Question

Determine if square root of, 2, end square root
2
2
​
is rational or irrational and give a reason for your answer.
Answer
Attempt 1 out of 2

The number square root of, 2, end square root
2
2
​
is
because

Answer 1

the square root of 2 is an irrational number.

To prove this, we can use proof by contradiction.

Assume that the square root of 2 is a rational number. This means it can be expressed as a fraction a/b, where a and b are integers and b is not equal to 0. Since the square root of 2 is positive, both a and b must be positive.

Therefore, we can write:

sqrt(2) = a/b

Squaring both sides of the equation, we get:

2 = (a^2)/(b^2)

Cross-multiplying, we get:

2b^2 = a^2

This implies that a^2 is even, and therefore a must also be even (since the square of an odd number is odd).

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

Substituting this into the equation, we get:

2b^2 = (2k)^2

2b^2 = 4k^2

Dividing both sides of the equation by 2, we get:

b^2 = 2k^2

This implies that b^2 is even, and therefore b must also be even.

However, if both a and b are even, then they have a common factor of 2. This contradicts our assumption that a/b is a fraction in its simplest form.

Therefore, our assumption that the square root of 2 is rational is false. Hence, the square root of 2 is an irrational number.