How can you tell if two complex numbers are equal? It's actually very simple. Two complex numbers are equal if their real parts are equal, and their imaginary parts are equal. Of course, the two numbers must be in a + bi form in order to do this comparison.

Example One
If a + bi = c + di, what must be true of a, b, c, and d?

Solution
a = c, b = d

Example Two
Are 3 + 2i -1 and 2 + 4i - 2i equal?

Solution
3 + 2i - 1 = 2 + 2i
2 + 4i - 2i = 2 + 2i

The two quantities have equal real parts, and equal imaginary parts, so they are equal.

Example Three
Find x if x + 2i = x + 2xi - 3i

Solution
Combine like terms on the right: x + 2i = x + (2x - 3)i

Since the imaginary parts must be equal, 2 = 2x - 3, so x = .

Example Four
Find x and y if x + yi = 3y - (2x - 4)i

Solution
This is interesting: we have only one equation, but two variables; it doesn't seem like there's enough information to solve. But since we can break this into a real part and an imaginary part, we can create two equations:

x = 3y, y = 2x - 4

Doing a substituion gives us y = 6y - 4, or y = , which gives x = .

1.

Are 5 + 2i and 3 + 2 - 5i + 3i equal? Why or why not?

2.

Are 3 + 2i and 3 - 2i³ equal? Why or why not?

3.

If x + xi = x + 4xi - 9i, find x.

4.

If 2 + 3i and 2 - 5xi are the same complex number, find x.

5.

If x + 1 + 3i = y - 1 + 2yi, find x and y.

6.

If 5 + 2i² - 3i = x + yi, find x and y