Question

Discrete Question:
how would you prove that the product of a nonzero rational number and an irrational number is an irrational number?

Answer 1

Here is a hint:
rational numbers are fractions like a/b with a and b integers, right?

Answer 2

yes

Answer 3

So rational/rational will be a rational number, right?

Answer 4

correct

Answer 5

So suppose we have a*b = c.
With a and c rational.
What kind of number must b be?

Answer 6

to get a rational it would need to be rational wouldn't it?

Answer 7

is rational closed under multiplication

Answer 8

Correct. So in our original problem ...c
... suppose for a moment we have a rational * irrational = possibly rational.
But we know we cannot have that ...
... because rational * ______ = rational requires a rational in the blank.

Answer 9

Absolutely.
Yes, it is closed.

Answer 10

so that would a good answer?

Answer 11

it is also closed under + - /
except for division by zero.

Answer 12

If you note the proof by contradiction, then yes.
(assume c is rational ...
... then b = c/a is rational ...
but that violates our assumption that b is irrational)

Answer 13

ok, thanks!

Answer 14

You are welcome. This is a very powerful technique:
assume the opposite of your conclusion ...
... and prove that nonsense results from that assumption.
Things like "b is both rational and irrational", "1 = 0" and such are nonsense.

Answer 15

that is a good idea, I'll keep that in mind on the rest of these!

Answer 16

Great, good luck! :)