Problem 51 - 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 \( 2\left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right) \).

Step 1: Identify the Real and Imaginary Parts

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

Step 2: Calculate the Magnitude

The magnitude of the complex number is calculated using the formula \( r = \sqrt{a^2 + b^2} \). Substitute \( a = \sqrt{2} \) and \( b = \sqrt{2} \) to get:\[r = \sqrt{(\sqrt{2})^2 + (\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2\]

Step 3: Find the Argument

The argument \( \theta \) is calculated using \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \). Since \( a = \sqrt{2} \) and \( b = \sqrt{2} \), we have:\[\theta = \tan^{-1}\left(\frac{\sqrt{2}}{\sqrt{2}}\right) = \tan^{-1}(1) = \frac{\pi}{4}\]

Step 4: Write the Polar Form

Now, express the complex number in polar form as \( r(\cos \theta + i \sin \theta) \). Substitute \( r = 2 \) and \( \theta = \frac{\pi}{4} \) to get:\[2\left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)\]Thus, the polar form of the complex number is \( 2\left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right) \).

Key Concepts

Complex Numbers

Complex numbers are an extension of the number system, opening up possibilities beyond real numbers by incorporating an "imaginary" component. At their core, a complex number is expressed in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, multiplied by \( i \), the imaginary unit. The imaginary unit \( i \) has the property that \( i^2 = -1 \). This allows complex numbers to capture magnitudes and directions in a plane, known as the complex plane.

In this representation:

• The horizontal axis is the real axis.
• The vertical axis is the imaginary axis.

Every point on this plane corresponds to a unique complex number. This graphical interpretation is fundamental in understanding how complex numbers function in different mathematical operations, such as addition, subtraction, and more importantly, in their conversion to polar form.

When working with the complex number \( \sqrt{2} + \sqrt{2}i \), we see this concept clearly, with the real part being \( \sqrt{2} \) and the imaginary part also being \( \sqrt{2} \). These components form a vector in the complex plane, originating from the origin \( (0, 0) \) to the point \( (\sqrt{2}, \sqrt{2}) \).

Magnitude of Complex Numbers

The magnitude of a complex number provides a measure of the distance from the origin to the point representing the complex number in the complex plane. It's similar to finding the length of a vector in a 2D coordinate system. The formula used to find it is derived from the Pythagorean theorem:

• Given a complex number \( a + bi \), the magnitude \( r \) is calculated as \( r = \sqrt{a^2 + b^2} \).

This magnitude tells you how "long" the vector is in the plane, essentially giving the absolute value or size of the complex number.

In our example, with the complex number \( \sqrt{2} + \sqrt{2}i \), substituting the values:

• \( r = \sqrt{(\sqrt{2})^2 + (\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2 \)

Here, the magnitude is \( 2 \), indicating the complex number's distance from the origin is exactly 2 units. Recognizing this helps when converting to polar form, where the magnitude forms part of the expression.

Argument of a Complex Number

The argument of a complex number is the angle it makes with the real axis, measured in radians. This angle, often denoted by \( \theta \), encapsulates the "direction" of the vector representing the complex number. It plays a crucial role in the polar representation of complex numbers.

To find this angle, we typically use the tangent function, recognizing that \( \tan \theta \) is the ratio of the imaginary part to the real part. So for a complex number \( a + bi \), the argument is given by:

• \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \).

In the case of \( \sqrt{2} + \sqrt{2}i \), we have:

• \( \theta = \tan^{-1}\left(\frac{\sqrt{2}}{\sqrt{2}}\right) = \tan^{-1}(1) = \frac{\pi}{4} \)

This means the angle between the complex number and the positive real axis is \( \frac{\pi}{4} \) radians. The argument is essential in polar form, as it helps describe the position of the complex number not just by its length (magnitude) but also by its orientation (angle). This angle is always taken to be in the range 0 to \( 2\pi \).