if r is any rational number s is any irrational number, then r/s is irrational. The sum of any two positive i?

Question

indicate whether the following statements are TRUE or FALSE. If the statement is FALSE, disprove by a counterexample.

If r is any rational number s is any irrational number, then r/s is irrational.

The sum of any two positive irrational numbers is irrational

The product of any two irrational numbers is irrational

Answer 1

1] false. r=0, s=π, therefore r/s=0 which is rational.

2] false. a=√10+4, b=4-√10. a+b=8 which is rational.

3] false. √2*√8= √16 = 4 which is rational.

Answer 2

If r is any rational number s is any irrational number, then r/s is irrational.

true

if r/s was rational then it would be equivalent to a/b where a and b are integers (b ≠ 0)

and GCD(a,b) = 1

this means

r b = s a

r is rational as well, then r = c/d (c and d integers, d non zero)

thus

(c/d) b = s a

that is

s = b c/(a d)



which means that s is rational, in contradiction with the hypothesis



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The sum of any two positive irrational numbers is irrational



false

take x = 2 + √3 and y = 2 - √3

x + y = 4



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The product of any two irrational numbers is irrational

false

take the numbers of the previous example



x = 2 + √3 and y = 2 - √3

xy = 4 - 3 = 1

Answer 3

I think the first two are TRUE.



The key to the second one is that both are positive. If any were allowed, you can have two opposite irrational numbers add together to get zero:



√2 - √2 = 0



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The last one definitely is FALSE. As a counterexample to disprove it:



√2 * √8



Two irrational numbers. Multiply them together:



√16 = 4



You have a rational number.