Question

prove that √2 is a irrational number​

Answer 1

Answer:

Assume ✓2 is rational

Then ✓2 = a/b where a,b belongs to z and b ≠ 0, and a, b are co primes

Squaring on both sides, we get

(✓2)² = (a/b)²

=> 2 = a² /b²

=> 2b² = a². -----> (A)

Since 2 divides 2b² it divides a² also

By theorem , is p divides a², then it divides a also

Hence 2 divides a ------(1)

Then there exists c such that a = 2c

Substitute a = 2c in (A)

Then 2b² = (2c)²

=>2b² = 4c²

=> b² = 2c²

Since 2 divides 2c² it divides b² also

Again by theorem, 2 divides b ------> (2)

From (1) and (2)

2 divides both a and b

But a and b are co primes

Hence our assumption is wrong

Therefore ✓2 is irrational

Hence proved

Step-by-step explanation: