AOS3 Topic 2: Polar Form

Definition

The polar form of a complex number is a way to represent it using its magnitude (modulus) and argument (angle) in the complex plane.

The diagram shows the point P corresponding to the complex number \( z = a + bi \). We see that \( a = r \cos \theta \) and \( b = r \sin \theta \), and so we can write

\( z = a + bi \)

\( = r \cos \theta + (r \sin \theta) i \)

\( = r \left( \cos \theta + i \sin \theta \right) \)

This is called the polar form of the complex number. The polar form is abbreviated to

\( z = r \, \text{cis} \, \theta \)

The distance \( r = \sqrt{a^2 + b^2} \) is called the modulus of \( z \) and is denoted by \( |z| \).

The angle \( \theta \), measured anticlockwise from the horizontal axis, is called an argument of \( z \) and is denoted by \( \text{arg} \, z \).

Polar form for complex numbers is also called modulus–argument form.

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Non-uniqueness of polar form

Each complex number has more than one representation in polar form.

Since \( \cos \theta = \cos(\theta + 2n\pi) \) and \( \sin \theta = \sin(\theta + 2n\pi) \), for all \( n \in \mathbb{Z} \), we can write

\[ z = r \text{cis} \theta = r \text{cis}(\theta + 2n\pi) \] for all \( n \in \mathbb{Z} \)

The convention is to use the angle \( \theta \) such that \( -\pi < \theta \leq \pi \)

Complex Conjugate in Polar Form

It is easy to show that the complex conjugate, z, is a reflection of the point z in the horizontal axis. Therefore, if \( z = r \text{ cis } \theta \), then \( z = r \text{ cis }(-\theta) \). This can be seen below.

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Principal value of the argument

For a non-zero complex number \( z \), the argument of \( z \) that belongs to the interval \( (-\pi, \pi] \) is called the principal value of the argument of \( z \) and is denoted by \( \text{Arg} \, z \). That is, \( -\pi < \text{Arg} \, z \leq \pi \)

Basic operations on complex numbers in polar form

Basic operations on complex numbers in polar form involve addition, subtraction, multiplication, and division. In polar form, a complex number is expressed as \( z = r(\cos \theta + i \sin \theta) \), where \( r \) is the modulus (magnitude) and \( \theta \) is the argument (angle).

1. Addition and Subtraction: To add or subtract complex numbers in polar form, you simply add or subtract their moduli and add or subtract their arguments.
2. 
   
3. Multiplication: Multiplying complex numbers in polar form involves multiplying their moduli and adding their arguments.
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5. Division: Dividing complex numbers in polar form entails dividing their moduli and subtracting their arguments.

Multiplication of Complex Numbers in Polar Form

Let \( z_1 = r_1(cos\theta_1 + isin\theta_1) \) and \( z_2 = r_2(cos\theta_2 + isin\theta_2) \) be nonzero complex numbers. Then, their product \( z_1 \cdot z_2 \) in the polar form is written as:

\( z_1 \cdot z_2 = r_1 r_2 \left( (cos\theta_1 + \theta_2) + isin(\theta_1 + \theta_2) \right) \).

Proof:

Let us prove this theorem. The product of \( z_1 \) and \( z_2 \) is written as:

\( z_1 \cdot z_2 = ( r_1 cos(\theta_1 + isin\theta_1) ) \cdot ( r_2 (cos\theta_2 + isin\theta_2) ) = r_1 \cdot r_2 (cos\theta_1 + isin\theta_1)(cos\theta_2 + isin\theta_2) \).

Multiplying through the parentheses, we obtain:

\( z_1 \cdot z_2 = r_1 \cdot r_2 \left( cos\theta_1\costheta_2 + icos\theta_1\sin\theta_2 + icos\theta_2sin\theta_1 + i^{2}sin\theta_1sin\theta_2 \right) \).

Using \( i^{2} = -1 \) and gathering the real and imaginary terms, we have:

\( z_1 \cdot z_2 = r_1 \cdot r_2 \left( (cos\theta_1cos\theta_2 - sin\theta_1sin\theta_2) + i(cos\theta_1sin\theta_2 + cos\theta_2sin\theta_1) \right) \). (1)

We state the summation formulae for the sine and cosine functions:

• \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)
• \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)

We can apply the cosine summation formula to the real part and the sine summation formula to the imaginary part inside the parentheses of equation (1). Then, we can rewrite:

\( z_1 \cdot z_2 = r_1 \cdot r_2 \left( (cos\theta_1 +\theta_2) + isin(\theta_1 + \theta_2) \right) \).

