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) About Me  (Show Source):
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Answer: False

Reason:
We just need one counter-example to disprove the claim. Through a bit of trial and error we can generate this sqrt%2820%29%2Asqrt%285%29+=+sqrt%2820%2A5%29+=+sqrt%28100%29+=+10

The sqrt%2820%29 and sqrt%285%29 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 a+=+sqrt%282%29 and b+=+sqrt%283%29 to see what happens).

Answer by ikleyn(52780) About Me  (Show Source):
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.

Let's  a = sqrt%282%29,  b = sqrt%282%29.


Both "a" and "b" are irrational,  but  a*b = sqrt%282%29%2Asqrt%282%29 = 2  is a rational number.

This example makes the claim FALSE.




Answer by greenestamps(13200) About Me  (Show Source):
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FALSE

Let a and b both be equal to the same irrational number of the form sqrt%28n%29, where n is any positive number. Then a%2Ab=sqrt%28n%29%2Asqrt%28n%29=n, which is rational.