Question

Let $$ \begin{aligned} z &=r(\cos \theta+i \sin \theta) \\ z^{\prime} &=r^{\prime}\left(\cos \theta^{\prime}+i \sin \theta^{\prime}\right), \text { and } \\ z^{\prime \prime} &=r^{\prime \prime}\left(\cos \theta^{\prime \prime}+i \sin \theta^{\prime \prime}\right) \end{aligned} $$ as in Corollary 2. Prove that \(z=z^{\prime} z^{\prime \prime}\) if $$ r=|z|=r^{\prime} r^{\prime \prime}, \text { i.e., }\left|z^{\prime} z^{\prime \prime}\right|=\left|z^{\prime}\right|\left|z^{\prime \prime}\right|, \text { and } \theta=\theta^{\prime}+\theta^{\prime \prime} . $$ Discuss the following statement: To multiply \(z^{\prime}\) by \(z^{\prime \prime}\) means to rotate \(\overrightarrow{0 z^{\prime}}\) counterclockwise by the angle \(\theta^{\prime \prime}\) and to multiply it by the scalar \(r^{\prime \prime}=\) \(\left|z^{\prime \prime}\right| .\) Consider the cases \(z^{\prime \prime}=i\) and \(z^{\prime \prime}=-1 .\)

Short answer

We show that \(z = z'z''\) by verifying the magnitude and angle conditions. Multiplying complex numbers rotates and scales vectors as described. For \(z'' = i\), it is a \(\frac{\pi}{2}\) rotation, and for \(z'' = -1\), it is a \(\pi\) rotation.

Step-by-step solution

Express Product in Polar Form

Start by expressing both \(z'\) and \(z''\) in polar form. We have \(z' = r' (\cos \theta' + i \sin \theta')\) and \(z'' = r''(\cos \theta'' + i \sin \theta'')\). The product \(z' z''\) in polar form is \(r' r'' (\cos (\theta' + \theta'') + i \sin (\theta' + \theta''))\).

Verify Magnitude Condition

According to the problem, \(r = |z| = |z' z''|\) and \(|z' z''| = |z'| |z''|\) which implies \(r = r' r''\). The magnitude condition is satisfied when \(z' z''\) is expressed as \(r' r'' (\cos (\theta' + \theta'') + i \sin (\theta' + \theta''))\).

Verify Angle Addition Condition

According to the problem condition, \(\theta = \theta' + \theta''\). This is directly from the properties of sine and cosine in the multiplication of complex numbers in polar form where the angles add up. Here, the given condition is satisfied: the angular part of \(z' z''\) matches \(\theta' + \theta''\).

Discuss the Rotation and Scaling

When multiplying \(z'\) by \(z''\), you rotate \(z'\) by \(\theta''\) (the angle of \(z''\)) clockwise, and then scale its magnitude by \(r''\) (the magnitude of \(z''\)). This aligns with multiplying complex numbers where the arguments (angles) add, and magnitudes multiply.

Consider Special Case: \(z'' = i\)

If \(z'' = i\), then \(r'' = 1\) and \(\theta'' = \frac{\pi}{2}\). This rotates \(z'\) by \(\frac{\pi}{2}\) counterclockwise on the complex plane without changing its magnitude, as \(r'' = 1\).

Consider Special Case: \(z'' = -1\)

If \(z'' = -1\), then \(r'' = 1\) and \(\theta'' = \pi\). This rotates \(z'\) by \(\pi\) radians (180 degrees) on the complex plane, effectively reflecting \(z'\) over the origin.

Key concepts

Polar Form

Complex numbers can be expressed quite uniquely in polar form, particularly useful for multiplication and division. A complex number \(z\) in polar form is represented as \(r(\cos \theta + i \sin \theta)\), where \(r\) is the magnitude of the complex number (distance from the origin in the complex plane), and \(\theta\) is the argument (angle from the positive x-axis).
The polar form showcases a marriage between trigonometry and complex numbers, making it easier to manage operations like multiplying two complex numbers. Instead of dealing with the real and imaginary parts separately, you multiply magnitudes and add angles.
So, when two complex numbers \(z'\) and \(z''\) are given in polar form as \(r'(\cos\theta' + i\sin\theta')\) and \(r''(\cos\theta'' + i\sin\theta'')\), their product \(z' z''\) becomes \(r' r'' (\cos(\theta' + \theta'') + i\sin(\theta' + \theta''))\).

• This simplifies the process of multiplying complex numbers.
• It transforms the cumbersome Cartesian form calculations into simple arithmetic operations on polar coordinates.

Magnitude Condition

In the realm of complex numbers, particularly in polar form, the magnitude condition reflects on how the sizes of the numbers relate. For multiplication, the magnitudes (or absolute values) of the numbers are multiplied together.
Given complex numbers \(z'\) and \(z''\) in polar form, the magnitude condition is articulated as:
\[ |z| = |z' z''| = |z'| |z''| = r' r'' \]
Here, \(|z' z''|\) expresses the result of multiplying magnitudes \(r'\) and \(r''\). This condition ensures that the magnitude of the product is a straightforward multiplication of each complex number's magnitude. This works seamlessly due to the properties of polar representation.

• Ensures that multiplication scales the number's size appropriately.
• Connects the product's magnitude directly to those of the factors.

This condition confirms that in the polar form, magnitudes are consistently handled, simplifying calculations remarkably.

Angle Addition

A wonderful property of polar form is the angle addition during multiplication of complex numbers. This involves summing up the angles—or arguments—of the numbers.
When complex numbers \(z'\) and \(z''\) are multiplied, their corresponding angles add up:
\[ \theta = \theta' + \theta'' \]
This property is crucial because it allows us to predict the direction in which the resulting product will point in the complex plane.
For instance, if \(z'\) has an angle \(\theta'\) and \(z''\) has an angle \(\theta''\), the product \(z' z''\) results in a new angle \(\theta' + \theta''\). This is a rotation by the angle \(\theta''\) applied to \(z'\), coupled with the magnitude scaling.

• Makes complex multiplication intuitive.
• The direction of the result is directed by the sum of the initial angles.

This rotational effect is deeply ingrained in the geometry of the complex plane, facilitating visualization and computation alike.