We have seen how to obtain a line whose length is a rational number. How do we obtain lines whose lengths are irrational?
Answer: Use right triangles. On the number line take OA = 1, draw AB = 1 perpendicular at A; then OB = √(12 + 12) = √2. An arc with centre O and radius OB cuts the number line at P, so OP = √2.
Step-by-step solution
Idea: The Baudhāyana–Pythagoras theorem turns two rational lengths into a hypotenuse whose square is their sum of squares. A compass then carries that length down onto the number line.
- On the number line mark O at 0 and A at 1, so OA = 1 unit. Draw a line perpendicular to OA through A.½ mark
- On this perpendicular mark B with AB = 1 unit and join OB.½ mark
- Triangle OAB is right-angled at A, so OB2 = OA2 + AB2 = 1 + 1 = 2, giving OB = √2 units.1 mark
- With centre O and radius OB, draw an arc to cut the number line at P. Then OP = OB = √2, so P marks the irrational number √2. Other irrational lengths are made the same way with suitable legs.1 mark
Build a right triangle with legs 1 and 1 on the number line; its hypotenuse is √2, and an arc centred at O transfers it to the point P = √2.
Check: OP ≈ 1.414, so P lies between 1 and 2, a little before 1.5 ✓.
Answer to write in the exam
OA = 1 unit on the number line; AB ⊥ OA, AB = 1 unit
OB2 = OA2 + AB2 = 1 + 1 = 2 (Baudhāyana–Pythagoras theorem)
OB = √2
Arc with centre O, radius OB cuts the number line at P
∴ OP = √2; P represents √2.
Common mistakes that cost marks
- Drawing AB at an angle that is not 90°. The theorem only works for a right angle.
- Using A (not O) as the centre of the arc. The distance must be measured from O, the zero point.
- Marking P at 2 because ‘one plus one is two’; the hypotenuse is √2 ≈ 1.414, shorter than the two legs together.
How this can come in the exam
MCQ (1 mark)
In a right triangle OAB with OA = 2 units and AB = 1 unit (right angle at A), OB =
- 3
- √3
- √5
- 5
Show answer
(C) √5
OB2 = 4 + 1 = 5.
Short answer (2 marks)
Describe how to mark √10 on the number line.
Show answer
Take OA = 3 units on the line and AB = 1 unit perpendicular at A; OB2 = 9 + 1 = 10 (1 mark). An arc with centre O and radius OB cuts the line at P, and OP = √10 (1 mark).Try one yourself
Which legs would you use to construct a length of √13?
Show answer
Legs 3 and 2: 32 + 22 = 13.
More questions like this
- 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.
- We know what it means to add 2, 100, or even a lakh terms. What does it mean to add an infinite number of terms?
- It terminates: The division eventually leaves a remainder of 0. The decimal stops. 38 = 0.375. (Can you tell for which rational numbers the decimal will be terminating?)
- It repeats: The division never reaches a remainder of 0, but the sequence of digits begins to loop infinitely. 511 = 0.454545 … = 0.45.
- Try to find the decimal expansions of 103 and 1112. What do you observe about the repetition of the digits after the decimal point?