Complex Numbers in Polar Form

Introduction to Complex Numbers in Polar Form

Complex numbers extend the real number system by introducing the imaginary unit, i, defined by:

i² = −1

A complex number is commonly written in rectangular form:

z = a + bi

where:

• a is the real part.

• b is the imaginary part.

• i is the imaginary unit.

For example:

z = 3 + 4i

This number can be represented as the point (3, 4) on the complex plane.

However, complex numbers can also be described using a distance from the origin and an angle. This is called polar form.

Polar form is particularly useful when multiplying, dividing, raising complex numbers to powers, and finding roots.

The Complex Plane

The complex plane, also called the Argand plane, is a coordinate system used to represent complex numbers geometrically.

It has two perpendicular axes:

• The horizontal axis represents the real part.

• The vertical axis represents the imaginary part.

For the complex number:

z = 3 + 4i

the corresponding point is:

(3, 4)

The number can also be represented by a vector extending from the origin to this point.

This geometric interpretation allows complex numbers to be described using polar coordinates.

Modulus and Argument

Two quantities are particularly important.

Modulus (r)

The modulus is the distance from the origin to the point representing the complex number.

It is written:

r = |z|

For z = a + bi:

|z| = √(a² + b²)

The modulus is always nonnegative.

Argument (θ)

The argument is the angle measured counterclockwise from the positive real axis to the vector representing the complex number.

It is written:

arg(z) = θ

The angle can be measured in degrees or radians.

The principal argument is commonly chosen within:

−π < θ ≤ π

although some courses use 0 ≤ θ < 2π.

Understanding Polar Form

A complex number can be expressed in polar form as:

z = r(cos θ + i sin θ)

where:

• r is the modulus.

• θ is an argument of the complex number.

• cos θ determines the horizontal component.

• sin θ determines the vertical component.

The relationships between rectangular and polar form are:

a = r cos θ

b = r sin θ

Therefore:

z = a + bi

can be rewritten as:

z = r(cos θ + i sin θ)

Why Polar Form Works

Consider a complex number represented by the point (a, b).

Using right-triangle trigonometry:

cos θ = a/r

sin θ = b/r

Rearranging:

a = r cos θ

b = r sin θ

Substituting into z = a + bi:

z = r cos θ + ir sin θ

Factorizing r:

z = r(cos θ + i sin θ)

This gives the polar representation.

Converting Rectangular Form to Polar Form

To convert a complex number from rectangular form to polar form:

1. Identify the real and imaginary components.

2. Calculate the modulus.

3. Determine the reference angle.

4. Identify the correct quadrant.

5. Write the polar expression.

Worked Example: Converting 3 + 4i

Convert:

z = 3 + 4i

Step 1: Identify the components.

a = 3

b = 4

Step 2: Calculate the modulus.

r = √(3² + 4²)

r = √25

r = 5

Step 3: Calculate the argument.

tan θ = 4/3

θ = tan⁻¹(4/3)

θ ≈ 53.13°

Step 4: Check the quadrant.

Both the real and imaginary parts are positive, so the number lies in Quadrant I.

Final answer:

z = 5(cos 53.13° + i sin 53.13°)

The rectangular and polar forms represent exactly the same complex number.

Worked Example: A Number in Quadrant II

Convert:

z = −3 + 3i

Step 1: Calculate the modulus.

r = √((-3)² + 3²)

r = √18

r = 3√2

Step 2: Determine the argument.

The reference angle is:

tan⁻¹(3/3) = 45°

Because the point lies in Quadrant II:

θ = 180° − 45°

θ = 135°

Final answer:

z = 3√2(cos 135° + i sin 135°)

In radians:

z = 3√2(cos(3π/4) + i sin(3π/4))

Converting Polar Form to Rectangular Form

To convert from polar form to rectangular form, use:

a = r cos θ

b = r sin θ

Then write:

z = a + bi

Worked Example

Convert:

z = 6(cos 60° + i sin 60°)

Step 1: Calculate the real part.

a = 6cos 60°

a = 6(1/2)

a = 3

Step 2: Calculate the imaginary part.

b = 6sin 60°

b = 6(√3/2)

b = 3√3

Final answer:

z = 3 + 3√3i

Worked Example: Negative Argument

Convert:

z = 4(cos(−30°) + i sin(−30°))

a = 4cos(−30°)

a = 2√3

b = 4sin(−30°)

b = −2

Final answer:

z = 2√3 − 2i

This complex number lies in Quadrant IV.

