Problem 15 - Algebra and Trigonometry

Question

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ \sqrt{3}-i $$

Step-by-Step Solution

Answer

\(2 (\cos(-\frac{\pi}{6}) + i \sin(-\frac{\pi}{6})) \) and \(2 e^{-i \frac{\pi}{6}}\).

Step 1: Identify the Real and Imaginary Parts

The given complex number is \( \sqrt{3} - i \). Here, the real part is \( \sqrt{3} \) and the imaginary part is \( -1 \).

Step 2: Plot the Complex Number

On the complex plane, plot the point corresponding to \( \sqrt{3} \) on the real axis and \( -1 \) on the imaginary axis. The point \( (\ \sqrt{3}, -1) \) represents the complex number.

Step 3: Calculate the Magnitude

The magnitude (or modulus) of the complex number is calculated as follows: \[ |z| = \sqrt{(\sqrt{3})^2 + (-1)^2} = \sqrt{3 + 1} = 2 \]

Step 4: Calculate the Argument

The argument (or angle) θ of the complex number is calculated using the formula \[ \theta = \tan^{-1} \left( \frac{-1}{\sqrt{3}} \right) \]. This simplifies to \[ \theta = \tan^{-1} \left( - \frac{1}{\sqrt{3}} \right) = \tan^{-1}(-\frac{1}{\sqrt{3}}) = -\frac{\pi}{6} \]

Step 5: Write the Complex Number in Polar Form

The polar form of the complex number is given by \[ z = r \left( \cos(\theta) + i \sin(\theta) \right) \], where \( r = 2 \) and \( \theta = -\frac{\pi}{6} \), thus \[ z = 2 (\cos(-\frac{\pi}{6}) + i \sin(-\frac{\pi}{6})) \]

Step 6: Write the Complex Number in Exponential Form

The exponential form of the complex number is given by \[ z = r e^{i \theta} \], where \( r = 2 \) and \( \theta = -\frac{\pi}{6} \), thus \[ z = 2 e^{-i \frac{\pi}{6}} \]

Key Concepts

Complex Plane

The complex plane is a visual way to represent complex numbers. Think of it as a grid. The horizontal axis represents the real part of the complex number, and the vertical axis represents the imaginary part. For the complex number \( \sqrt{3} - i \), you plot \( \sqrt{3} \) on the real axis and \( -1 \) on the imaginary axis. This brings you to the point \( (\sqrt{3}, -1) \) on the complex plane. Visualizing complex numbers this way helps you see their geometric properties.

Polar Form

Polar form is another way to represent complex numbers. Instead of using a rectangular grid (real and imaginary), you describe the number using its magnitude and angle. The magnitude \( r \) is the distance from the origin to the point, and the angle \( \theta \) is measured from the positive real axis. For our example \( \sqrt{3} - i \), the magnitude is calculated as 2, and the angle is \( -\frac{\pi}{6} \). So in polar form, it is written as \( 2 (\cos( -\frac{\pi}{6}) + i \sin( -\frac{\pi}{6})) \).

Exponential Form

Exponential form uses Euler's formula to represent complex numbers. Euler's formula states that \( e^{i \theta} = \cos( \theta) + i \sin( \theta) \). By substituting the magnitude \( r \) and angle \( \theta \), we can write complex numbers in a compact form. For \( \sqrt{3} - i \), the magnitude is 2 and the angle is \( -\frac{\pi}{6} \), so in exponential form, it becomes \( 2 e^{-i \frac{\pi}{6}} \). This form is especially useful in advanced mathematics and engineering.

Magnitude of Complex Numbers

The magnitude (or modulus) of a complex number measures its distance from the origin in the complex plane. For any complex number \( a + bi \), the magnitude is calculated as \( \sqrt{a^2 + b^2} \). For the number \( \sqrt{3} - i \), the magnitude is \( \sqrt{(\sqrt{3})^2 + (-1)^2} = \sqrt{3 + 1} = 2 \). This tells us how 'long' the number is when viewed as a vector on the complex plane.

Argument of Complex Numbers

The argument of a complex number is the angle it makes with the positive real axis. For a complex number \( a+bi \), the argument \( \theta \) is found using \( \tan^{-1} \left(\frac{b}{a}\right) \). In our example \( \sqrt{3} - i \), the argument is calculated as \( \tan^{-1} \left( -\frac{1}{\sqrt{3}} \right) = -\frac{\pi}{6} \). The argument helps in converting the complex number to polar and exponential forms, giving us insights into its geometric orientation.