Question

1. Give an example of 2 irrational numbers whose sum is rational.
2. Give an example of 2 irrational numbers whose product is rational.

Answer 1

Hint: We will think of any 2 irrational numbers, we will find their sum and see if it is a rational number. We will think of any 2 irrational numbers, we will find their product and see if it is a rational number.
Formula used: $$\left(a+b\right)\left(a-b\right)=a^2-b^2$$

Complete step-by-step answer:
We will look at some properties of rational and irrational numbers that will help us in solving the question:
We know that a number is a rational number if it can be expressed in the form $${\displaystyle \frac{p}{q}}$$ where $$p$$and $$q$$ are integers with no common factor and $$q\ne 0$$.
We know that a number is an irrational number if it cannot be expressed in the form $${\displaystyle \frac{p}{q}}$$ where $$p$$and $$q$$ are integers with no common factor and $$q\ne 0$$.
We know that the sum of a rational and an irrational number is always an irrational number.
We know that the difference of a rational and an irrational number is always an irrational number.
We know that $$\sqrt{7}$$ is an irrational number.
We can conclude from the 3rd property that $$2+\sqrt{7}$$ is an irrational number.
We can conclude from the 4th property that $$2-\sqrt{7}$$ is an irrational number.
We will take the first number as $$2+\sqrt{7}$$ and the second number as $$2-\sqrt{7}$$.
We will find the sum of the 2 numbers:
$(2+\sqrt{7}+(2-\sqrt{7}$
=$2+\sqrt{7}+2-\sqrt{7}$
=$2+2$+$\sqrt{7}-\sqrt{7}$
= 4
4 is a rational number.
We will find the product of the 2 numbers. We will substitute 2 for $$a$$ and $$\sqrt{7}$$ for $$b$$in the formula:
$(2+\sqrt{7})$.$(2-\sqrt{7})$
=$2^2-(\sqrt{7}$
= 4 - 7
 = - 3
$$-3$$ is a rational number.
$$\therefore 2+\sqrt{7}$$ and $$2-\sqrt{7}$$ are 2 irrational numbers whose sum as well as the product are rational numbers.

Note: We must know that the square roots of all non-negative integers except the perfect squares are irrational numbers. This will help us in choosing the correct irrational number. For example, $$\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7},$$etc are irrational numbers whereas $$\sqrt{9},\sqrt{16},\sqrt{25},\sqrt{49},$$ etc are rational numbers.