same tail.

    • If you chose A: 10x = 2.7272... and x = 0.2727... have tails that are one digit out of step. Subtracting does not cancel them. Go to M4.1.6.
    • If you chose C: Multiplying by 2 gives 2x = 0.5454..., which does not move the repeat along. The point must shift 2 places, and that means multiplying by 10², which is 100. Go to M4.1.6.

  6. From an earlier chapter

    Work out (2 × 10³) × (3 × 10⁴) in standard form. What is the power of ten?

    See the answer

    7

    2 × 3 = 6 and 3 + 4 = 7, so the answer is 6 × 10⁷.

  7. Write 0.4444... as a fraction in its simplest form.

    See the answer

    4/9

    The repeat has 1 digit, so multiply by 10. Then 10x − x = 4, so 9x = 4 and x = 4/9.

  8. How many of these numbers are irrational: √16, √20, π, 0.5, √7?

    See the answer

    3

    √16 = 4 is rational. 0.5 = 1/2 is rational. 20 and 7 are not square numbers, so √20 and √7 are irrational, and π is irrational. That makes 3.

  9. Let x = 0.999... What whole number is x equal to?

    See the answer

    1

    The repeat has 1 digit, so multiply by 10: 10x = 9.999... Then 10x − x = 9, so 9x = 9 and x = 1.

Keep it

Copy this into your exercise book.

Rational: a fraction of whole numbers. Repeating decimals are rational: multiply by 10 to the repeat length, subtract x. Irrational: no such fraction, like √2 and π.

Draw: x = 0.2727..., then 100x = 27.2727..., then 99x = 27, and x = 3/11.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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