Question

How do you find the values of m and n that make the equation `8+15i=2m+3ni` true?

Answer 1

Please see below.

Explanation:

Complex numbers are equal exactly when the real parts are equal and the imaginary parts are equal.

`a+bi = c+di` if and only of `a=c` and `b=d`.

So `8+15i = 2m+3ni` if and only if

`8 = 2m` and `15 = 3n`. And this happens only when

`m=4` and `n=5`.

Note Students sometimes wonder why mathematicians bother to state when two complex numbers are equal. After we state it, some react with "Well, obviously!".

But consider another kind of two-part number -- fractions.

When is `a/b=c/d`?

Since `3/6 = 7/14` the answer is NOT only when `a=c` and `b=d`.

(In case you've forgotten, the answer is `a/b=c/d` if and only if `ad=bc`.)