Find three rational numbers between 3.1415 and 3.1416.
Answer: 3.14151, 3.14153 and 3.14155 (each is a terminating decimal, so rational; e.g. 3.14151 = 314151100000).
Step-by-step solution
Idea: Add a fifth decimal place: 3.1415 = 3.14150 and 3.1416 = 3.14160. Any of 3.14151, …, 3.14159 lies strictly between.
- Write both with five decimal places: 3.14150 and 3.14160.½ mark
- Numbers strictly between: 3.14151, 3.14152, …, 3.14159. Choose 3.14151, 3.14153, 3.14155.1 mark
- They are terminating decimals, so they are rational, e.g. 3.14153 = 314153100000.½ mark
3.14151, 3.14153 and 3.14155 (any three of 3.14151 to 3.14159, or others such as 3.141505, are correct).
Answer to write in the exam
3.1415 = 3.14150, 3.1416 = 3.14160
3.14150 < 3.14151 < 3.14153 < 3.14155 < 3.14160
∴ Three rational numbers: 3.14151, 3.14153, 3.14155
Common mistakes that cost marks
- Saying there is nothing between 3.1415 and 3.1416 because they differ only in the last digit. Adding decimal places always makes room.
- Giving π as an answer. π ≈ 3.14159 lies between them but is irrational, not rational.
- Writing 3.14165, which is bigger than 3.1416.
How this can come in the exam
MCQ (1 mark)
Which number lies between 2.71 and 2.72?
- 2.7
- 2.721
- 2.715
- 2.0715
Show answer
(C) 2.715
2.71 = 2.710 < 2.715 < 2.720.
Short answer (2 marks)
Write two rational numbers between 0.1 and 0.11 and express one of them in the form pq.
Show answer
0.100 < 0.103 < 0.107 < 0.110 (1 mark). 0.103 = 1031000 (1 mark).Try one yourself
Find three rational numbers between 1.4142 and 1.4143.
Show answer
For example 1.41421, 1.41425, 1.41428.
More questions like this
- Can you think of other way(s) to find a rational number between any two rational numbers?
- Can √2 be written as a rational number pq?
- Try to prove the irrationality of √3 using the approach of proof by contradiction. Will the same approach work for √5, √7, or √10?
- We have seen how to obtain a line whose length is a rational number. How do we obtain lines whose lengths are irrational?
- Try to extend this method for constructing line segments of lengths √3 and √5 using a ruler and a compass. Generalise this method to construct a line segment of any length of the form √n, where n is a positive integer.