# Prove or disprove that the product of two irrational numbers is irrational.

The aim of this question is to understand deductive logic and the concept of irrational and rational numbers.

A number (N) is said to be rational if it can be written in the form of a fraction such that the numerator and denominator both belong to a set of integers. Also it’s a necessary condition that the denominator has to be non-zero. This definition can be written in the mathematical form as follows:

$N \ = \ \dfrac{ P }{ Q } \text{ where } P, \ Q \ \in Z \text{ and } Q \neq 0$

Where $N$ is the rational number while $P$ and $Q$ are the integers belonging to the set of integers $Z$. On similar lines, we can conclude that any number that can’t be written in the form of a fraction (with numerator and denominator being integers) is called an irrational number.

An integer is such a number that does not have any fractional part or doesn’t have any decimal. An integer can be both positive and negative. Zero is also included in the set of integers.

$Z \ = \ \{ \ …, \ -3, \ -2, \ -1, \ 0, \ +1, \ +2, \ +3, \ … \ \}$

Now to prove the given statement, we can prove the contraposition. The contraposition statement of the given statement can be written as follows:

“A product of two rational numbers is also a rational number.”

Let us say that:

$\text{ 1st rational number } \ = \ A$

$\text{ 2nd rational number } \ = \ B$

$\text{ Product of two rational numbers } \ = \ C \ = \ A \times B$

By definition of rational numbers as described above, $C$ can be written as:

$\text{ A rational number } \ = \ C$

$\text{ A rational number } \ = \ A \times \ B$

$\text{ A rational number } \ = \ \dfrac{ A }{ 1 } \times \dfrac{ 1 }{ B }$

$\text{ A rational number } \ = \ \text{ Product of two rational numbers }$

Now we know that $\dfrac{ A }{ 1 }$ and $\dfrac{ 1 }{ B }$ are rational numbers. Hence proved that a product of two rational numbers $A$ and $B$ is also a rational number $C$.

So the contrapositive statement must also be true, that is, the product of two irrational numbers must be an irrational number.

## Numerical Result

The product of two irrational numbers must be an irrational number.

## Example

Is there a condition where the above statement does not hold true. Explain with the help of example.

Let’s consider an irrational number $\sqrt{ 2 }$. Now if we multiply this number with itself:

$\text{ Product of two irrational numbers } \ = \ \sqrt{ 2 } \ \times \ \sqrt{ 2 }$

$\text{ Product of two irrational numbers } \ = \ ( \sqrt{ 2 } )^2$

$\text{ Product of two irrational numbers } \ = \ 2$

$\text{ Product of two irrational numbers } \ = \text{ a rational number }$

Hence, the statement does not hold true when we multiply an irrational number with itself.