Problem 26 - Algebra and Trigonometry

Question

In Exercises \(11-26,\) plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$ 1-i \sqrt{5} $$

Step-by-Step Solution

Answer

The polar form of the given complex number \(1 - i\sqrt{5}\) is \(\sqrt{6} (\cos(-1.381) + i \sin(-1.381))\) or \(\sqrt{6}e^{i(-1.381)}\)

Step 1: Identify real and imaginary parts

In a complex number, the part without 'i' is the real part and the part with 'i' is the imaginary part. For this complex number \(1 - i\sqrt{5}\), the real part is 1 and the imaginary part is \( -\sqrt{5}\). Plot a point on the complex plane with x-value 1 (Real) and y-value \(-\sqrt{5}\) (Imaginary).

Step 2: Calculate the modulus (r)

The modulus (r) of a complex number is the distance from the origin to the point representing the complex number on the plane. It is computed using the equation: \( r = \sqrt{Re(z)^2 + Im(z)^2} \) Where Re(z) is the real part of the complex number and Im(z) is the imaginary part. In this case, Re(z) = 1, Im(z) = \( -\sqrt{5}\). Therefore, \( r = \sqrt{(1)^2 + (-\sqrt{5})^2} = \sqrt{1 + 5} = \sqrt{6} \).

Step 3: Calculate the argument \(\Theta\)

The argument (also known as the phase) of a complex number, denoted as \(\Theta\), is the angle that a line drawn from the origin to the point representing the complex number on the plain makes with the positive real axis. It can be computed using the equation: \( \Theta = atan2(Im(z), Re(z)) \) where Re(z) is the real part of the complex number and Im(z) is the imaginary part. For our case, \(\Theta = atan2(-\sqrt{5}, 1)\) which equals to \(-79.193 degrees or -1.381 radians \) (we can use either degrees or radians as per the exercise's instructions).

Step 4: Express the number in polar form

The polar form of a complex number is written as \(z = r(\cos\Theta + i \sin\Theta) \). Remembering the equivalence with Euler’s formula, this can also be written as \(z = re^{i\Theta} \). Given that we've already found that r = \(\sqrt{6} \), and \(\Theta\) = \(-1.381 radians \), the polar form expression of the complex number is: \( z = \sqrt{6} (\cos(-1.381) + i \sin(-1.381)) \) or \( z = \sqrt{6}e^{i(-1.381)} \)

Key Concepts

Complex Plane

The complex plane is a visual representation where complex numbers are plotted. Think of it like a graph, but instead of only x and y axes, it has real and imaginary axes. The horizontal axis represents the real part, while the vertical axis represents the imaginary part. For instance, if you have the complex number \(1 - i\sqrt{5}\), the number 1 is plotted on the real axis and \(-\sqrt{5}\) on the imaginary axis. This point marks the location of the complex number on the plane.
Understanding the complex plane helps us visualize complex numbers as points or vectors. It bridges the gap between algebra and geometry by providing a graphical interpretation of complex operations.

Modulus of Complex Numbers

The modulus of a complex number, often referred to as the magnitude or absolute value, is the distance from the origin to the point in the complex plane. It's like the length of a vector. To find the modulus, we use the formula:
\[ r = \sqrt{Re(z)^2 + Im(z)^2} \]
Here, \(Re(z)\) is the real part and \(Im(z)\) is the imaginary part. For \(1 - i\sqrt{5}\), the modulus is \( r = \sqrt{1 + 5} = \sqrt{6} \). This tells us how far the point is from the origin if we measured it like a straight-line distance.

Argument of Complex Numbers

The argument of a complex number is the angle it makes with the positive real axis, measured in radians or degrees. This angle indicates the direction of the line connecting the origin to the complex number on the plane. It’s calculated using:
\[ \Theta = \text{atan2}(Im(z), Re(z)) \]
For our example \(1 - i\sqrt{5}\), the argument is \(-79.193\) degrees or \(-1.381\) radians. This angle shows where the line tilts as it extends from the origin to \(1 - i\sqrt{5}\).
The argument is essential because it tells us the orientation of the complex number in the plane.

Euler’s Formula

Euler's formula is a key link between complex numbers and trigonometry. It states that:
\[ e^{i\Theta} = \cos\Theta + i\sin\Theta \]
This relationship lets us express any complex number in polar form. The polar form uses the modulus and argument, allowing us to write a complex number as:
\[ z = re^{i\Theta} \]
For \(1 - i\sqrt{5}\), already calculated as \(\sqrt{6}e^{i(-1.381)}\), this expression gives a neat representation using both radius and angle.
Euler’s formula simplifies complex multiplication and division, aligning them with geometric transformations like rotations.