The Argument and Quadrants

Determining the correct argument is essential when converting complex numbers into polar form.

The inverse tangent function may return a reference angle rather than the correct argument.

For principal arguments, Quadrants III and IV are often represented using negative angles.

For example:

225° = −135° + 360°

Both describe the same direction.

Special Cases

For a positive real number:

arg(z) = 0°

For a negative real number:

arg(z) = 180° (or π radians under the stated principal-argument convention)

For a positive imaginary number:

arg(z) = 90°

For a negative imaginary number:

arg(z) = −90°

The complex number zero has modulus zero, but its argument is undefined.

Multiplying Complex Numbers in Polar Form

Multiplication is one of the main advantages of polar form.

Suppose:

z₁ = r₁(cos θ₁ + i sin θ₁)

z₂ = r₂(cos θ₂ + i sin θ₂)

Their product is:

z₁z₂ = r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]

This means:

• Multiply the moduli.

• Add the arguments.

Geometric Interpretation

Multiplying two complex numbers combines scaling and rotation.

• The modulus determines how much the magnitude changes.

• The argument determines the amount of rotation.

Worked Example: Multiplying in Polar Form

Given:

z₁ = 3(cos 30° + i sin 30°)

z₂ = 2(cos 45° + i sin 45°)

Find z₁z₂.

Step 1: Multiply the moduli.

r = 3 × 2

r = 6

Step 2: Add the arguments.

θ = 30° + 45°

θ = 75°

Final answer:

z₁z₂ = 6(cos 75° + i sin 75°)

This is much simpler than expanding both complex numbers in rectangular form.

Dividing Complex Numbers in Polar Form

Division follows a similar rule.

For z₂ ≠ 0:

z₁/z₂ = (r₁/r₂)[cos(θ₁ − θ₂) + i sin(θ₁ − θ₂)]

This means:

• Divide the moduli.

• Subtract the arguments.

Worked Example: Division in Polar Form

Given:

z₁ = 12(cos 100° + i sin 100°)

z₂ = 3(cos 40° + i sin 40°)

Find z₁/z₂.

Step 1: Divide the moduli.

r = 12/3

r = 4

Step 2: Subtract the arguments.

θ = 100° − 40°

θ = 60°

Final answer:

z₁/z₂ = 4(cos 60° + i sin 60°)

In rectangular form:

z₁/z₂ = 2 + 2√3i

Geometric Meaning of Division

Dividing complex numbers reverses the scaling and rotation associated with multiplication.

For example, dividing by a complex number with:

• Modulus 2 scales distances by 1/2.

• Argument 30° rotates the result by −30°.

This geometric interpretation is useful in transformations and engineering.

Trigonometric and Exponential Forms

Complex numbers in polar form can also be written using exponential notation.

Euler's formula states:

e^(iθ) = cos θ + i sin θ

Therefore:

z = r(cos θ + i sin θ)

can be written as:

z = re^(iθ)

This is called the exponential form of a complex number.

For example:

5(cos 60° + i sin 60°)

can be written as:

5e^(iπ/3)

Here, the angle is expressed in radians.

Why Exponential Form Is Useful

Exponential notation makes multiplication and division especially compact.

Multiplication:

r₁e^(iθ₁) × r₂e^(iθ₂) = r₁r₂e^(i(θ₁ + θ₂))

Division:

(r₁e^(iθ₁))/(r₂e^(iθ₂)) = (r₁/r₂)e^(i(θ₁ − θ₂))

Exponential form is widely used in advanced mathematics, physics, and electrical engineering.

De Moivre's Theorem

De Moivre's Theorem provides a method for raising complex numbers in polar form to integer powers.

It states:

[r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ)

where n is an integer.

In exponential form:

(re^(iθ))ⁿ = rⁿe^(inθ)

The theorem shows that raising a complex number to a power involves:

• Raising the modulus to that power.

• Multiplying the argument by that power.

Worked Example: Squaring a Complex Number

Given:

z = 2(cos 30° + i sin 30°)

Find z².

Step 1: Apply De Moivre's Theorem.

z² = 2²(cos(2 × 30°) + i sin(2 × 30°))

Step 2: Simplify.

z² = 4(cos 60° + i sin 60°)

Step 3: Convert to rectangular form.

z² = 4(1/2 + i√3/2)

Final answer:

z² = 2 + 2√3i

Worked Example: Raising to the Fourth Power

Given:

z = 3(cos 45° + i sin 45°)

Find z⁴.

Step 1: Raise the modulus to the fourth power.

3⁴ = 81

Step 2: Multiply the argument by four.

