Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 8: write 7/20 as a decimal.
Answer
0.35. Multiply the top and bottom by 5 to get 35/100. - From Year 9: is √45 rational or irrational, and between which two whole numbers does it lie?
Answer
Irrational. 6² = 36 and 7² = 49, so it lies between 6 and 7. - Simplify 18/48.
Answer
3/8. Divide the top and bottom by 6.
Words used in this lesson
- recurring decimal
- a decimal whose digits repeat in a block for ever. For example: 0.8333….
- closed
- an operation on members of a set always gives a member of the same set. For example: whole numbers are closed under +.
- contradiction
- two statements that cannot both be true. For example: p is even and p is odd.
- proof by contradiction
- assume the opposite of what you want to show and reach a contradiction. For example: assume √5 = p/q.
- lowest terms
- a fraction with no common factor in its top and bottom. For example: 7/12.
- counter-example
- one case that shows a statement is not always true. For example: 2, for “every prime number is odd”.
An endless decimal that is exactly a fraction
On a calculator, 1 ÷ 7 shows 0.142857142857… and the digits seem to go on for ever. A decimal that never stops sounds as if it cannot be a fraction, yet 1/7 is one. What decides whether an endless decimal is rational is whether its digits repeat.
Predict first: would you expect 0.636363… to have a denominator of 11, 63 or 99? And is √3 + (−√3) rational or irrational? Use the two figures to test your prediction.
x = 0.777777…; 10x − x = 7, so 9x = 7 and x = 7/9.
Show me the steps
- 0.777777…: one digit repeats, so multiply by 10. The digits after the point in 10x and in x are the same endless string of 7s, so 10x − x = 7 and x = 7/9.
- 0.454545…: two digits repeat, so multiply by 100. 100x − x = 45, so 99x = 45 and x = 45/99, which is 5/11 in lowest terms.
- 0.123123…: three digits repeat, so multiply by 1000. 1000x − x = 123, so x = 123/999, which is 41/333.
- 0.166666…: only the 6 repeats, and the 1 does not. Multiply by 10 and by 100 so that the tails line up: 100x − 10x = 15, so 90x = 15 and x = 15/90 = 1/6.
3 + 1/2 = 7/2, which is rational.
Show me the steps
- Start with 3 + 1/2. Both are rational, and the sum 7/2 is rational: ℚ is closed under addition.
- Set a to 3 and b to √3 and add. A rational plus an irrational number is irrational: if the sum were a rational r, then √3 = r − 3 would be rational.
- Set a to √3 and b to −√3 and add. Two irrational numbers, and the sum is 0, which is rational. The irrational numbers are not closed under addition.
- Press a × b with a = √3 and b = 2√3. Two irrational numbers again, and the product is 6, a rational number.
A recurring decimal is a fraction. Let x be the decimal and let k be the length of its repeating block. Multiply by 10k, so the tails line up, subtract x, and divide. If some digits come before the block, multiply by two powers of ten that line up the tails.
Every rational number is a terminating or recurring decimal, and every such decimal is rational. Irrational numbers have decimals that neither stop nor repeat. A set is closed under an operation when the result always stays in the set. ℚ is closed under +, −, × and ÷ (by a non-zero number); the irrational numbers are not closed under any of them.
To prove, not just notice, assume the opposite and follow it to a contradiction.
Two traps. “An irrational plus an irrational is irrational” is false: √7 and 1 − √7 are both irrational and they add to 1. And a pattern in the first few digits proves nothing: 0.1010010001… looks regular, but the number of zeros keeps growing, so no block repeats and the number is irrational.
Worked examples
1. Write 0.4333… as a fraction in lowest terms.
- The 4 does not repeat; the 3 does. Let x = 0.4333…, so 10x = 4.333… and 100x = 43.333…
- The tails match, so subtract: 100x − 10x = 43.333… − 4.333… = 39.
- 90x = 39, so x = 39/90 = 13/30.
2. Prove that √5 is irrational.
- Suppose √5 = p/q, a fraction in lowest terms, so p and q have no common factor.
- Squaring: 5 = p²/q², so 5q² = p². Then 5 divides p², and since 5 is prime, 5 divides p. Write p = 5k.
- Then 5q² = 25k², so q² = 5k². Now 5 divides q² and so 5 divides q.
- Both p and q are multiples of 5, which contradicts “lowest terms”. So √5 is irrational.
3. Show that 3√2 is irrational, given that √2 is.
- Suppose 3√2 = r, a rational number.
- Then √2 = r/3, and ℚ is closed under division, so √2 would be rational: a contradiction.
- So 3√2 is irrational.
Your go. The first step is given. Write 0.606060… as a fraction in lowest terms.
- Two digits repeat, so multiply by 100: if x = 0.606060…, then 100x = 60.606060…
- Now subtract x and solve for x. Simplify the fraction.
Check the answer
100x − x = 60, so 99x = 60 and x = 60/99 = 20/33.
Check before you go on
1. x = 0.272727… (the block 27 repeats). By which number should you multiply x so that subtracting x leaves a whole 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. Write 0.212121… (the block 21 repeats) as a fraction in lowest terms. What is its denominator?
3. Let x = 0.72222… (the 2 repeats). What whole number is 100x − 10x?
Level 2 · Building
4. A calculator shows 17 ÷ 41 = 0.4146341463414634…, and the block 41463 repeats. What is the 100th digit after the decimal point?
5. Write 0.58888… (the 8 repeats) as a fraction in lowest terms. What is its denominator?
6. A spreadsheet for the 990 students at a school in Lima gives the share who walk to school as 0.444… (the 4 repeats) and the share who cycle as 0.272727… (the block 27 repeats). Write the share who walk or cycle as a single fraction in lowest terms. What is its numerator?
Level 3 · Stretch
7. To prove √7 is irrational, suppose √7 = p/q in lowest terms, so 7q² = p². Then 7 divides p, so write p = 7k. Substituting gives q² = ?k². What number goes in place of ?
8. How many of these six numbers are rational? 2 + √3, √3 × √12, (√5)², 3√2 − √18, π − π, 1/√2.
Something to think about
A friend says: "The product of two irrational numbers is always irrational." Find a counter-example. Then find two irrational numbers whose product is irrational. What does that tell you about the word "always"? Finally, π is irrational: is 1/π?
One way to think about it
√2 × √2 = 2 is a counter-example, and √2 × √3 = √6 is irrational, so both outcomes happen and "always" is false: the honest statement is "sometimes". For 1/π, suppose it equalled a rational r. Then π = 1/r would be rational, which it is not. So 1/π is irrational, and π × (1/π) = 1 is a product of two irrational numbers that is rational.
Exit ticket
9. p is a non-zero rational number and s is an irrational number. Which of these must be irrational?