Question

How do you determine weather the square root of 3 is irrational

Answer 1

To determine whether the square root of 3 is irrational, we need to determine if it can be expressed as a fraction of two integers (a rational number) or not.

Assume, for contradiction, that the square root of 3 is rational. This means it can be written as the fraction a/b, where a and b are integers that have no common factors other than 1 and b is not equal to 0.

Taking the square of both sides, we get:
(sqrt(3))^2 = (a/b)^2
3 = (a^2)/(b^2)

Cross multiplying, we get:
3b^2 = a^2

This implies that a^2 is a multiple of 3. As a result, a must also be a multiple of 3. Let a = 3k, where k is an integer.

Substituting back into the equation, we get:
3b^2 = (3k)^2
3b^2 = 9k^2
b^2 = 3k^2

This implies that b^2 is also a multiple of 3, and therefore b must also be a multiple of 3.

Now, we have a contradiction, because both a and b are multiples of 3. However, we initially assumed that a and b have no common factors other than 1, which contradicts our assumption that the square root of 3 is rational.

Hence, we conclude that the square root of 3 is irrational.