4 × 45° = 180°

Step 3: Apply the theorem.

z⁴ = 81(cos 180° + i sin 180°)

Since:

cos 180° = −1

sin 180° = 0

Final answer:

z⁴ = −81

This example demonstrates how a complex number can produce a real number when raised to a power.

Finding Roots of Complex Numbers

De Moivre's Theorem can also be used to find roots of complex numbers.

Suppose:

z = r(cos θ + i sin θ)

The n distinct nth roots, for z ≠ 0, are:

wₖ = r^(1/n)[cos((θ + 360°k)/n) + i sin((θ + 360°k)/n)]

where:

k = 0, 1, 2, ..., n − 1

When using radians, replace 360° with 2π.

Worked Example: Finding Cube Roots

Find the cube roots of:

z = 8(cos 0° + i sin 0°)

Step 1: Calculate the cube root of the modulus.

r = 8^(1/3)

r = 2

Step 2: Determine the arguments.

θₖ = (0° + 360°k)/3

For k = 0:

θ₀ = 0°

For k = 1:

θ₁ = 120°

For k = 2:

θ₂ = 240°

Step 3: Write the roots.

w₀ = 2(cos 0° + i sin 0°) = 2

w₁ = 2(cos 120° + i sin 120°) = −1 + √3i

w₂ = 2(cos 240° + i sin 240°) = −1 − √3i

Final answer:

2, −1 + √3i, −1 − √3i

Geometric Interpretation of Roots

The nth roots of a nonzero complex number:

• Have equal modulus.

• Are equally spaced around a circle.

• Have angular separation of 360°/n.

• Form the vertices of a regular n-sided polygon when n ≥ 3.

For example:

• Square roots appear at opposite ends of a diameter.

• Cube roots form an equilateral triangle.

• Fourth roots form a square.

• Fifth roots form a regular pentagon.

This provides a powerful connection between complex numbers and geometry.

Geometric Transformations Using Complex Numbers

Complex multiplication can represent transformations in the plane.

Rotation

Multiplying a complex number by:

cos θ + i sin θ

rotates the corresponding point counterclockwise by θ.

For example, multiplying by i rotates a point 90° counterclockwise.

Consider:

z = 2 + 3i

Multiply by i:

iz = i(2 + 3i)

iz = 2i + 3i²

iz = −3 + 2i

The point (2, 3) becomes (−3, 2).

This is exactly a 90° counterclockwise rotation about the origin.

Scaling

Multiplying by a positive real number changes the modulus without changing the argument.

For example:

2(3 + 4i) = 6 + 8i

The distance from the origin doubles.

Rotation and Scaling Together

Multiplying by a complex number with modulus 3 and argument 45°:

• Triples the distance from the origin.

• Rotates the point 45° counterclockwise.

These transformations are important in computer graphics, geometry, and engineering.

Applications of Complex Numbers in Polar Form

Complex numbers in polar form are useful whenever magnitude and direction are important.

Electrical Engineering

Alternating-current circuits often involve voltages and currents that oscillate sinusoidally.

Complex numbers can represent both:

• The amplitude or magnitude of an electrical quantity.

• Its phase angle relative to another oscillating quantity.

For example, an electrical impedance might be represented as:

Z = 10(cos 30° + i sin 30°) Ω

This indicates an impedance magnitude of 10 Ω and a phase angle of 30°.

Polar form makes multiplication and division of these quantities more convenient.

Rotations in Computer Graphics

Complex multiplication provides an efficient method for rotating two-dimensional points.

A point represented by z can be rotated by an angle θ using:

z' = z(cos θ + i sin θ)

This is useful in animation, game development, and graphical transformations.

Waves and Oscillations

Complex numbers can represent the amplitude and phase of oscillating quantities.

For example:

z = Ae^(iθ)

can represent a phasor with magnitude A and phase θ.

This approach simplifies calculations involving waves, vibrations, and periodic motion.

Common Mistakes

Confusing modulus and argument

The modulus describes distance from the origin, while the argument describes direction.

Using the wrong quadrant

Inverse tangent calculations must be adjusted according to the signs of the real and imaginary parts.

Forgetting to add arguments during multiplication

Multiply moduli and add arguments.

Forgetting to subtract arguments during division

Divide moduli and subtract arguments.

Applying De Moivre's Theorem incorrectly

When raising a complex number to a power, raise the modulus to that power and multiply the argument by the power.

Mixing degrees and radians

Ensure that angles are expressed consistently, particularly when using exponential notation.

