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Complex Numbers on
the Complex Plane

Visualize addition, subtraction, multiplication, and conjugation of complex numbers as geometric transformations: translations, rotations, and reflections.

Algebra 2

What Is the Complex Plane?

Every complex number \( z = a + bi \) maps to a unique point \( (a, b) \) on a two-dimensional plane. The horizontal axis represents the real part, and the vertical axis represents the imaginary part.

Re Im -2 -1 1 2 3
\( i \)
\( 2i \)
\( -i \)
\( 3+2i \)
\( b=2 \)
\( a=3 \)
 0

Modulus and Argument

Modulus \( |z| \)
The distance from the origin to point \( z \): $$ |z| = \sqrt{a^2 + b^2} $$
Argument \( \theta \)
The angle from the positive real axis to the line segment \( \overline{Oz} \): $$ \theta = \arg(z) = \arctan\!\left(\frac{b}{a}\right) $$

Why Geometry Matters

Algebra tells you what the answer is. Geometry tells you why. Each complex-number operation has a visual meaning:

Addition
Vector translation: tip-to-tail parallelogram
Subtraction
Reverse translation: vector difference
Multiplication
Rotate + Scale: multiply moduli, add arguments
Conjugation
Reflection across the real axis

Addition: Vector Translation

Re Im
\( z_1 = 2+i \)
\( z_2 = 1+2i \)
\( z_1+z_2=3+3i \)
 2 1 3
\( i \)
\( 2i \)
\( 3i \)

Subtraction: Reverse Translation

Re Im
\( z_1 = 3+i \)
\( z_2 = 1+2i \)
\( z_1-z_2=2-i \)
 2 3 1
\( i \)
\( 2i \)
\( -i \)

Multiplication: Rotate and Scale

The fundamental rule of complex multiplication in polar form:

Scale
The distance from the origin becomes \( |z_1| \cdot |z_2| \).
Rotate
The angle from the real axis becomes \( \theta_1 + \theta_2 \).

Visualizing Multiplication

Re Im
\( z_1: r=2, \theta=60^\circ \)
\( 60^\circ \)
\( z_2: r=1.5, \theta=45^\circ \)
\( z_1 z_2: r=3, \theta=105^\circ \)
\( 105^\circ \)
 1 2 3

Conjugation: Reflection Across Re

Re Im
\( z = 3+2i \)
\( \bar{z} = 3-2i \)
 mirror line (Re axis) 1 2 3
\( i \)
\( 2i \)
\( -i \)
\( -2i \)

Rectangular vs Polar Form

Rectangular \( a + bi \) Polar \( r\,\text{cis}\,\theta \)
Best for Addition, Subtraction Multiplication, Division, Powers
Addition Add real & imaginary parts Convert to rectangular first
Multiplication FOIL then simplify \( i^2 = -1 \) Multiply moduli, add arguments
Powers Repeated multiplication \( z^n = r^n\,\text{cis}(n\theta) \)
Conjugate Negate imaginary part Negate argument

How to Apply the Geometric View

1
Identify the operation: addition, subtraction, multiplication, power, or conjugation.
2
Choose the best form: rectangular for \( \pm \), polar for \( \times, \div, \) or powers.
3
Convert if needed: \( r = \sqrt{a^2+b^2} \), \( \theta = \arctan(b/a) \), adjusting for quadrant.
4
Apply the geometric rule: translate (add), rotate+scale (multiply), reflect (conjugate).
5
Verify visually: sketch the result on the complex plane to check direction and distance.

Example 1: Addition on the Plane

Compute \( (2 + 3i) + (-1 + i) \) and represent the result geometrically.

1
Add real parts: \( 2 + (-1) = 1 \).
2
Add imaginary parts: \( 3 + 1 = 4 \).
3
Result: \( 1 + 4i \), located at the point (1, 4).

Example 2: Multiplication via Polar

Compute \( (1 + i)(1 + i) \) geometrically.

1
Convert to polar: \( 1 + i \) has \( r = \sqrt{1^2+1^2} = \sqrt{2} \), \( \theta = 45^\circ \).
2
Multiply moduli: \( \sqrt{2} \cdot \sqrt{2} = 2 \).
3
Add arguments: \( 45^\circ + 45^\circ = 90^\circ \).
4
Result: \( 2\,\text{cis}\,90^\circ = 2(\cos 90^\circ + i\sin 90^\circ) = 0 + 2i = 2i \).

Example 3: \( (-1+\sqrt{3}\,i)^3 = 8 \)

Re Im
\( z: r=2, 120^\circ \)
\( z^2: r=4, 240^\circ \)
\( z^3 = 8 \)
\( 120^\circ \)
 8 2 4

Common Mistakes

✕ Adding Moduli Instead of Multiplying
Multiplication means multiply moduli and add arguments, not add moduli.
✕ Wrong quadrant for argument
\( \arctan(b/a) \) alone ignores quadrant. Use signs of \( a \) and \( b \) to place \( \theta \) correctly.
✕ Confusing conjugate with negation
\( \bar{z} = a - bi \) (flip imaginary). \( -z = -a - bi \) (flip both). They are different operations.

Summary: Operations at a Glance

Operation Algebraic Rule Geometric Meaning
Addition \( (a+c) + (b+d)i \) Parallelogram (vector sum)
Subtraction \( (a-c) + (b-d)i \) Vector difference
Multiplication \( r_1 r_2\,\text{cis}(\theta_1+\theta_2) \) Scale + Rotate
Powers \( r^n\,\text{cis}(n\theta) \) Repeated rotation + scaling
Conjugation \( a - bi \) Reflect across Re axis

Question 1 of 5

What is the geometric interpretation of adding two complex numbers on the complex plane?

Question 2 of 5

If \( z \) has modulus 3 and argument \( 40^\circ \), what is the modulus and argument of \( z^2 \)?

Question 3 of 5

The conjugate \( \bar{z} \) of a complex number \( z \) is obtained geometrically by:

Question 4 of 5

Compute \( (-1 + \sqrt{3}\,i)^3 \) using the geometric (polar) approach.

Question 5 of 5

If \( z_1 \) has modulus 5 and argument \( 30^\circ \), and \( z_2 \) has modulus 2 and argument \( 60^\circ \), what is the argument of \( z_1 \cdot z_2 \)?