There are two possible answers:
- 'a' is irrational and b is rational, OR
- 'a' is rational and b is irrational
You can only pick one of those two cases. In short, exactly one of the 'a' or b is irrational and the other is rational. Both 'a' and b cannot be rational together, and they both cannot be irrational together.
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Explanation:
For a moment, let's say that 'a' and b are both rational numbers. This means they would be ratios of integers. So,
a = p/q
b = r/s
where p,q,r,s are integers. The denominators q & s cannot be zero.
Multiplying those rational numbers 'a' and b leads to
a*b = (p/q)*(r/s) = (p*r)/(q*s)
We get a result in the form of integer/integer. Recall that multiplying any two integers leads to some other integer. Since (p*r)/(q*s) is a ratio of integers, it is rational and that makes a*b = c to be rational. However, your teacher said that c is irrational which is the opposite of rational.
In short, if both 'a' and b are rational, then a*b = c is rational. This means we know for certain that both 'a' and b cannot be both rational for c to be irrational.
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Next, we'll consider 'a' and b to both be irrational. Is it possible to have a*b = c to be irrational as well? It depends.
If we have the following
a = sqrt(3)
b = sqrt(12)
then,
a*b = sqrt(3)*sqrt(12) = sqrt(3*12) = sqrt(36) = 6
Showing that a*b is rational because 6 = 6/1 is a ratio of integers. This is a bit strange considering how 'a' and b themselves were irrational, but they led to a rational product.
This doesn't always happen since something like
sqrt(3)*sqrt(5) = sqrt(3*5) = sqrt(15)
is irrational. So if both 'a' and b are irrational together, then a*b could be rational or it could be irrational. We would need more information about the values of 'a' and b.
Therefore, we know that we can rule out the case that 'a' and b are irrational together.
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Lastly, we'll consider the case when one of 'a' or b is rational while the other is irrational. The order doesn't matter. So we'll make 'a' to be irrational and b to be rational.
Since b is rational, this means b = p/q for some integers p,q where q is nonzero. The value of 'a' cannot be written as a ratio like this because it is irrational.
Multiplying any nonzero rational number with an irrational one always leads to an irrational result. If a*b was rational, then we could say
a*b = r/s
a*(p/q) = r/s
a = (r/s)*(q/p)
a = (r*q)/(s*p)
Showing that 'a' is rational, but this contradicts the fact we made 'a' to be irrational at the top of this section. Therefore, a*b is always irrational if exactly one of 'a' or b is irrational and the other is rational.
In other words, if,
- 'a' is rational and b is irrational, or
- 'a' is irrational and b is rational
then a*b = c is irrational
