Real Numbers — Challenge
Every number you meet this year is either rational — a fraction — or irrational. Knowing which tells you how it behaves.
What these problems 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.
- Integer — A whole number, positive, negative, or zero.
- π — The circumference of any circle divided by its diameter — about 3.14159.
- Perfect square — A number that is some whole number squared: 1, 4, 9, 16, 25, …
- Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
- 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.
Problem 1
Which of these numbers is irrational?
- √2
- 0.75
- 73
- -4
- √9
Worked answer
√2
A rational number can be written as a fraction of two whole numbers; an irrational one cannot.
0.75 = 34, 73 is already a fraction, -4 = -41 and √9 = 3, so all four are rational.
√2 cannot be written as such a fraction, so it is irrational.
Problem 2
Between which two consecutive whole numbers does √30 lie?
- 4 and 5
- 5 and 6
- 6 and 7
- 7 and 8
- 14 and 15
Worked answer
5 and 6
Find the perfect squares on either side of 30: 25 and 36.
Since 25 < 30 < 36, taking square roots gives 5 < √30 < 6.
So √30 lies between 5 and 6.
Problem 3
Write the repeating decimal 0.6 as a fraction in lowest terms.
- 23
- 34
- 610
- 16
- 29
Worked answer
23
Let x = 0.666…, so 10x = 6.666….
Subtracting gives 9x = 6, so x = 69 = 23.
The option 610 is 0.6 exactly, which stops rather than repeating.
Problem 4
Which of these numbers is a whole number?
- √16
- √10
- 12
- -0.5
- π
Worked answer
√16
√16 = 4, because 4· 4 = 16.
None of the others is a whole number: √10≈ 3.16, and the rest are fractions or π≈ 3.14.
Problem 5
Which of these numbers is the greatest?
- 58
- 0.6
- 23
- 0.58
- 711
Worked answer
23
Write them all as decimals to compare them on one scale.
58 = 0.625, 23≈ 0.667 and 711≈ 0.636.
The list is 0.625, 0.6, 0.667, 0.58, 0.636, so 23 is the greatest.
Problem 6
What is √64?
- 6
- 7
- 8
- 32
- 64
Worked answer
8
Ask which number multiplied by itself gives 64.
8· 8 = 64, so √64 = 8. Halving would give 32 instead.
Problem 7
Which statement about √25 is true?
- It is rational
- It is irrational
- It is negative
- It is not a real number
- It lies between 5 and 6
Worked answer
It is rational
√25 = 5 exactly, because 5· 5 = 25.
Since 5 = 51, it is a ratio of whole numbers and so rational.
A square root is not automatically irrational; it is irrational only when the number under it is not a perfect square.
Problem 8
Which of these is an integer?
- -7
- 13
- 2.5
- √3
- 0.1
Worked answer
-7
The integers are the whole numbers together with their negatives: …,-2,-1,0,1,2,…
-7 is on that list. The others are fractions, decimals or an irrational root.
Problem 9
Write 38 as a decimal.
- 0.125
- 0.375
- 0.38
- 0.625
- 2.667
Worked answer
0.375
Divide 3 by 8.
8 goes into 30 three times with 6 left over, into 60 seven times with 4 left, and into 40 five times exactly.
So 38 = 0.375, a decimal that terminates because 8 is a power of 2.
Problem 10
Which of these numbers is the least?
- -2.5
- -2
- -12
- 0
- 12
Worked answer
-2.5
The least number is the one furthest left on the number line.
Among the negatives, the one with the greatest size is furthest left, and 2.5 > 2 > 12.
So -2.5 is the least.