Polar Form of Complex Numbers

Polar Form of Complex Numbers

A complex number can be represented in different forms. The standard form is a + bi, where a is the real part and b is the imaginary part. However, complex numbers can also be expressed in polar form, which is useful for multiplication, division, and finding powers and roots of complex numbers.

Understanding Polar Coordinates

In polar coordinates, a complex number is represented using a magnitude (or modulus) and an angle (or argument). The magnitude is the distance from the origin to the point represented by the complex number in the complex plane, while the angle is the direction of that point from the positive real axis.

Magnitude of a Complex Number

The magnitude of a complex number z = a + bi is calculated using the formula:

|z| = √(a² + b²)

For example, consider the complex number z = 3 + 4i. The magnitude is:

|z| = √(3² + 4²) = √(9 + 16) = √25 = 5

Argument of a Complex Number

The argument of a complex number is the angle θ formed with the positive real axis. It can be found using the arctangent function:

θ = arctan(b/a)

For the complex number z = 3 + 4i, the argument is:

θ = arctan(4/3)

Using a calculator, you find that:

θ ≈ 0.93 radians

Converting to Polar Form

The polar form of a complex number is expressed as:

z = r(cos θ + i sin θ)

where r is the magnitude and θ is the argument. Using the previous example, we can convert z = 3 + 4i to polar form:

z = 5(cos(0.93) + i sin(0.93))

This can also be written using Euler's formula:

z = re^(iθ)

Thus, the polar form of z = 3 + 4i can also be expressed as:

z = 5e^(i0.93)

Example of Conversion

Convert the complex number z = -1 + √3i to polar form.

1. Calculate the magnitude:

|z| = √((-1)² + (√3)²) = √(1 + 3) = √4 = 2

2. Calculate the argument:

θ = arctan(√3 / -1)

The complex number is in the second quadrant, so we add π to the angle:

θ = π + arctan(√3 / -1) = π - π/3 = 2π/3

3. Write in polar form:

z = 2(cos(2π/3) + i sin(2π/3))

or

z = 2e^(i(2π/3))

Multiplication and Division in Polar Form

Multiplying and dividing complex numbers in polar form is simpler than in rectangular form. When multiplying two complex numbers, you multiply their magnitudes and add their angles:

If z_1 = r_1e^(iθ_1) and z_2 = r_2e^(iθ_2), then:

z = z_1 × z_2 = r_1r_2e^(i(θ_1 + θ_2))

For example, if z_1 = 2e^(iπ/4) and z_2 = 3e^(iπ/3), then:

z = (2 × 3)e^(i(π/4 + π/3)) = 6e^(i(3π/12 + 4π/12)) = 6e^(i(7π/12))

Division in Polar Form

When dividing two complex numbers, you divide their magnitudes and subtract their angles:

If z_1 = r_1e^(iθ_1) and z_2 = r_2e^(iθ_2), then:

z = z_1 / z_2 = (r_1 / r_2)e^(i(θ_1 - θ_2))

For example, if z_1 = 4e^(iπ/6) and z_2 = 2e^(iπ/3), then:

z = (4 / 2)e^(i(π/6 - π/3)) = 2e^(i(π/6 - 2π/6)) = 2e^(-iπ/6)

Finding Powers and Roots in Polar Form

To find powers of a complex number in polar form, use De Moivre's theorem. If z = re^(iθ), then:

z^n = r^n e^(inθ)

For example, to find (1 + i)^3:

1. Convert 1 + i to polar form:

r = √(1² + 1²) = √2, θ = arctan(1/1) = π/4

2. Apply De Moivre's theorem:

(1 + i)^3 = (√2)^3 e^(i(3π/4)) = 2√2 e^(i(3π/4))

To find roots, use the formula:

z^(1/n) = r^(1/n) e^(i(θ + 2kπ)/n) where k = 0, 1, ..., n-1.

Example of Finding Roots

Find the square roots of the complex number z = -1.

1. Convert to polar form:

r = 1, θ = π

2. Apply the root formula:

z^(1/2) = 1^(1/2) e^(i(π + 2kπ)/2), k = 0, 1

3. Calculate for k = 0:

z_1 = e^(iπ/2) = i

4. Calculate for k = 1:

z_2 = e^(i(π + 2π)/2) = e^(i(3π/2)) = -i

Common Mistakes

Summary

• A complex number can be expressed in polar form as z = r(cos θ + i sin θ) or z = re^(iθ).
• The magnitude |z| = √(a² + b²) and the argument θ = arctan(b/a) determine the polar representation.
• Multiplication and division of complex numbers in polar form are simplified by multiplying/dividing magnitudes and adding/subtracting angles.
• De Moivre's theorem is used to find powers and roots of complex numbers in polar form.

Check your understanding

1. Convert the complex number z = 1 - i to polar form.
2. What is the argument of the complex number z = -2 - 2i?
3. Multiply the complex numbers z_1 = 1 + i and z_2 = 2e^(iπ/4) in polar form.
4. Find the cube roots of the complex number z = 8.