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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.

Respuesta :

mhanifa

Answer:

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

Answer:

First we have to find product

[tex]\\ \sf\longmapsto (m+ni)(p+qi)[/tex]

[tex]\\ \sf\longmapsto m(p+qi)+ni(p+qi)[/tex]

[tex]\\ \sf\longmapsto mp+mqi+npi+nqi^2[/tex]

[tex]\\ \sf\longmapsto mp+mqi+nqi-nq[/tex]

[tex]\\ \sf\longmapsto mp-nq+(mq+nq)i[/tex]

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

[tex]\\ \sf\longmapsto mp-nq=0[/tex]

[tex]\\ \sf\longmapsto mp=nq[/tex]