This means that there are infinitely many rational numbers between any two points. It feels as though the rational numbers must completely fill the number line, leaving no gaps whatsoever. But do they?
Step-by-step solution
Idea: Being dense (always another number in between) is not the same as filling every point. One point that is not rational is enough to show there are gaps.
- Draw a square of side 1 unit. By the Baudhāyana–Pythagoras theorem its diagonal d satisfies d2 = 12 + 12 = 2, so d = √2. With a compass this length can be marked on the number line from 0.1 mark
- But √2 is not rational (assuming √2 = pq in lowest terms forces both p and q to be even, a contradiction). So the point √2 is a gap among the rational numbers: the rational numbers do not fill the line. The missing points are the irrational numbers.1 mark
Answer to write in the exam
Square of side 1: d2 = 12 + 12 = 2 ⇒ d = √2
√2 can be marked on the number line, but √2 is not of the form pq
∴ No; the rational numbers leave gaps (irrational numbers such as √2).
Common mistakes that cost marks
- Thinking ‘infinitely many between any two’ means ‘every point is covered’. Density does not rule out gaps.
- Using 1.414 as √2. 1.414 is rational and only approximately √2; 1.4142 = 1.999396 ≠ 2.
- Giving π ≈ 227 as ‘proof’ that π is rational. 227 is only an approximation.
How this can come in the exam
Which of these numbers is a point on the number line that is NOT rational?
- 227
- 1.414
- √2
- √4
Show answer
(C) √2
√4 = 2 and the others are fractions or terminating decimals; √2 is irrational.
Assertion (A): The rational numbers fill the whole number line.
Reason (R): Between any two rational numbers there is another rational number.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(D) A is false but R is true.
A is false (√2 is a point that is not rational). R is true: rational numbers are dense.
Try one yourself
A right triangle has legs 1 and 2. Is its hypotenuse a rational length?
Show answer
Hypotenuse = √(1 + 4) = √5, which is irrational, so no.
More questions like this
- Represent the rational numbers 23, −54 and 112 on a single number line.
- Find three distinct rational numbers that lie strictly between −12 and 14.
- Simplify the expression: (−14) + (512).
- A tailor has 1534 metres of fine silk. If making one kurta requires 214 metres of silk, exactly how many kurtas can he make?
- Find three rational numbers between 3.1415 and 3.1416.