SOLUTION: there are infinitely many irrational numbers.give reason

Question 974888: there are infinitely many irrational numbers.give reason

Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

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.

By the way it is also true that between any two rational numbers there is an infinity of irrational numbers.

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

John

My calculator said it, I believe it, that settles it

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