Is the square root of 2 rational or irrational?
The Correct Answer and Explanation is:
The square root of 2 is an irrational number.
To explain why, we can use a proof by contradiction. Let’s assume that the square root of 2 is rational. This means it can be expressed as the ratio of two integers, say:2=ab\sqrt{2} = \frac{a}{b}2=ba
where aaa and bbb are integers, and aaa and bbb have no common factors (i.e., the fraction is in its simplest form). Squaring both sides, we get:2=a2b22 = \frac{a^2}{b^2}2=b2a2
Multiplying both sides by b2b^2b2, we get:2b2=a22b^2 = a^22b2=a2
This equation tells us that a2a^2a2 is an even number because it is equal to 2b22b^22b2. If a2a^2a2 is even, then aaa must also be even, since the square of an odd number is odd.
Let’s now express aaa as a=2ka = 2ka=2k, where kkk is an integer. Substituting this back into the equation 2b2=a22b^2 = a^22b2=a2, we get:2b2=(2k)2=4k22b^2 = (2k)^2 = 4k^22b2=(2k)2=4k2
Dividing both sides by 2:b2=2k2b^2 = 2k^2b2=2k2
This shows that b2b^2b2 is also even, and therefore, bbb must also be even.
Now, we have shown that both aaa and bbb are even, which contradicts our assumption that they have no common factors (because both are divisible by 2). Therefore, the assumption that 2\sqrt{2}2 is rational must be false.
Since 2\sqrt{2}2 cannot be written as the ratio of two integers, it is irrational.
