Question

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ \sqrt{5}-i $$

Short answer

\[ re^{i\theta} = \textbackslash sqrt{6} e^{\left(i\ \textbackslash arctan\ (\frac{{-1}}{\textbackslash sqrt{5}}\right)} \] Polar Form: \ \textbackslash sqrt{6}\left( \textbackslash cos\left( \textbackslash arctan\left(\frac{-1}{\textbackslash sqrt{5}}\right) - i \textbackslash sin\right)\right)

Step-by-step solution

Identify the Complex Number

The given complex number is \(\textbf{\textbackslash sqrt{5} - i} \).

Plot the Complex Number in the Complex Plane

Plot the complex number \(\textbf{\textbackslash sqrt{5} - i} \) in the complex plane: The real part is \(\textbf{\textbackslash sqrt{5} \) (point along the x-axis) and the imaginary part is \(\textbf{-1}\textbackslashsqrt{5}}\textbackslash i \) along the y-axis. Locate the point \(\textbf{(\textbackslash sqrt{5}, -1)}\) on the complex plane.

Calculate the Magnitude of the Complex Number

The magnitude \(|z| \) is calculated using the formula for the modulus: \[ |z| = \textbackslash sqrt{[\text{\textbackslash real}^2 + \text{\textbackslash imaginary}^2 ]} = \textbackslash sqrt{(\textbackslash sqrt{5})^2 + (-1)^2} = \textbackslash sqrt{5 + 1} = \textbackslash sqrt{6} \]

Calculate the Argument of the Complex Number

The argument \(\theta \) is the angle formed with the positive real axis, calculated as follows (considering the complex number is located in the fourth quadrant): \[ \tan(\theta ) = \frac{{\textbackslash imaginary}}{{\textbackslash real}} = \frac{{-1}}{{\textbackslash sqrt{5}}} \] \(\theta \) is thus: \[ \theta = \textbackslash arctan\left(\frac{{-1}}{{\textbackslash sqrt{5}}}\right) \]

Write the Number in Polar Form

The polar form of \(\textbackslash sqrt{5} - i}\) is: \[ |z|\left( \textbackslash cos(\theta) + i \sin(\theta) \right) = \textbackslash sqrt{6}[\textbackslash cos(\arctan(\frac{{-1}}{{\textbackslash sqrt{5}}})) + i\sin(\arctan(\frac{-1}{\textbackslash sqrt{5}})) \]

Write the Number in Exponential Form

Finally, in the exponential form, the complex number can be represented using Euler's formula \( z = re^{i\theta}\), thus it becomes: \[ z = \textbackslash sqrt{6} e^{ i\ \textbackslash arctan \left( \frac{-1}}{\textbackslash sqrt{5}}}\right) \]

Key concepts

Polar Form

Polar form is a way to represent complex numbers using the modulus and argument rather than the real and imaginary parts. This can be very useful when it comes to multiplying or dividing complex numbers. In polar form, a complex number is written as:

\t
• \(z = |z|(\text{cos}(\theta) + i\text{sin}(\theta))\)

Here, \(|z|\) is the modulus, and \(\theta\) is the argument. This is also often written in shorthand using the notation \(z = |z| \text{cis} \theta\). Polar form illustrates the complex number as a vector from the origin to a point in the complex plane.

Exponential Form

Exponential form is another way to express complex numbers, using Euler's formula:

\t
• \(e^{i\theta} = \text{cos}(\theta) + i\text{sin}(\theta)\)

In this form, a polar-form complex number \(|z|(\text{cos}(\theta) + i\text{sin}(\theta))\) can be written as:

\t
• \(z = |z| e^{i\theta}\)

This form is highly useful in advanced mathematics, especially in areas involving differentiation and integration of complex functions. In fact, breaking down complex multiplications and divisions becomes simpler in exponential form.

Complex Plane

The complex plane is a two-dimensional plane used to visually represent complex numbers. The horizontal axis, called the real axis, represents the real part of a complex number, while the vertical axis, called the imaginary axis, represents the imaginary part.

\t
• A complex number \(z = a + bi\) is represented by the point \((a, b)\) on the complex plane.
\t
• This point captures both the real component \(a\) and the imaginary component \(b\).

By plotting complex numbers on this plane, we can easily perform arithmetic operations and visualize their magnitudes and directions.

Modulus

The modulus of a complex number \(z = a + bi\) is a measure of its distance from the origin on the complex plane. It is denoted by \(|z|\) and calculated using the Pythagorean theorem:

\t
• \(|z| = \sqrt{a^2 + b^2}\)

For the number \(\sqrt{5} - i\):

\t
• Real part = \(\sqrt{5}\)
\t
• Imaginary part = \(-1\)

So, the modulus is:

\t
• \(|z| = \sqrt{(\sqrt{5})^2 + (-1)^2} = \sqrt{6}\)

The modulus helps in both the polar and exponential forms, reflecting the 'length' of the complex number vector from the origin.

Argument

The argument of a complex number is the angle \(\theta\) between the positive real axis and the line segment representing the number in the complex plane. This angle can be found using the tangent function:

\t
• For a complex number \(a + bi\), \(\theta = \arctan(\frac{b}{a})\)

For the number \(\sqrt{5} - i\):

\t
• The real part = \(\sqrt{5}\)
\t
• The imaginary part = \(-1\)
\t
• \(\tan(\theta) = \frac{-1}{\sqrt{5}}\)

So, the argument is:

\t
• \(\theta = \arctan(\frac{-1}{\sqrt{5}})\)

This argument tells you the angle required to rotate the positive real axis to coincide with the number's position vector in the complex plane.