But it turns out that books disproves the statement saying $sqrt2cdotsqrt2=2$ which is a rational number and hence Product of two irrational number need not always be irrational. Which I find convincing.

Đang xem: Is the product of two irrational numbers always irrational

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

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edited Jan 11 “15 at 17:35

Martin Sleziak

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asked Sep 25 “13 at 4:48

Amit TomarAmit Tomar

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$egingroup$ The contrapositive is $

eg q implies

eg p$. $endgroup$–user61527 Sep 25 “13 at 4:50

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## 3 Answers 3

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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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edited Sep 25 “13 at 8:03

answered Sep 25 “13 at 5:27

DidDid

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Disprove:

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

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answered Sep 28 “14 at 6:57

Hassan MuhammadHassan Muhammad

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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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edited Jul 25 “15 at 18:26

Peter Woolfitt

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answered Jul 25 “15 at 17:59

shrutishruti

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