Operations with Complex Numbers

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Add, subtract, and multiply complex numbers.
  • Divide complex numbers using conjugates.
  • Simplify expressions involving \(i\) and \(i^2 = -1\).
  • Rewrite any complex expression in the form \(a + bi\).

Key Ideas

A complex number has the form: \[ a + bi \] where \(a\) and \(b\) are real numbers.

  • Real part = \(a\)
  • Imaginary part = \(b\)

Imaginary unit identity: \[ i^2 = -1 \]

Visualizing Complex Numbers

Complex numbers can be represented on the complex plane.

  • Horizontal axis = Real part
  • Vertical axis = Imaginary part

The complex number

\[ 3 + 2i \]

is represented by the point \((3,2)\).

Complex Conjugate

For \(a + bi\), the conjugate is: \[ a - bi \]

Useful for division: \[ \frac{a+bi}{c+di} \cdot \frac{c-di}{c-di} \]

Common Problem Types

Adding & Subtracting

Combine like terms (real with real, imaginary with imaginary).

Example:
\((3 + 5i) + (2 - 7i) = 5 - 2i\)

Multiplying Complex Numbers

Use distributive property or FOIL; replace \(i^2\) with \(-1\).

Example:
\((4 + i)(3 - 2i)\)
Real: \(4 \cdot 3 = 12\)
Outer: \(4 \cdot (-2i) = -8i\)
Inner: \(i \cdot 3 = 3i\)
Last: \(i \cdot (-2i) = -2i^2 = 2\)
Total: \(14 - 5i\)

Dividing Complex Numbers

Multiply top and bottom by the conjugate of the denominator.

Example:
\[ \frac{5 + i}{2 - 3i} \cdot \frac{2 + 3i}{2 + 3i} \]

Simplifying Using \(i^2 = -1\)

Replace \(i^2\) immediately.

Example:
\(6 - 4i + i^2 = 6 - 4i - 1 = 5 - 4i\)

Converting to Rectangular Form (\(a+bi\))

Even complicated expressions can be rewritten this way.

Example:
\(\frac{8}{1 - i}\) → multiply by \((1 + i)\).

Strategies

  • Treat \(i\) like a variable, but replace \(i^2\) with \(-1\).
  • Combine real parts and imaginary parts separately.
  • For division: conjugate, multiply, simplify, separate real/imag parts.
  • Always write your final answer in \(a + bi\) form.

Worked Examples

Example 1 — Add & Subtract

Compute: \[ (7 - 3i) - (2 + 5i) \]

Real: \(7 - 2 = 5\)
Imaginary: \(-3i - 5i = -8i\)
Final: \[ 5 - 8i \]

Example 2 — Multiply

Compute: \[ (3 + 2i)(4 - i) \]

FOIL:
\(3 \cdot 4 = 12\)
\(3 \cdot (-i) = -3i\)
\(2i \cdot 4 = 8i\)
\(2i \cdot (-i) = -2i^2 = 2\)

Add: \[ 14 + 5i \]

Example 3 — Divide

Compute: \[ \frac{6 - i}{3 + 2i} \]

Multiply by the conjugate: \[ \frac{(6 - i)(3 - 2i)}{(3 + 2i)(3 - 2i)} \]

Denominator: \[ 3^2 - (2i)^2 = 9 - (-4) = 13 \]

Numerator: \(6 \cdot 3 = 18\)
\(6 \cdot (-2i) = -12i\)
\(-i \cdot 3 = -3i\)
\(-i \cdot (-2i) = 2i^2 = -2\)

Combine: \[ (18 - 2) + (-12i - 3i) = 16 - 15i \]

Final: \[ \frac{16}{13} - \frac{15}{13}i \]

WarningCommon Mistakes
  • Forgetting \(i^2 = -1\).
  • Not multiplying by the conjugate when dividing.
  • Mixing real and imaginary parts incorrectly.
  • Leaving final answers not in \(a + bi\) form.
  • Dropping negative signs in multiplication.

Practice Problems

  1. \((4 + 7i) + (3 - 2i)\)
  2. \((5 - i)(2 - 3i)\)
  3. \(\frac{7 + 4i}{1 - 2i}\)
  4. Simplify: \(9 + 6i - 3i^2\)
  5. Put in \(a+bi\) form: \(\frac{10}{3 + i}\)

1. \(7 + 5i\)

2.
\((5 - i)(2 - 3i)\)
\(= 10 -15i -2i + 3i^2\)
\(= 10 -17i -3\)
\(= 7 - 17i\)

3.
Multiply by conjugate \((1 + 2i)\):
\[ \frac{(7+4i)(1+2i)}{1^2 - (2i)^2} \]
Denominator: \(1 - (-4) = 5\)
Numerator: \(7 + 14i + 4i + 8i^2 = 7 + 18i - 8\)
So: \(-1 + 18i\)
Final:
\[ -\frac{1}{5} + \frac{18}{5}i \]

4.
\(9 + 6i - 3i^2 = 9 + 6i + 3 = 12 + 6i\)

5.
Multiply by conjugate \((3 - i)\):
\[ \frac{10(3 - i)}{3^2 - i^2} = \frac{30 - 10i}{9 + 1} \]
Final:
\[ 3 - i \]

Summary

  • Add/subtract by combining like parts.
  • Multiply using FOIL; replace \(i^2\) with \(-1\).
  • Divide by multiplying by the conjugate.
  • Always express answers in \(a + bi\).
  • Real with real, imaginary with imaginary.
  • Conjugate for division every time.
  • \(i^2\) is your shortcut to simplify.
  • Keep track of negative signs!