Question

the product and QUOTIENT of a non-zero rational and irrational number is
irrational. meaning...​

Answer 1

Answer:

"The product of a non-zero rational number and an irrational number is irrational."

If you multiply any irrational number by the rational number zero, the result will be zero, which is rational. Any other situation, however, of a rational times an irrational will be irrational.

Now, as for the quotient, let's see this example

Let’s assume the opposite, that you could divide a rational, R1, by an irrational, Irr, and get another rational, R2. That would mean:

R1/Irr = R2

Which in turn would mean (muliplying both sides by Irr/R2):

R1/R2 = Irr

And that also means:

(a/b) / (c/d) = Irr

(ad) / (cd) = Irr

And that would lead to a contradiction, namely that a rational number, (ad)/(cd), would be equal to an irrational.

But there is an exception. I have implicitly assumed that R2 is not zero. If R2 is zero, then you cannot muliply both sides by Irr/R2, because that would mean dividing by zero.

Moreover, if R1 is 0, then dividing it by any irrational produces 0, which would make R2 equal to 0. Which, once again, would make the proof invalid.

Thus, a rational number divided by an irrational must produce another irrational, assuming that the rational number is not zero.

I hope this helps....