# Is The Product Of Two Irrational Numbers Always Irrational, Sum And Product Rationals Irrationals

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.

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

discrete-wtbblue.comematics logic propositional-calculus irrational-numbers
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edited Jan 11 “15 at 17:35

Martin Sleziak
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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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

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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
answered Jul 25 “15 at 17:59
shrutishruti
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