Problem 10 - Precalculus

Question

Express each complex number in polar form. $$2+2 i$$

Step-by-Step Solution

Answer

The polar form is \(2\sqrt{2}(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})\).

Step 1: Identify the Real and Imaginary Parts

The given complex number is expressed in the form \(a + bi\), where \(a = 2\) and \(b = 2\).

Step 2: Calculate the Modulus

The modulus \(r\) is the distance of the complex number from the origin in the complex plane, calculated using the formula \(r = \sqrt{a^2 + b^2}\). Substituting \(a = 2\) and \(b = 2\), we get \(r = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}\).

Step 3: Determine the Argument

The argument \(\theta\) represents the angle the complex number makes with the positive real axis. This is calculated using \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). Here, \(\theta = \tan^{-1}\left(\frac{2}{2}\right) = \tan^{-1}(1) = \frac{\pi}{4}\).

Step 4: Write the Polar Form

The polar form of a complex number is expressed as \(r(\cos \theta + i\sin \theta)\). With \(r = 2\sqrt{2}\) and \(\theta = \frac{\pi}{4}\), the polar form is \(2\sqrt{2}(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})\).

Key Concepts

Complex Number

A complex number is a fundamental concept in mathematics, introduced to extend the real numbers. It is usually written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. This structure allows complex numbers to represent points on a plane, which is called the complex plane. Here, the horizontal axis represents the real part, while the vertical axis represents the imaginary part.

• Real Part \( (a) \): Represents the horizontal component and is analogous to the typical real numbers.
• Imaginary Part \( (bi) \): Represents the vertical component, denoted with \(i\), where \(i\) is the imaginary unit with the property \(i^2 = -1\).

Understanding complex numbers is crucial because they allow for solutions to polynomial equations that do not have real number solutions. They also play an important role in various fields such as engineering, physics, and applied mathematics. For example, in the original exercise, the complex number is \(2 + 2i\), where both the real and imaginary components are 2.

Modulus

The modulus of a complex number is a measure of its size or distance from the origin in the complex plane. Denoted by \(r\), it is calculated using the formula \(r = \sqrt{a^2 + b^2}\), derived from the Pythagorean theorem.

• This formula essentially measures the hypotenuse of a right-angled triangle formed by the real and imaginary components.
• The modulus is always a non-negative real number.

The modulus gives insights into how far a complex number is from the origin, which is crucial when converting from rectangular to polar form. For the complex number \(2 + 2i\), the modulus is calculated as \(2\sqrt{2}\). Understanding and calculating the modulus helps in various applications such as signal processing, control systems, and physics.

Argument

The argument of a complex number is the angle formed with the positive real axis in the complex plane. It is denoted by \(\theta\) and is typically measured in radians. The formula to compute the argument is \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\).

• The argument gives the direction of the complex number from the origin.
• It identifies the position by measuring the angle counterclockwise from the positive real axis.

The argument can vary between \(-\pi\) and \(\pi\), making it essential to know the quadrant in which the angle falls. It plays a significant role in trigonometry, physics, and engineering. In the given exercise of the complex number \(2 + 2i\), the argument calculated is \(\frac{\pi}{4}\). Knowing how to derive the argument aids in the conversion between rectangular and polar forms, simplifying multiplication and division of complex numbers.