Assuming the argument is unique

Arguments differing by multiples of 360° or 2π represent the same direction.

Forgetting multiple roots

A nonzero complex number has exactly n distinct nth roots.

Using a negative modulus in standard polar form

The modulus should be nonnegative. A negative radial factor can be converted to a positive modulus by adding 180° to the argument.

Trying to find an argument for zero

The complex number 0 has no defined argument.

Key Terms

• Complex number: A number written in the form a + bi, where a and b are real numbers and i² = −1.

• Imaginary unit: The number i, defined by i² = −1.

• Rectangular form: A complex number written as a + bi.

• Polar form: A complex number written as r(cos θ + i sin θ).

• Complex plane: A coordinate plane representing real and imaginary components.

• Argand plane: Another name for the complex plane.

• Real part: The horizontal component a of a complex number.

• Imaginary part: The coefficient b of i in a complex number.

• Modulus: The distance of a complex number from the origin, written |z|.

• Argument: The angle of a complex number measured from the positive real axis.

• Principal argument: The selected argument within a standard interval, commonly −π < θ ≤ π.

• Exponential form: A complex number written as re^(iθ).

• Euler's formula: The identity e^(iθ) = cos θ + i sin θ.

• De Moivre's Theorem: A theorem used to raise complex numbers in polar form to integer powers.

• Complex root: A complex number that produces a given number when raised to a specified power.

• Rotation: A transformation that changes the direction of a point about a fixed centre.

• Scaling: A transformation that changes the distance of a point from the origin.

• Phase angle: An angular measure describing the relative position within an oscillation.

• Phasor: A complex representation of the magnitude and phase of a sinusoidal quantity.

• Geometric transformation: A change in the position, orientation, or size of a geometric figure.

Key Takeaways

• Complex numbers can be represented in both rectangular and polar form.

• Rectangular form describes real and imaginary components, while polar form describes magnitude and direction.

• The modulus of z = a + bi is |z| = √(a² + b²).

• The argument is determined from the real and imaginary components, taking the correct quadrant into account.

• Polar form is written as z = r(cos θ + i sin θ).

• Conversion to rectangular form uses a = r cos θ and b = r sin θ.

• Multiplying complex numbers in polar form involves multiplying their moduli and adding their arguments.

• Dividing complex numbers in polar form involves dividing their moduli and subtracting their arguments.

• De Moivre's Theorem provides an efficient method for raising complex numbers to integer powers.

• The nth roots of a nonzero complex number are equally spaced around a circle in the complex plane.

• Complex multiplication can represent rotation and scaling geometrically.

• Exponential form, z = re^(iθ), provides a compact representation based on Euler's formula.

• Polar complex numbers are important in electrical engineering, wave analysis, computer graphics, and advanced mathematics.

Check Your Understanding

Recall and Understanding

1. What is the difference between rectangular form and polar form of a complex number?

2. Define the modulus and argument of a complex number.

3. Explain why the argument must be checked against the quadrant.

4. State the rules for multiplying and dividing complex numbers in polar form.

5. State De Moivre's Theorem and explain its purpose.

Application

6. Convert z = 5 + 5i into polar form.

7. Convert z = −4 + 4i into polar form.

8. Convert z = 8(cos 30° + i sin 30°) into rectangular form.

9. Multiply 3(cos 20° + i sin 20°) by 4(cos 50° + i sin 50°).

10. Divide 12(cos 120° + i sin 120°) by 3(cos 45° + i sin 45°).

Analysis and Challenge

11. Use De Moivre's Theorem to calculate [2(cos 30° + i sin 30°)]³.

12. Find the fourth roots of 16, expressing your answers in rectangular form.

13. Explain geometrically why multiplying a complex number by i rotates its position by 90° counterclockwise.

14. A complex number has modulus 6 and argument 150°. Determine its rectangular form and identify its quadrant.

15. A point represented by z = 3 + 4i is multiplied by a complex number with modulus 2 and argument 60°. Determine the resulting modulus and argument, and explain the geometric transformation.

Final Summary

Complex numbers in polar form provide a powerful connection between algebra, trigonometry, and geometry.

By representing complex numbers using modulus and argument, operations such as multiplication, division, powers, and roots become easier to understand and calculate.

De Moivre's Theorem extends these relationships by providing a systematic method for calculating powers and roots. The geometric interpretation of complex multiplication also reveals how complex numbers can represent rotations and scaling transformations.

These ideas form an important foundation for advanced mathematics, electrical engineering, physics, signal processing, and computer graphics.