Proving/Disproving Product of two irrational number is irrational

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I saw this question where I had to prove/disprove that:

Ques. Product of two irrational number is irrational.

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I tried 'Proof by Contraposition'.

Product of two irrational number is irrational.

p : Product of two irrational number

q : Irrational number.

Thus, given statement is : p -> q

Contraposition of p : ¬q -> ¬p

Rational number -> Can be broken down into product of two rational number.

Proof :

Let m be a rational number such that m = p/q.

Then I can always write m as (p/1)*(1/q)

where (p/1) and (1/q) are both rational numbers. Hence proved.

But it turns out that books disproves the statement saying $\sqrt2\cdot\sqrt2=2$ which is a rational number and hence Product of two irrational number need not always be irrational. Which I find convincing.

Can someone please point out where am I going wrong in my proof?

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3 Answers

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The negation of the assertion [Is the product of two irrational numbers] is the assertion [Is not the product of two irrational numbers]. There is no a priori reason to expect that the assertion [Is not the product of two irrational numbers] is equivalent to the assertion [Is the product of two rational numbers] (and in fact these last two are not equivalent).

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Disprove:

Let $\sqrt{2}$ be the irrational number. Then $\sqrt{2}\times \sqrt{2}=|2|$, which is rational. So, the product of two irrational numbers is not always irrational

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Some more examples are

$$\sqrt{8}×\sqrt2=\sqrt{16} =4$$

$$\sqrt2×\sqrt{32}=\sqrt{64} =8$$

$$\sqrt5×\sqrt5=\sqrt{25} =5$$

In this way product of two irrational number is rational.

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