Question

(a) Using Euler's relation (2.76), prove that any complex number \(z=x+\) iy can be written in the form \(z=r e^{i \theta},\) where \(r\) and \(\theta\) are real. Describe the significance of \(r\) and \(\theta\) with reference to the complex plane. (b) Write \(z=3+4 i\) in the form \(z=r e^{i \theta} .\) (c) Write \(z=2 e^{-i \pi / 3}\) in the form \(x+i y.\)

Short answer

Any complex number can be expressed as \( z = re^{i\theta} \). For \( z = 3+4i \), \( z = 5e^{i0.927} \), and for \( z = 2e^{-i\pi/3} \), \( z = 1 - i\sqrt{3} \).

Step-by-step solution

Understanding Euler's Relation

Euler's relation is given by the formula \( e^{i\theta} = \cos\theta + i\sin\theta \). This relation connects complex exponentials with trigonometric functions.

Expressing a Complex Number

To express any complex number \( z = x + iy \) in the form \( z = re^{i\theta} \), we first need to identify \( r \) as the modulus of \( z \) and \( \theta \) as the argument. The modulus is \( r = \sqrt{x^2 + y^2} \), and the argument \( \theta \) can be found using \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).

Applying to Specific Values

(a) For any complex number \( z = x + iy \), substituting into Euler's formula gives \( z = r(\cos\theta + i\sin\theta) = re^{i\theta} \). This shows that any complex number can be expressed in polar form using Euler's relation.

Transforming z = 3+4i

For the complex number \( z = 3 + 4i \):- Calculate \( r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 \).- Calculate \( \theta = \tan^{-1}\left(\frac{4}{3}\right) \).Therefore, \( z = 5 e^{i \theta} \), where \( \theta \approx 0.927 \) radians.

Transforming Polar to Rectangular

(b) For \( z = 2 e^{-i \pi / 3} \), the rectangular form is found using \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \):- Here \( r = 2 \), \( \theta = -\pi/3 \).- \( x = 2 \cos(-\pi/3) = 2 \times 1/2 = 1 \).- \( y = 2 \sin(-\pi/3) = -\sqrt{3} \).Thus, \( z = 1 - i\sqrt{3} \).

Key concepts

Euler's Relation

Euler's Relation is an important formula in mathematics that relates complex numbers and trigonometry. It is expressed as \( e^{i\theta} = \cos\theta + i\sin\theta \). This elegantly bridges real and imaginary components of a complex number through exponential functions.
In Euler’s form, any complex number can be represented as \( re^{i\theta} \), where \( r \) is the modulus of the number, and \( \theta \) is the argument. This can be visualized by placing the complex number on a unit circle in the complex plane.
Understanding Euler's relation is crucial because it simplifies complex multiplication and powers by transforming them into exponential operations. This transformation opens the door to advanced topics in math and physics by providing a simple yet powerful way to handle complex numbers.

Polar Form of Complex Numbers

The Polar Form provides a way to express complex numbers emphasizing their magnitude and angle, rather than their algebraic form of \( x + iy \). In the polar form, a complex number \( z \) is written as \( r e^{i\theta} \).

• \( r \) is the modulus or magnitude of the complex number.
• \( \theta \) is the argument, which is the angle formed with the positive real axis.

To convert a complex number from its algebraic form to polar form:

• Calculate the modulus \( r \) using \( r = \sqrt{x^2 + y^2} \).
• Find the argument \( \theta \) using \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).

This form is particularly useful in simplifying calculations involving powers and roots of complex numbers. It provides an intuitive insight into the geometric representation of complex numbers on the Argand diagram.

Modulus and Argument of Complex Numbers

The modulus and argument are fundamental components when working with complex numbers, providing depth to their geometric and mathematical analysis.

• Modulus: The modulus \( r \) of a complex number \( z = x + iy \) is its distance from the origin in the complex plane, computed as \( r = \sqrt{x^2 + y^2} \).
• Argument: The argument \( \theta \) is the angle measured between the positive real axis and the line representing the complex number, given by \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).

Together, these components allow complex numbers to be represented in their polar form, bridging algebraic and geometric perspectives. Knowing the modulus and argument facilitates advanced operations, such as multiplication and division, which become much more manageable in polar form.