Surds & irrational numbers · Ages 13–14
Rational and irrational numbers
A ratio of integers is rational; n is rational only when n is a perfect square.
How to do it
Method
A ratio of integers is rational; √n is rational only when n is a perfect square
49 = 7 is rational, 50 is irrational
Another way
Between two perfect squares
49 < 50 < 64, so 7 < 50 < 8
In full
A number that can be written as a ratio of two integers, pq (q not 0), is rational: integers, fractions, decimals that end (0.25 = 14) and recurring decimals (0.3̇ = 13) all are. A number that can't is irrational: its decimal never ends and never repeats, like 2 = 1.41421356…, π = 3.14159265… and 0.1010010001… (more and more 0s between the 1s, never repeating). To judge n: when n is a perfect square (1, 4, 9, 16, …), n is a whole number, so rational; otherwise n is irrational. 49 = 7 is rational, 50 irrational. Fractions too: 49 = 23 is rational. To estimate n, find the perfect squares either side: 49 < 50 < 64, so 7 < 50 < 8. A calculator can show whether a decimal seems to repeat, but it shows only so many digits; the reasoning above decides.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Which of these is rational?
- A. 25
- B. π
- C. 35
- D. 33
Choose the right answer
Show the answer and working
Answer: A. 25
25 = 5, which is 51: rational.
Example 2
376 lies between two whole numbers in a row. The smaller is .
Show the answer and working
Answer: 19
361 < 376 < 400, that is 192 < 376 < 202, so 19 < 376 < 20.
Example 3
Fill in two whole numbers in a row: < 116 <
Fill in each blank
Show the answer and working
Answer: 10, 11
Find the perfect squares either side of 116: 102 < 116 < 112.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.Which of these is irrational?
- A. 6
- B. 4
- C. 67
- D. 0.75
Choose the right answer
Answer:
2.139 is rational.
True or false?
Answer:
3.Which of these is rational?
- A. π
- B. 64
- C. 77
- D. 78
Choose the right answer
Answer:
4.2549 is rational.
True or false?
Answer:
5.Fill in two whole numbers in a row: < 226 <
Fill in each blank
Answer:
All the answers
- 1. A. 6
- 2. False
- 3. B. 64
- 4. True
- 5. 15, 16
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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