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 1

First we have to find product

`\\ \sf\longmapsto (m+ni)(p+qi)`

`\\ \sf\longmapsto m(p+qi)+ni(p+qi)`

`\\ \sf\longmapsto mp+mqi+npi+nqi^2`

`\\ \sf\longmapsto mp+mqi+nqi-nq`

`\\ \sf\longmapsto mp-nq+(mq+nq)i`

• We have to keep imaginary parts i.e Im(z)

`\\ \sf\longmapsto mp-nq=0`

`\\ \sf\longmapsto mp=nq`

Answer 2

Given complex numbers:

• m+ni
• p+qi

Their product is:

• (m + ni)(p + qi) =
• mp + npi + mqi + nqi² =
• mp - nq + (np + mq)i

In order to have only an imaginary part we need:

• mp - nq = 0

or

• mp = nq