Equality of Two Complex Numbers, and the Two Equations It Gives You

In one line

Two complex numbers are equal when their real parts are equal and their imaginary parts are equal, so one complex equation gives you two real ones.

The definition

Two complex numbers z1 = a + ib and z2 = c + id are equal if and only if a = c and b = d.

That is it. There is no other way for them to be equal. It looks so obvious that students skip it, and then cannot start half the questions in the topic.

Why it is a technique and not a definition

Look at what the definition actually gives you. If somebody tells you that two complex expressions are equal, they have told you two separate facts about real numbers, for the price of one statement.

That is the engine of every "find the values of x and y" question in this part of the paper. The procedure is always the same three steps.

1. Get both sides into the form (something real) + i (something real).
2. Set the two real parts equal. That is your first equation.
3. Set the two imaginary parts equal. That is your second equation.

Then you have two ordinary simultaneous equations in two unknowns and the complex numbers have done their job.

Worked: the standard question

Find real x and y such that the equation below holds.

The left side is already separated. Compare real parts, then imaginary parts.

Real parts: x + 2y = 7. Imaginary parts: 3x minus y equals 1.

Solve them together. From the second, y = 3x minus 1. Substituting into the first gives x plus 6x minus 2 equals 7, so 7x = 9 and therefore x = 9/7, and then y = 27/7 minus 1 = 20/7.

Now check, which is the step worth building a habit of.

Both parts match, so the answer is right.

Worked: when the separating has to be done first

This is the harder version, and the commoner one in an examination. Find x and y from the equation below.

Nothing can be compared yet, because the left side is not in standard form. Multiply it out first.

Now compare. Real: 2x minus 3y equals 13. Imaginary: 3x plus 2y equals 1.

Multiply the first by two and the second by three, then add: 4x minus 6y equals 26, and 9x plus 6y equals 3, so 13x = 29. Hmm, that does not give whole numbers, which usually means checking the arithmetic. Multiply the first by 2: 4x - 6y = 26. Multiply the second by 3: 9x + 6y = 3. Adding gives 13x = 29, so x = 29/13 and then from the second equation 2y = 1 - 3x = 1 - 87/13 = -74/13, so y = -37/13.