Problem 11 - Precalculus

Question

Express each complex number in polar form. $$1+\sqrt{3} i$$

Step-by-Step Solution

Answer

In polar form, the complex number is expressed as \( 2(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}) \).

Step 1: Identify Real and Imaginary Parts

A complex number is typically written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. In this case, for the number \( 1 + \sqrt{3}i \), we identify \( a = 1 \) and \( b = \sqrt{3} \).

Step 2: Calculate the Magnitude (r)

The magnitude (also known as the modulus) of a complex number \( z = a + bi \) can be calculated using the formula \( r = \sqrt{a^2 + b^2} \). Substituting the values, we get: \[ r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2. \]

Step 3: Determine the Argument (θ)

The argument \( \theta \) is found using the formula \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \). Here, \( a = 1 \) and \( b = \sqrt{3} \), so: \[ \theta = \tan^{-1}\left(\frac{\sqrt{3}}{1}\right) = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3}. \]

Step 4: Write in Polar Form

The polar form of a complex number is expressed as \( z = r(\cos\theta + i\sin\theta) \). Using the magnitude \( r = 2 \) and argument \( \theta = \frac{\pi}{3} \), the polar form is: \[ z = 2 \left( \cos\frac{\pi}{3} + i\sin\frac{\pi}{3} \right). \]

Key Concepts

Complex Numbers

Complex numbers are fascinating because they extend our concept of numbers beyond just real numbers. A complex number is typically written in the form of \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. Imaginary numbers include the term \( i \), which represents the square root of -1. For example, in the complex number \( 1 + \sqrt{3}i \), \( 1 \) is the real part, while \( i\sqrt{3} \) is the imaginary part. This means complex numbers are all about combining real and imaginary dimensions.

• The real part is the horizontal component when visualized on a plane.
• The imaginary part is the vertical component on this plane.

This way of representing numbers allows for much more complex operations and solutions to equations where real numbers fall short.

Magnitude

The magnitude of a complex number gives us an idea about its 'size' when plotted on a complex plane. It is also known as the modulus. For a complex number \( z = a + bi \), the magnitude is calculated using the formula:\[r = \sqrt{a^2 + b^2}.\]This is analogous to finding the hypotenuse of a right triangle in trigonometry. The real part \( a \) and imaginary part \( b \) form the two legs of the triangle, while \( r \) is like the hypotenuse. For our example, with \( a = 1 \) and \( b = \sqrt{3} \), the magnitude calculated is:\[r = \sqrt{1^2 + (\sqrt{3})^2} = 2.\]This gives insight into how far the complex number is from the origin of the complex plane.

Argument of a Complex Number

The argument of a complex number, often represented as \( \theta \), is the angle formed between the positive real axis and the line that connects the complex number with the origin in the complex plane.To determine \( \theta \), we use:\[\theta = \tan^{-1}\left(\frac{b}{a}\right).\]For \( 1 + \sqrt{3}i \), where \( a = 1 \) and \( b = \sqrt{3} \), the argument is:\[\theta = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3}.\]This discusses how the complex number is positioned with respect to the positive real axis. The argument can vary from \(-\pi\) to \(\pi\), providing a full 360-degree perspective from standard position in the coordinate system.

Polar Coordinates

Turning a complex number into its polar form involves using polar coordinates. This conversion is useful in many mathematical and engineering applications.In polar form, a complex number \( z = a + bi \) can be expressed as:\[z = r(\cos\theta + i\sin\theta).\]Here:

• \( r \) is the magnitude
• \( \theta \) is the argument

For \( 1 + \sqrt{3}i \), its polar form calculates to:\[z = 2\left( \cos \frac{\pi}{3} + i\sin \frac{\pi}{3} \right).\]This transformation helps in simplifying multiplication and division operations of complex numbers, and it also simplifies handling powers and roots of complex numbers, utilizing Euler's formula for more advanced equations.