Question

State whether the following statements are true or false. Give reasons for your answers.
For any real number x, ${{x}^{2}}\ge 0$.
(a) True
(b) False

Answer 1

Hint: We start solving the problem by assuming x as 0 and finding the square of it to check whether it is satisfying ${{x}^{2}}\ge 0$ or not. We then assume x as a positive integer and find the square of it to check whether it is satisfying ${{x}^{2}}\ge 0$ or not. We then assume x as a negative integer and find the square of it to check whether it is satisfying ${{x}^{2}}\ge 0$ or not. We then perform similar calculations by assuming x as a positive rational number, negative rational number, irrational number of the form $p+\sqrt{q}$ and $p-\sqrt{q}$ to get the required answer.

Complete step by step answer:
According to the problem, we are given that we need to state whether the given statement is true or false with reason.
We have a statement given as For any real number x, ${{x}^{2}}\ge 0$.
Let us assume that $x=0$, we get ${{x}^{2}}=0$ which satisfies ${{x}^{2}}\ge 0$.
Now, let us assume that x is a positive integer. Let $x=4$, then ${{x}^{2}}=16>0$. So, this satisfies ${{x}^{2}}\ge 0$.
Now, let us assume that x is a negative integer. Let $x=-2$, then ${{x}^{2}}=4>0$. So, this satisfies ${{x}^{2}}\ge 0$.
Now, let us assume x is a negative rational number. Let $x=\dfrac{-5}{7}$, then ${{x}^{2}}=\dfrac{25}{49}>0$. So, this satisfies ${{x}^{2}}\ge 0$.
Now, let us assume x is a positive rational number. Let $x=\dfrac{3}{5}$, then ${{x}^{2}}=\dfrac{9}{25}>0$. So, this satisfies ${{x}^{2}}\ge 0$.
Now, let us assume x is an irrational number in the form $p+\sqrt{q}$. Let $x=2+\sqrt{3}$, then ${{x}^{2}}=4+3+4\sqrt{3}=7+4\sqrt{3}>0$. So, this satisfies ${{x}^{2}}\ge 0$.
Now, let us assume x is an irrational number in the form $p-\sqrt{q}$. Let $x=1-\sqrt{5}$, then ${{x}^{2}}=1+5-2\sqrt{5}=6-2\sqrt{5}>0$. So, this satisfies ${{x}^{2}}\ge 0$.
We can see that every form of real number x is satisfying the property ${{x}^{2}}\ge 0$.
So, the given statement is true.

The correct option for the given problem is (a).

Note:
Whenever we get this type of problem, we take an example to represent the real numbers for proving the given point. We should consider every form of number present on the number line to complete the required proof otherwise the proof will be wrong. Similarly, we can expect the problem to check the given proof for a complex number.