Problem 90 - Precalculus

Question

Graphing calculators are able to convert complex numbers from rectangular to polar form using the \(\overline{\mathbf{a b s}}\) command to find the modulus and the angle command to find the angle. Find abs \((1-i) .\) Find angle \((1-i) .\) Write \(1-i\) in polar form.

Step-by-Step Solution

Answer

The polar form of \(1-i\) is \(\sqrt{2} (\cos (-\pi/4) + i \sin (-\pi/4))\).

Step 1: Determine the modulus

To find the modulus of the complex number \(1-i\), we use the formula for the modulus \(|z| = \sqrt{a^2 + b^2}\). Here, \(a = 1\) and \(b = -1\). So, \(|1-i| = \sqrt{1^2 + (-1)^2} = \sqrt{2}\).

Step 2: Find the angle (argument)

The angle or argument of a complex number can be found using \( \theta = \tan^{-1}(\frac{b}{a}) \). For \(1-i\), this becomes \(\theta = \tan^{-1}\left(\frac{-1}{1}\right) = \tan^{-1}(-1)\). Since \(1-i\) is located in the fourth quadrant, \(\theta = -\frac{\pi}{4}\) or equivalent in degrees, \(-45^\circ\).

Step 3: Express in polar form

The polar form of a complex number is given by \( z = r (\cos \theta + i \sin \theta) \), where \(r\) is the modulus and \(\theta\) is the argument. Substituting the found values, \(1-i\) in polar form is \(\sqrt{2} (\cos (-\pi/4) + i \sin (-\pi/4))\).

Key Concepts

Polar Form

Polar form is a way of representing complex numbers that highlights both their magnitude and direction. Unlike the rectangular form, which expresses a complex number as a sum of a real part and an imaginary part, polar form uses the modulus and argument to give us a clear picture of where the number is located in the complex plane.

• The general formula for polar form is: \[ z = r (\cos \theta + i \sin \theta) \] Here, \( r \) is the modulus (length from the origin) and \( \theta \) is the argument (angle with respect to the positive x-axis).
• This form is especially useful for multiplication and division of complex numbers, as it simplifies the process to manipulating their moduli and arguments directly.

For complex number \( 1-i \), its polar form is given by \( \sqrt{2} (\cos (-\pi/4) + i \sin (-\pi/4)) \), indicating a magnitude of \( \sqrt{2} \) and direction \(-\pi/4\) radians.

Rectangular Form

Rectangular form is the most straightforward way to represent complex numbers. It shows a complex number as a point in the complex plane using its horizontal (real) and vertical (imaginary) components.

• The general expression is \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.
• For the complex number \( 1-i \), \( a = 1 \) and \( b = -1 \).

This form is easy for addition and subtraction of complex numbers because it allows you to independently handle the real and imaginary parts. However, it becomes cumbersome for multiplication and division, where polar form is more advantageous.

Modulus

The modulus of a complex number measures its distance from the origin on the complex plane. It's essentially the 'length' of the vector representing the complex number from the origin. Calculating the modulus tells us how far away the number is from the origin, regardless of direction.

• For any complex number \( z = a + bi \), the modulus is \( |z| = \sqrt{a^2 + b^2} \).
• In the example of \( 1-i \), the modulus is calculated as \( |1-i| = \sqrt{1^2 + (-1)^2} = \sqrt{2} \).

Understanding the modulus is essential for converting between rectangular and polar forms, as it forms the radius part of the polar form. Larger moduli indicate values farther from the origin, impacting how we interpret and use these numbers in operations like scaling.

Argument

The argument of a complex number is the angle it forms with the positive x-axis in the complex plane. It indicates the direction of the vector representing the complex number.

• The argument is commonly denoted by \( \theta \) and is calculated using \( \theta = \tan^{-1}(\frac{b}{a}) \).
• For \( 1-i \), with \( a = 1 \) and \( b = -1 \), the argument is \( \theta = \tan^{-1}(-1) = -\frac{\pi}{4} \) radians or \(-45^\circ\).

It's important to consider the quadrant in which the complex number lies, as this affects the angle's sign and actual value. For \( 1-i \), which is in the fourth quadrant, this consideration ensures the argument is correctly assigned. This concept is vital for understanding the orientation of complex numbers.