Question

Two complex numbers are given, where m, n, p, and q are real numbers.

m+ni
p+qi

For what relationship among m, n, p, and q, will be the product of these two complex numbers have only an imaginary part?

This is Algebra 2.
Answer choices:
np mq
mp-np=0
mp+nq=-1

Answer 1

`\displaystyle pm - qn = 0\text{ and } pn + qm \\eq 0`

Or:

`pm = qn \text{ and } pn + qm \\eq 0`

Explanation:

We are given two complex numbers:

`\displaystyle m + ni \text{ and } p + qi`

Where m, n, p, and q are real numbers.

And we want to determine the relationship among m, n, p, and q such that the product of the two complex numbers will only have an imaginary part.

Find the product:

`\displaystyle \begin{aligned} (m+ni)(p+qi) &= p(m+ni) + qi(m+ni) \\ &= (pm + pni) + (qmi + qni^2) \\ &= (pm + pni) + (qmi - qn) \\ &= (pm - qn) + i(pn + qm) \end{aligned}`

Therefore, for the product to have only an imaginary part, the real part must be zero and the imaginary part must not be zero. That is:

`\displaystyle pm - qn = 0\text{ and } pn + qm \\eq 0`

In conclusion, the relationship between m, n, p, and q is:

`\displaystyle pm - qn = 0\text{ and } pn + qm \\eq 0`

Or:

`pm = qn \text{ and } pn + qm \\eq 0`

(Note: I couldn't understand the provided answer choices, but the above answer is indeed correct (and the second equation can be ignored).)