document.write( "Question 974888: there are infinitely many irrational numbers.give reason\r
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Algebra.Com's Answer #596724 by solver91311(24713) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Pick a set that you know has countably infinite elements, for example the set

of positive integers. Then pick your favorite irrational number, say

. Since we know that the product of a rational number and an irrational number is irrational, every product of the form

where

is irrational and the number of such products is infinite because

has infinite elements. Since a subset of the set of irrational numbers has infinite elements, the entire set must, perforce, be infinite.\r
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\n" ); document.write( "\n" ); document.write( "By the way it is also true that between any two rational numbers there is an infinity of irrational numbers.\r
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\n" ); document.write( "\n" ); document.write( "If you are in doubt about the assertion made as to the irrationality of a product of a rational and an irrational number, then read: Dr. Math Proof that the product of a rational and an irrational is irrational\r
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\n" ); document.write( "\n" ); document.write( "John
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\n" ); document.write( "My calculator said it, I believe it, that settles it\r
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