Prove that 1/root2 is irrational?

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Proving that 1/√2 is irrational

Introduction:
In mathematics, a rational number is any number that can be expressed as the ratio of two integers, whereas an irrational number is a number that cannot be expressed as the ratio of two integers. In this question, we need to prove that 1/√2 is an irrational number.

Assumption:
Let's assume that 1/√2 is a rational number.

Proof:
If 1/√2 is a rational number, then it can be expressed as the ratio of two integers, say a and b.

So, 1/√2 = a/b, where a and b are integers with no common factors.

Squaring both sides of the equation, we get:

1/2 = a^2/b^2

Multiplying both sides of the equation by b^2, we get:

b^2/2 = a^2

This implies that a^2 is even, which means a is even, say a = 2c.

Substituting the value of a in the equation, we get:

b^2/2 = (2c)^2

b^2/2 = 4c^2

b^2 = 8c^2

This implies that b^2 is even, which means b is even, say b = 2d.

Substituting the value of b in the equation, we get:

(2d)^2/2 = a^2

4d^2 = a^2

This implies that a^2 is even, which means a is even, which contradicts our assumption that a and b have no common factors.

Therefore, our assumption that 1/√2 is a rational number is false.

Hence, 1/√2 is an irrational number.

Conclusion:
Thus, we have proved that 1/√2 is an irrational number by contradiction.

Answer

1/root2=root1/root2=root (1/2)
If it is rational, it can be expressed in the form p/q (both whole numbers)
therefore, 1/2=p^2/q^2
Cross multiply

2p^2=q^2
Both have to be whole numbers (p and q have to be whole numbers), and so when factorized each number should have an even exponent.

But because of the odd exponent of 2 (1), the left side would always have an odd exponent.

That means that p and q cannot be expressed in whole numbers, and hence 1/root2 is irrational.

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