Question

show that 2+root2 is a not a rational​

Answer 1

Answer:

Let us assume that √2 is a rational and.

Then, √2 = p/q ( equation 1.)

Where p and q are integers, co-prime to each other and q is not equal to 0

2 = p^2/q^2 => p^2 = 2q^2 ( equation 2.)

By equation (2), we can say that p^2 is an even integer.

Therefore, p is also an even integer ( since, the square of an even integer is always even)

Let p = 2k, where k is an integer.

From (2),

p^2 = 2q^2

(2k)^2 = 2q^2

4k^2 = 2q^2

=> q^2 = 2k^2

q^2 is an even integer, q is also an even integer.

Thus, p and q have a common factor 2 which contradicts the hypothesis that p,q are co- prime to each other.

Therefore,√2 is an irrational number.

Step-by-step explanation:

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