Represent Complex Numbers on the Complex Plane

Lesson 4 of 9: Complex Number System

In this lesson:

• Plot complex numbers on the complex plane
• Convert between rectangular and polar form
• Explain why both forms describe the same number

What You Will Learn Today

1. Plot complex numbers on the complex plane using real and imaginary axes
2. Convert from rectangular form to polar form
3. Convert from polar form to rectangular form
4. Explain why rectangular and polar forms represent the same number

Recall: Coordinate Plane and Plotting Points

• Two axes: horizontal -axis and vertical -axis
• Plotting : 3 units right, 4 units up — unique point
• Quadrant rule: signs of and determine the quadrant

Can you plot and without hesitation?

The Complex Plane Is the Coordinate Plane

Start with a familiar coordinate plane. Rename the axes:

• Horizontal axis: real axis (instead of x-axis)
• Vertical axis: imaginary axis (instead of y-axis)

plots at the point .

You already know how to do this.

Five Complex Numbers on the Plane

Rectangular Form: Two Perpendicular Moves

The form gives two coordinates:

• : real part → horizontal position
• : imaginary part → vertical position

Axis rules:

• : on the real axis
• : on the imaginary axis

Reading and Plotting in Both Directions

Quick Check: Plot and Read

1. Plot on the complex plane: , ,
2. A point is at on the complex plane. Write it as .

Problem 2: real part is 0.

Two Addresses for the Same Location

One location, two descriptions:

• Rectangular: "3 blocks east, 4 blocks north" → position
• Polar: "5 blocks at 53° from east" → same point

Both are correct. Neither is more right.

Complex numbers have the same two systems.

Polar Coordinates: Modulus and Argument

Every nonzero complex number has:

• Modulus : distance from origin
• Argument : angle from positive real axis

For : ,

always.

Polar Form of a Complex Number

Conversion Formulas and the Quadrant Rule

Rectangular → Polar:

Polar → Rectangular: ,

returns values in only.
Plot the point first. If , add to the arctan result.

Rectangular to Polar: Worked Example

Convert to polar form.

Step 1:

Step 2: — first quadrant, no adjustment

Polar form:

Check: , ✓

Polar to Rectangular: Worked Example

Convert to rectangular form.

Step 1:

Step 2:

Rectangular form:

Quadrant Trap: Second Quadrant Example

Convert .

. Naive arctan: ✗

is in quadrant II → add : ✓

Quick Check: Convert Both Ways

1. Convert to polar form. Exact .
2. Convert to rectangular form.

Problem 1: identify the quadrant before computing .

Are These the Same Number?

vs.

Convert : ,

So ✓

Different forms. Same number.

Each Form Has Its Strongest Use

Preview:

Polar Form Makes Squaring One Line

in rectangular: — four steps

in polar: ,

Square: , → ✓

Polar squaring: multiply the modulus, double the angle.

Three Computations: No Help Provided

1. Plot . State the quadrant.
2. Convert to polar form. Give exact .
3. Convert to rectangular form.

Watch for These Three Common Errors

1. Axis labels: Imaginary axis is numbered 1, 2, 3 — not
2. Quadrant error: returns — plot first, add when
3. Negative : — sign belongs in , not

What You Now Know About the Complex Plane

✓ plots at — real axis horizontal
✓ , from real axis
✓ Polar form:
✓ Both forms: same number, different description

ignores quadrant — add when
always

Next Lesson: Multiplication as Rotation

In HSN.CN.B.5, you'll multiply complex numbers geometrically.

The product's modulus multiplies. The product's argument adds.