Problem 2 - Precalculus Mathematics for Calculus

Question

Let \(z=a+b i\) (a) The modulus of \(z\) is \(r=\) _____ and an argument of \(z\) is an angle \(\theta\) satisfying \(\tan \theta=\)_____. (b) We can express \(z\) in polar form as \(z=\)_____ where \(r\) is the modulus of \(z\) and \(\theta\) is the argument of \(z\).

Step-by-Step Solution

Answer

(a) \(r=\sqrt{a^2+b^2}\), \(\tan \theta=\frac{b}{a}\); (b) \(z=r(\cos \theta+i\sin \theta)\).

Step 1: Understand the Complex Number Notation

A complex number is expressed as \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part of the complex number.

Step 2: Calculate the Modulus

The modulus of a complex number \( z = a + bi \) is calculated as \( r = \sqrt{a^2 + b^2} \). This gives the magnitude of the complex number.

Step 3: Determine the Argument

The argument \( \theta \) of the complex number is the angle made with the positive real axis, calculated as \( \tan \theta = \frac{b}{a} \). This provides the angular measure of the number.

Step 4: Express in Polar Form

The polar form of the complex number is given by \( z = r (\cos \theta + i \sin \theta) \) or equivalently as \( z = re^{i\theta} \), where \( r \) is the modulus and \( \theta \) is the argument.

Key Concepts

Modulus of Complex Numbers

The modulus of a complex number is like measuring the distance from the origin to a point on the complex plane. If you imagine the complex number as a position on this plane, where the horizontal axis is the real part, and the vertical axis is the imaginary part, you can think of the modulus as the length of the line connecting the origin to this point.
For a complex number expressed as \( z = a + bi \), the modulus is calculated using the formula: \[ r = \sqrt{a^2 + b^2} \] Here,

• \(a\) is the real part of the complex number.
• \(b\) is the imaginary part.

The modulus \( r \) gives us the absolute value or the magnitude of the complex number, helping us understand its size, regardless of direction. It is always a non-negative number, giving us a clear metric to compare complex numbers by their magnitudes.

Polar Form of Complex Numbers

To express a complex number in polar form, we utilize both the modulus and the argument of the number. Polar form is an alternative representation that emphasizes the magnitude and direction rather than horizontal and vertical components.
For a complex number \( z = a + bi \), the polar form is represented as: \[ z = r (\cos \theta + i \sin \theta) \] This can also be elegantly written using Euler's formula as: \[ z = re^{i\theta} \] In this form,

• \( r \) is the modulus, representing the distance from the origin to the number.
• \( \theta \) is the argument, indicating the angle or direction from the positive real axis.

Using polar form can simplify many complex number operations, like multiplication and division, by allowing us to focus on angles and magnitudes instead of dealing with rectangular coordinates.

Argument of Complex Numbers

The argument of a complex number is a crucial concept that indicates its angular direction relative to the positive real axis. For a complex number \( z = a + bi \), the argument \( \theta \) is calculated using the inverse tangent function:\[ \tan \theta = \frac{b}{a} \]

• \(b\) is the imaginary part of the complex number.
• \(a\) is the real part.

The argument tells us how much to "turn" from the positive real axis to reach the direction where our complex number lies. This makes it an important property in applications involving rotations and oscillations.
However, since \( \tan \theta \) is periodic, the argument can have multiple valid values, typically expressed either within the range of 0 to \(2\pi\) (radians) or \(-\pi\) to \(\pi\) to ensure uniqueness. Understanding the argument can be significantly useful in tasks like plotting complex numbers on the complex plane or converting from polar to cartesian forms.