SOLUTION: True or false? If a and b are both irrational then a*b must always be irrational?

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Question 1208935: True or false? If a and b are both irrational then a*b must always be irrational?
Found 3 solutions by math_tutor2020, ikleyn, greenestamps:
Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

Answer: False

Reason:
We just need one counter-example to disprove the claim. Through a bit of trial and error we can generate this

The and are each irrational since they individually cannot be expressed as a fraction of two integers.
But the result 10 = 10/1 is rational since it can be written as a ratio of integers 10 over 1.

This disproves the template irrational*irrational = irrational as there are some exceptions.

Note that irrational*irrational = rational is false as well (try and to see what happens).
Answer by ikleyn(52780)   (Show Source): You can put this solution on YOUR website!
.

Let's  a = ,  b = .


Both "a" and "b" are irrational,  but  a*b =  = 2  is a rational number.

This example makes the claim FALSE.




Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


FALSE

Let a and b both be equal to the same irrational number of the form , where n is any positive number. Then , which is rational.
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