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Question
Prove that the following is irrational:
`1/sqrt2`
Solution
`1/sqrt2`
`1/sqrt2 xx sqrt2/sqrt2 = sqrt2/2`
Let a = `(1/2)sqrt2` be a rational number.
∴ `1/2 (sqrt2)` is rational.
Let `1/2 (sqrt2) = a/b`, such that a and b are co-prime integer and b ≠ 0.
∴ `sqrt2 = (2a)/b` ...(1)
Since the division of two integers is rational.
∴ `(2a)/b` is rational.
From (1), `sqrt2` is rational, which contradicts the fact that `sqrt2` is irrational.
∴ Our assumption is wrong.
Thus, `1/sqrt2` is irrational.
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Classroom activity (Constructing the ‘square root spiral’): Take a large sheet of paper and construct the ‘square root spiral’ in the following fashion. Start with a point O and draw a line segment OP1 of unit length. Draw a line segment P1 P2 perpendicular to OP1 of unit length. Now draw a line segment P2 P3 perpendicular to OP2. Then draw a line segment P3 P4 perpendicular to OP3. Continuing in this manner, you can get the line segment Pn–1Pn by drawing a line segment of unit length perpendicular to OPn–1. In this manner, you will have created the points P2, P3,...., Pn,.... ., and joined them to create a beautiful spiral depicting `sqrt2, sqrt3, sqrt4,` ...
