Problem 2 - Algebra and Trigonometry

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) \(z = r(\cos \theta + i\sin \theta)\) or \(z = re^{i\theta}\).

Step 1: Modulus of Complex Number

Given the complex number \( z = a + bi \), the modulus \( r \) of \( z \) is calculated using the formula \( r = \sqrt{a^2 + b^2} \). This provides the distance from the origin to the point \( (a, b) \) on the complex plane.

Step 2: Argument of Complex Number

The argument \( \theta \) of a complex number is the angle made with the positive x-axis. It can be found using \( \tan \theta = \frac{b}{a} \), which is derived from the fact the tangent of an angle in a right triangle is the opposite side over the adjacent side.

Step 3: Expressing in Polar Form

The polar form of a complex number is expressed as \( z = r(\cos \theta + i\sin \theta) \) or \( z = re^{i\theta} \). To express \( z \) in polar form, identify the modulus \( r \) and compute the argument \( \theta \). Substitute these into the polar form equation.

Key Concepts

Modulus of Complex Number

The modulus of a complex number is a key concept in understanding the properties of these intriguing numbers. The complex number is typically denoted as \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The modulus, represented as \( r \), is essentially the "size" or "absolute value" of this complex number. It tells us how far our complex number is from the origin on the complex plane.

To find this modulus, we use the formula \( r = \sqrt{a^2 + b^2} \). This formula might remind you of the Pythagorean theorem because it calculates the hypotenuse (modulus in this case) of a right triangle formed by the real and imaginary parts. For example, if we have the complex number \( 3 + 4i \), its modulus would be \( \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \).

The modulus helps us get a geometric picture of the complex number, making it easier to handle in further calculations or transformations.

Argument of Complex Number

The argument of a complex number is equally important as its modulus. This concept refers to the angle \( \theta \) that the line drawn from the origin to the point \( (a, b) \) makes with the positive x-axis (or real axis) in the plane.

To determine the argument, we use the relation \( \tan \theta = \frac{b}{a} \). This formula stems from basic trigonometry, where the tangent of an angle in our right triangle is calculated as the length of the opposite side (imaginary part \( b \)) over the adjacent side (real part \( a \)).

• If \( a > 0 \), \( \theta \) can be determined directly from \( \tan^{-1}\left(\frac{b}{a}\right) \).
• If \( a < 0 \), we need to adjust \( \theta \) by adding \( \pi \) radians to get the correct direction.

For example, for the complex number \( 1 + i \sqrt{3} \), \( \tan \theta = \sqrt{3}/1 \) so \( \theta = \frac{\pi}{3} \) radians.

Understanding the argument allows us to gain insights into the orientation of the complex number, an aspect that becomes particularly useful in complex multiplication and division.

Polar Form of Complex Numbers

The polar form is another way to represent complex numbers, offering a powerful perspective that relates closely to their geometric nature. In polar form, the complex number \( z = a + bi \) is expressed using its modulus \( r \) and argument \( \theta \) as \( z = r(\cos \theta + i\sin \theta) \). This can also be written in exponential form as \( z = re^{i\theta} \), leveraging Euler's formula.

Converting a complex number into its polar form involves two main steps:

• Calculate the modulus \( r \) using \( \sqrt{a^2 + b^2} \).
• Determine the argument \( \theta \) using \( \tan \theta = \frac{b}{a} \).

Once you have \( r \) and \( \theta \), simply substitute them into the polar form equation. For instance, for \( 3 + 4i \), with \( r = 5 \) and \( \theta = \tan^{-1}(\frac{4}{3}) \), the polar form is \( 5(\cos \theta + i\sin \theta) \) or \( 5e^{i\theta} \).

The polar form is incredibly useful for multiplying and dividing complex numbers, as multiplication becomes simply multiplying moduli and adding arguments, while division requires dividing moduli and subtracting arguments.