Problem 31 - Precalculus Mathematics for Calculus

Question

Write the complex number in polar form with argument \(\theta\) between 0 and \(2 \pi\). $$\sqrt{2}-\sqrt{2} i$$

Step-by-Step Solution

Answer

The polar form is \( 2(\cos\frac{7\pi}{4} + i\sin\frac{7\pi}{4}) \).

Step 1: Identify the Complex Number

The given complex number is \( \sqrt{2} - \sqrt{2}i \). Here, the real part \( a = \sqrt{2} \) and the imaginary part \( b = -\sqrt{2} \).

Step 2: Calculate the Magnitude

The magnitude \( r \) of a complex number \( a + bi \) is calculated using the formula \( r = \sqrt{a^2 + b^2} \). Thus, for \( \sqrt{2} - \sqrt{2}i \), we have:\[ r = \sqrt{(\sqrt{2})^2 + (-\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2 \]

Step 3: Determine the Argument

The argument \( \theta \) of \( a + bi \) is found using \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \). For \( \sqrt{2} - \sqrt{2}i \), the formula becomes:\[ \theta = \tan^{-1}\left(\frac{-\sqrt{2}}{\sqrt{2}}\right) = \tan^{-1}(-1) \]Since the point \( (\sqrt{2}, -\sqrt{2}) \) lies in the fourth quadrant, \( \theta \) is actually \( \frac{7\pi}{4} \) as it measures the angle from the positive x-axis.

Step 4: Write in Polar Form

The polar form of a complex number is given by \( r(\cos\theta + i\sin\theta) \). Now substitute the values of \( r \) and \( \theta \):\[ 2\left( \cos\left(\frac{7\pi}{4}\right) + i\sin\left(\frac{7\pi}{4}\right) \right) \]

Key Concepts

Complex Number Representation

Complex numbers are a combination of a real and an imaginary part, often written as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. They provide essential tools in mathematics to solve equations that do not have solutions within the real numbers alone.

• The real part \( a \) dictates the position on the horizontal axis.
• The imaginary part \( b \) influences the position on the vertical axis.

For example, in the complex number \( \sqrt{2} - \sqrt{2}i \):

• \( a = \sqrt{2} \) is the real part.
• \( b = -\sqrt{2} \) is the imaginary part.

Expressing numbers in the complex plane helps visualize and understand operations such as addition, subtraction, and multiplication.

Magnitude of Complex Numbers

The magnitude, or modulus, of a complex number measures its distance from the origin in the complex plane.
This is calculated using the formula \[ r = \sqrt{a^2 + b^2} \]where \( a \) and \( b \) are the real and imaginary parts, respectively.

• For \( \sqrt{2} - \sqrt{2}i \), plug in \( a = \sqrt{2} \) and \( b = -\sqrt{2} \) into the formula to find the magnitude.
• The calculation follows as \( r = \sqrt{(\sqrt{2})^2 + (-\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2 \).

A magnitude of \( 2 \) indicates that the point lies at a distance of \( 2 \) units from the origin.

Argument of Complex Numbers

The argument of a complex number is the angle it forms with the positive x-axis in the complex plane.
It is represented by \( \theta \) and typically calculated using the formula \[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \].

• For the number \( \sqrt{2} - \sqrt{2}i \), we calculate \( \theta = \tan^{-1}\left(\frac{-\sqrt{2}}{\sqrt{2}}\right) = \tan^{-1}(-1) \).
• Since this point resides in the fourth quadrant, \( \theta \) is adjusted to \( \frac{7\pi}{4} \) to ensure it fits between \( 0 \) and \( 2\pi \).

This specific value of \( \theta \) shows the counterclockwise rotation needed to reach the point from the positive x-axis.

Polar Coordinates

Polar coordinates provide an alternative way to express complex numbers, emphasizing a number's magnitude and angle.
A complex number \( a + bi \) in polar form is expressed as \[ r(\cos\theta + i\sin\theta) \].

• This form utilizes the magnitude \( r \) and the argument \( \theta \) to convey the same information.
• For \( \sqrt{2} - \sqrt{2}i \), we've already calculated \( r = 2 \) and \( \theta = \frac{7\pi}{4} \).
• Substitute these values into the polar form to get \( 2(\cos\left(\frac{7\pi}{4}\right) + i\sin\left(\frac{7\pi}{4}\right)) \).

Polar coordinates facilitate understanding rotations and scaling in transformations, essential in fields like physics and engineering.