Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Year 7, Unit 5, lesson 7) Write 1/4 as a decimal.
Answer
0.25, because 1 ÷ 4 = 0.25. - (Year 7, Unit 1, lesson 2) What is 25?
Answer
32, because 2 × 2 × 2 × 2 × 2 = 32. - (Year 7, Unit 1, lesson 3) Is 91 a prime number?
Answer
No: 91 = 7 × 13, so it has factors other than 1 and itself.
Words used in this lesson
- rational number
- a number that can be written as a fraction of two integers. For example: 3/8, −5 and 0.333….
- irrational number
- a number that cannot be written as a fraction; its decimal never ends or repeats. For example: π and √2.
- real numbers
- all the rational and irrational numbers together. For example: every point on the number line.
- terminating decimal
- a decimal that ends. For example: 0.375.
- repeating decimal
- a decimal in which a block of digits repeats forever. For example: 0.142857142857….
- integer
- a whole number, positive, negative or zero. For example: −7, 0 and 12.
- natural number
- a counting number; the IB Diploma includes 0, and some courses start at 1. For example: 0, 1, 2, 3.
A decimal that never ends
Divide 1 by 7 and you get 0.142857142857… The digits go on and on. The square root of 2 is 1.41421356… and again the digits go on. A square of side 1 m has a diagonal of √2 m. Can any ruler marked in fractions of a metre land exactly on it? By the end of the lesson you can say. Predict first: is 1/6 as a decimal one that ends or one that goes on forever? Look at the 6 on the bottom, then move the sliders to test your guess on other fractions.
3/8 = 0.375: the decimal ends after 3 places.
Show me the steps
- 3/8 = 0.375. 8 = 2 × 2 × 2, only 2s, so the decimal ends after 3 places.
- 1/3 = 0.333...: 3 is not a 2 or a 5, so the block 3 repeats forever.
- 1/7 = 0.142857142857...: a block of 6 digits repeats forever.
- 1/6 = 0.1666...: 6 = 2 × 3, and the 3 makes it repeat, after the digit 1.
- 7/20 = 0.35. 20 = 2 × 2 × 5, only 2s and 5s, so it ends.
- 9/11 = 0.8181...: the block 81 repeats forever.
The pattern is not luck. Dividing by a number b can leave only the remainders 1 up to b − 1, so sooner or later a remainder comes round again, and from there the digits repeat. Dividing 1 by 6 leaves remainder 4, then 4 again, so the 6 repeats. A decimal ends only when the remainder reaches 0, and that happens exactly when the denominator, in simplest form, is built from 2s and 5s, the factors of 10: 1/6 = 0.1666… because 6 = 2 × 3 and the 3 never divides into a power of 10.
A rational number is a number that can be written as a fraction p/q of two integers, with q not 0. Its decimal always ends or repeats. In simplest form the decimal ends exactly when the denominator has no prime factor other than 2 and 5.
An irrational number cannot be written as a fraction. Its decimal neither ends nor repeats: π, √2 and √5 are irrational. The rational and irrational numbers together make the real numbers.
Where the numbers sit
Every integer is rational, because 4 = 4/1. Every natural number is an integer, and every rational number is real. The sets nest, and the irrational numbers are the real numbers left outside the rational ones.
The exception to watch: a square root is not automatically irrational. √49 = 7 is a natural number, and √2 is not. The square root of a whole number is rational exactly when that whole number is a perfect square. So √2 is not a fraction, and no ruler marked in fractions of a metre can land exactly on the diagonal of the 1 m square. Some courses start the natural numbers at 1; the IB Diploma includes 0, as this diagram does.
A common mistake: “It goes on forever, so it is irrational.” 0.333… goes on forever and is the fraction 1/3. Going on forever is not the test. The test is whether a block of digits repeats: a repeating decimal is rational, and a decimal with no repeating block is irrational.
Worked example: will it end?
Does 7/40 end or repeat? Write it as a decimal.
- 7/40 is in simplest form, because 7 and 40 share no factor. 40 = 2 × 2 × 2 × 5: only 2s and 5s, so it ends.
- Make the bottom a power of 10: 7/40 = 175/1000, because 7 × 25 = 175 and 40 × 25 = 1000.
- So 7/40 = 0.175, which ends after 3 places.
Does 11/30 end or repeat?
- 30 = 2 × 3 × 5. The factor 3 is not a 2 or a 5, so the decimal repeats.
- Divide: 11 ÷ 30 = 0.3666…, with the block 6 repeating after the digit 3.
Your go. The first step is given. Does 13/50 end or repeat? If it ends, write it as a decimal.
- 50 = 2 × 5 × 5: only 2s and 5s, so it ends.
- Now make the bottom 100, then check.
Check the answer
13/50 = 26/100, because 13 × 2 = 26 and 50 × 2 = 100. So it ends: 0.26.
Check before you go on
1. Which of these is a rational number?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A recipe uses 7/16 of a cup of oil. Write 7/16 as a decimal.
3. Which of these fractions, all in simplest form, has a decimal that ends?
Level 2 · Building
4. 5/37 = 0.135135135… How many digits are in the repeating block?
5. 11/80 is in simplest form, and 80 = 2 × 2 × 2 × 2 × 5. Its decimal ends. How many places does it have after the point?
6. Sana says: “0.2666… goes on forever, so it must be an irrational number.” Which reply is right?
Level 3 · Stretch
7. How many of these six numbers are irrational? 6, √16, √7, 0.2424… (the block 24 repeats), π, √12
8. For how many whole numbers d from 2 to 20 does 1/d have a decimal that ends?
Something to think about
Find a fraction whose decimal repeats a block of exactly 2 digits. What did you look for in the denominator?
One way to think about it
1/11 = 0.0909..., 1/33 = 0.0303... and 1/99 = 0.0101... all repeat a 2-digit block. Each denominator divides 99 but does not divide 9. A denominator such as 3 or 9 divides 9 and gives a 1-digit block, as in 1/3 = 0.333.... Check by long division.
Exit ticket
9. Which of these is NOT a rational number?