Prove that 1/root 2 is a irrational number?

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Proof that √2 is an irrational number

Introduction:
In mathematics, an irrational number is a real number that cannot be expressed as a ratio of two integers. In other words, an irrational number cannot be written in the form of p/q, where p and q are integers and q is not equal to zero. The square root of 2, denoted as √2, is one such number that cannot be expressed as a ratio of two integers. In this article, we will prove that √2 is an irrational number.

Proof:
To prove that √2 is an irrational number, we will use the method of proof by contradiction. That is, assume that √2 is a rational number, and then show that this assumption leads to a contradiction.

Assumption:
Let us assume that √2 is a rational number. That is, there exist two integers p and q, such that √2 = p/q, where q is not equal to zero, and p and q have no common factors.

Squaring both sides:
If we square both sides of the above equation, we get:

2 = p^2/q^2

Multiplying both sides by q^2, we get:

2q^2 = p^2

This implies that p^2 is an even number, since it is equal to 2q^2. Now, we know that the square of an odd number is always odd, and the square of an even number is always even. Therefore, p must be an even number.

Rewriting p as 2m:
Let us rewrite p as 2m, where m is an integer. Then, we have:

p^2 = (2m)^2 = 4m^2

Substituting this value in the equation 2q^2 = p^2, we get:

2q^2 = 4m^2

Dividing both sides by 2, we get:

q^2 = 2m^2

This implies that q^2 is an even number, since it is equal to 2m^2. Therefore, q must also be an even number.

Contradiction:
We have assumed that p and q have no common factors. However, we have now shown that both p and q are even numbers. This means that p and q have a common factor of 2, which contradicts our assumption. Therefore, our initial assumption that √2 is a rational number must be false.

Conclusion:
We have proved that √2 is an irrational number. This means that it cannot be expressed as a ratio of two integers. The proof by contradiction is a powerful tool in mathematics, which allows us to prove theorems by assuming the opposite and showing that it leads to a contradiction.

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