Problem 73 - Contemporary Precalculus

Question

Explain what is meant by saying that multiplying a complex number \(z=r(\cos \theta+i \sin \theta)\) by \(i\) amounts to rotating \(z 90^{\circ}\) counterclockwise around the origin. [Hint: Express \(i\) and \(i z\) in polar form. What are their relative positions in the complex plane?]

Step-by-Step Solution

Answer

Question: Prove that multiplying a complex number in polar form by \(i\) results in a rotation of \(90^{\circ}\) counterclockwise around the origin. Answer: Multiplying a complex number \(z = r(\cos\theta + i \sin\theta)\) in polar form by \(i\) gives us \(i \cdot z = r(-\sin\theta + 1i\cos\theta)\). Comparing the angles of \(z\) and \(i \cdot z\), we see that the new angle is \((\theta + \frac{\pi}{2})\), which represents a \(90^{\circ}\) counterclockwise rotation around the origin.

Step 1: Express complex number z in polar form

The given complex number \(z\) is already in polar form: \(z = r (\cos\theta + i \sin\theta)\).

Step 2: Multiply z by i and express the result in polar form

We multiply \(z\) by \(i\), which can be expressed as \(0+1i\): $$ i \cdot z = (0+1i) \cdot \left[r (\cos\theta + i \sin\theta)\right]\\ i \cdot z = r(\cos\theta \cdot 1i + i^2 \sin\theta) \\ i \cdot z = r (-\sin\theta +1i\cos\theta) $$ The equation above is the polar form of \(i \cdot z\).

Step 3: Compare the angles of z and i*z

We need to prove that the angle between the two complex numbers is \(90^{\circ}\) counterclockwise. For the complex number z, the angle is \(\theta\). For the complex number \(i \cdot z\), the angle is \((90^{\circ}+\theta)\), since it represents a rotation of \(90^{\circ}\) counterclockwise. To represent this in radians, it should be \((\frac{\pi}{2} + \theta)\). If we rotate the complex number \(z\) by \(90^{\circ}\) or \(\frac{\pi}{2}\) radians counterclockwise, the new angle would be \((\theta+\frac{\pi}{2})\), which is the angle of the complex number \(i \cdot z\). So, multiplying \(z\) by \(i\) amounts to rotating \(z\) by \(90^{\circ}\) counterclockwise around the origin.

Key Concepts

Polar Form of Complex Numbers

The polar form of a complex number is a different way to represent a complex number in terms of its magnitude and direction. In contrast to the standard rectangular form, which is based on the x (real) and y (imaginary) axes, the polar form uses the length of the vector (referred to as the modulus) and the angle it makes with the positive real axis (referred to as the argument).

The polar form is often written as follows: \[ z = r (\cos\theta + i \sin\theta) \], where \( r \) is the modulus of the complex number and \( \theta \) is the argument, or the angle in radians.

In practical applications, this form is particularly useful for multiplying or dividing complex numbers because it simplifies the process to multiplying their moduli and adding their arguments. This is much more straightforward than performing the same operations in rectangular form, where you would have to deal with binomial products or quotients.

Rotation in the Complex Plane

When we talk about the multiplication of complex numbers, an interesting phenomenon in the geometric interpretation is the rotation effect in the complex plane. A complex number can be thought of as a vector originating from the origin (0,0) to the point (a,b) in a 2-dimensional plane. This plane is known as the complex plane, with the horizontal axis representing the real component and the vertical axis representing the imaginary component.

Multiplying a complex number by another, notably by a unit modulus complex number like \(i\), can be visualized as a rotation of the vector representing the initial complex number. Depending on the argument of the multiplier, the vector rotates counterclockwise (positive angle) or clockwise (negative angle) without changing its length.

Example of Rotation

If we multiply a complex number by \(i\), which has an argument of \(\frac{\pi}{2}\) or \(90^\circ\), the complex number rotates 90 degrees counterclockwise, as shown in the exercise.

Complex Number Multiplication by i

Multiplying a complex number by \(i\) may seem unintuitive at first, but when visualized in the complex plane, it becomes a straightforward operation. The complex unit \(i\) is defined by its property \(i^2 = -1\). When a complex number \(z\) is multiplied by \(i\), each term of \(z\) is effectively rotated 90 degrees in the complex plane.

The step-by-step solution demonstrates this idea: \[ i \cdot z = r (-\sin\theta + i \cos\theta) \]. The original angle \(\theta\) of the complex number \(z\) is increased by \(90^{\text{\circ}}\). This is because multiplying by \(i\) is equivalent to a rotation by \(\frac{\text{\pi}}{2}\) radians which is a quarter turn counterclockwise. In essence, the real part becomes the new imaginary part (with a sign change), and the imaginary part becomes the new real part.

Angle of Complex Numbers

The concept of angle in complex numbers is pivotal to understanding the complex plane and operations such as multiplication and division. The angle, or the argument, of a complex number is the measure of rotation needed to reach its position from the positive real axis.

For a given complex number \( z = r (\cos\theta + i \sin\theta) \), the angle \( \theta \) provides insight into the geometric significance of the number. It is important to note that while the modulus (or magnitude) of a complex number is always non-negative, the angle can take on any value, although it is often normalized to be within the range \( -\pi < \theta \leq \pi \).

The angle helps to describe the behavior of complex number multiplication, as seen in our exercise. When a complex number is multiplied by another (like \(i\)), the angles add up. If the angles are visualized as rotations, the net effect of a multiplication can be seen as the sum of the individual rotations from the positive real axis.