Adding and Subtracting Complex Numbers

A complex number is written in standard form \( a + bi \), where \( a \) is the real part and \( b \) is the coefficient of the imaginary part. If you need a refresher on where \( i \) comes from and what it represents, take a look at the lesson on introduction to imaginary numbers first. Once you're comfortable with the idea that \( i = \sqrt{-1} \), adding and subtracting complex numbers turns out to be very similar to combining like terms in a polynomial expression.

To add two complex numbers, add their real parts together and add their imaginary parts together:

\( (a + bi) + (c + di) = (a + c) + (b + d)i \)

To subtract, distribute the negative sign across both parts of the second complex number, then combine like terms:

\( (a + bi) - (c + di) = (a - c) + (b - d)i \)

The key idea is that \( 1 \) and \( i \) act like two different "types" of terms, similar to \( x \) and a constant. You can only combine a real number with another real number, and an imaginary term with another imaginary term. You never add a real part directly to an imaginary part.

Example 1: Adding complex numbers

Simplify \( (4 + 3i) + (2 - 5i) \).

Group the real parts and the imaginary parts: \( (4 + 2) + (3i - 5i) \). This gives \( 6 - 2i \).

Example 2: Subtracting complex numbers

Simplify \( (7 - 2i) - (3 + 6i) \).

Distribute the negative sign first: \( 7 - 2i - 3 - 6i \). Now combine like terms: \( (7 - 3) + (-2i - 6i) = 4 - 8i \).

Example 3: A longer expression

Simplify \( (-5 + i) + (2 - 3i) - (4 - i) \).

Distribute the subtraction: \( -5 + i + 2 - 3i - 4 + i \). Combine the real parts: \( -5 + 2 - 4 = -7 \). Combine the imaginary parts: \( i - 3i + i = -i \). The result is \( -7 - i \).

Every complex number can be plotted as a point on the complex plane, with the real part on the horizontal axis and the imaginary part on the vertical axis. Adding two complex numbers behaves just like adding two vectors: you can picture it as placing one arrow at the tip of the other and reading off the coordinates of the resulting point.

The most frequent error is forgetting to distribute the negative sign to both terms when subtracting. Writing \( (a + bi) - (c + di) \) as \( a + bi - c + di \) (dropping the sign on \( di \)) will give a wrong imaginary part. Always rewrite the subtraction as adding the opposite of every term inside the second parentheses before combining. This is the same distribution idea used when working through subtraction of functions, just applied to real and imaginary terms instead of function expressions.

Another common slip is mixing a real part with an imaginary part, such as writing \( 6 - 2i \) as \( 4i \). Keep the two types of terms separate all the way to the final answer.

Try simplifying \( (9 - 4i) - (-3 + 4i) + (1 + i) \) on your own before checking the steps below.

Distribute the negative sign: \( 9 - 4i + 3 - 4i + 1 + i \). Combine real parts: \( 9 + 3 + 1 = 13 \). Combine imaginary parts: \( -4i - 4i + i = -7i \). The simplified result is \( 13 - 7i \).

Once you're confident combining complex numbers this way, related skills like complex conjugates build directly on the same idea of pairing up real and imaginary parts, this time to help you divide complex numbers.

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