Problem 32 - Algebra and Trigonometry

Question

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 1-i $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Real and Imaginary Parts

The given complex number is written in the form \(a + bi\). Here \(a = 1\) (the real part) and \(b = -1\) (the imaginary part).

Step 2: Calculate the Magnitude

The magnitude \(r\) of a complex number \(a + bi\) is given by \(r = \sqrt{a^2 + b^2}\). Substitute \(a = 1\) and \(b = -1\): \[r = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}.\]

Step 3: Determine the Argument \(\theta\)

The argument \(\theta\) is found using \(\tan(\theta) = \frac{b}{a}\). Substitute \(a = 1\) and \(b = -1\): \[\tan(\theta) = \frac{-1}{1} = -1.\] The arctangent of -1 is \(-\frac{\pi}{4}\), but \(\theta\) must be in the range \([0, 2\pi)\). Since the number \(1-i\) lies in the fourth quadrant, adjust the angle: \[\theta = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4}.\]

Step 4: Write in Polar Form

A complex number in polar form is expressed as \(r(\cos \theta + i\sin \theta)\). From Steps 2 and 3, substitute \(r = \sqrt{2}\) and \(\theta = \frac{7\pi}{4}\): \[1-i = \sqrt{2}(\cos \frac{7\pi}{4} + i\sin \frac{7\pi}{4}).\]

Key Concepts

Polar Form

Complex numbers can be represented in more than just Cartesian form, which is the familiar format of real and imaginary parts, \(a + bi\). In polar form, a complex number is expressed as \(r(\cos \theta + i\sin \theta)\). This representation is often more intuitive when dealing with multiplication, division, and finding powers or roots of complex numbers. Moreover, it provides a visual understanding of the complex number as a point in the plane.

Instead of focusing on horizontal and vertical placements (real and imaginary axes, respectively), polar coordinates give us a radial distance, or magnitude \(r\), and an angular displacement \(\theta\), called the argument. Translating a complex number like \(1 - i\) into polar form involves identifying these two key components. Doing so allows us to articulate how far and in what direction from the origin this number is situated on the complex plane.

Magnitude

The magnitude of a complex number is akin to its distance from the origin of the complex plane. Mathematically, the magnitude is denoted as \(r\), and you find it using the formula:

• \(r = \sqrt{a^2 + b^2}\)

Here, \(a\) and \(b\) are the coefficients of the real and imaginary parts, respectively.

For the complex number \(1 - i\), substitute \(a = 1\) and \(b = -1\) into the formula:

• \(r = \sqrt{1^2 + (-1)^2} = \sqrt{2}\)

This gives the radial distance or magnitude of our complex number from the origin, ensuring a clear step towards the polar form.

Argument

The argument of a complex number, often represented as \(\theta\), tells us the angle at which the number is located with respect to the positive real axis on the complex plane. This functional aspect is calculated using the formula:

• \(\tan(\theta) = \frac{b}{a}\)

For \(1 - i\), substituting the values \(a = 1\) and \(b = -1\) yields:

• \(\tan(\theta) = -1\)

Here, when calculating \(\theta\), recognition of the quadrant in which the complex number resides on the plane becomes crucial.

Since our number lies in the fourth quadrant (where both sine and cosine of the angle are positive in real terms), adjust by adding \(2\pi\) to account for the negative angle:

• \(\theta = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4}\)

Thus, the argument correctly aligns with the angle's actual position within the standard range \([0, 2\pi)\).

Real and Imaginary Parts

Every complex number can be dissected into two fundamental components: its real part and its imaginary part. Typically expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, these parts provide the basic framework for calculating the magnitude and argument.

The real part \(a\) lies along the horizontal axis, while the imaginary part \(b\) stretches vertically in the complex plane. For our example, \(1 - i\), it's easy to see that:

• Real part: \(a = 1\)
• Imaginary part: \(b = -1\)

Understanding these parts does more than just establish the number’s location in rectangular form. They are integral in transitioning to the polar form, affecting both the magnitude and the argument calculations.