This proves the theorem.

Relationship between Product of Complex Numbers and Their Moduli and Arguments

For any pair of nonzero complex numbers \( z_1 \) and \( z_2 \), we have:

• \( |z_1 \cdot z_2| = |z_1| \cdot |z_2| \)
• \( arg(z_1 \cdot z_2) = arg(z_1) + arg(z_2) \)







Quotient of Complex Numbers in Polar Form

Given a pair of nonzero complex numbers in the polar form \( z_1 = r (cos\theta_1 + isin\theta_1) \) and \( z_2 = r (\theta_2 + i\theta_2) \), their quotient can be written in the polar form as:

\( \frac{z_1}{z_2} = r \left( cos(\theta_1 - \theta_2) + isin(\theta_1 - \theta_2) \right) \).




Proof:

Let us prove this theorem. We begin with the quotient of \( z_1 \) and \( z_2 \) written as:

\( \frac{z_1}{z_2} = \frac{r_1 (cos\theta_1 + isin\theta_1)}{r_2 (cos\theta_2 + isin\theta_2)} = \frac{r_1}{r_2} = \frac{cos\theta_1 + isin\theta_1}{cos\theta_1 + isin\theta_2} \).

To make the denominator of this fraction real, we multiply the numerator and denominator by the conjugate of the denominator to obtain:

\( \frac{z_1}{z_2} = \frac{r_1}{r_2}\frac{(\cos\theta_1 + i\sin\theta_1)(\cos\theta_2 - i\sin\theta_2)}{(\cos\theta_2 + i\sin\theta_2)(\cos\theta_2 - i\sin\theta2)} \).

Multiplying through the parentheses, we obtain:

\( \frac{z_1}{z_2} = \frac{r_1}{r_2}\frac{\cos\theta_1\cos\theta_2 - i\cos\theta_1\sin\theta_2 + i\sin\theta_1\cos\theta_2 - i^{2}\sin\theta_1\sin\theta_2}{\cos&^{2}\theta_2 + \sin^{2}\theta_2} \).

Using \( i^{2} = -1 \) and gathering the real and imaginary terms, we have:

\( \frac{z_1}{z_2} = \frac{r_1}{r_2}\frac{(\cos\theta_1\cos\theta_2 + \sin\theta_1\sin\theta_2) + i(\sin\theta_1\cos\theta_2 - \cos\theta_1\sin\theta_2)}{\cos^{2}\theta_2 + \sin^{2}\theta_2} \).

Using the trigonometric identity \( \sin^{2}\theta_2\cos^{2}\theta_2 = 1 \), we can simplify this expression to:

\( \frac{z_1}{z_2} = \frac{r}{r}(\theta + i\theta) = r (\theta + i\theta) \).

Finally, we state the difference identities for sine and cosine:

\( \cos(\theta_1 - \theta_2) = \cos\theta_1\cos\theta_2 + \sin\theta_1\sin\theta_2 \), \( \sin(\theta_1 - \theta_2) = \sin\theta_1\cos\theta_2 - \cos\theta_1\sin\theta_2 \).

We apply the cosine difference identity in the real part and the sine difference identity in the imaginary part of the complex number within the parentheses on the right-hand side of equation (2). Then, we can rewrite:

\( \frac{z_1}{z_2} = r (\theta_1 - \theta_2 + i(\theta_1 - \theta_2)) \).

This proves the theorem.




Relationship between Quotient of Complex Numbers and Their Moduli and Arguments

For any pair of nonzero complex numbers \( z_1 \) and \( z_2 \), we have:

• The modulus of the quotient of \( z_1 \) and \( z_2 \) is equal to the quotient of the moduli of \( z_1 \) and \( z_2 \), i.e., \( \left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|} \).
• The argument of the quotient of \( z_1 \) and \( z_2 \) is equal to the difference of the arguments of \( z_1 \) and \( z_2 \), i.e., \( \text{arg} \left( \frac{z_1}{z_2} \right) = \text{arg}(z_1) - \text{arg}(z_2) \).




Polar Form of a Reciprocal of a Complex Number

Given a nonzero complex number in the polar form \( z = r(cos\theta + isin\theta) \), the reciprocal can be written in the polar form as:

\( \frac{1}{z} = \frac{1}{r} \left( cos(-\theta) + isin(-\theta) \right) \).