Problem 38 - Precalculus Mathematics for Calculus

Question

Write the complex number in polar form with argument \(\theta\) between 0 and \(2 \pi\). $$4$$

Step-by-Step Solution

Answer

The polar form is \(4e^{i0}\).

Step 1: Understanding the Complex Number

The complex number given is 4. A complex number can be expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. In this case, 4 is actually \(4 + 0i\), with 4 as the real part and 0 as the imaginary part.

Step 2: Convert to Polar Coordinates

In polar form, a complex number is represented as \(r(\cos\theta + i\sin\theta)\), or \(re^{i\theta}\), where \(r\) is the magnitude and \(\theta\) is the argument. For the number \(4 + 0i\), we calculate \(r = \sqrt{a^2 + b^2}\). Substituting \(a = 4\) and \(b = 0\), we get \(r = \sqrt{4^2 + 0^2} = 4\).

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

The argument \(\theta\) is calculated using \(\tan^{-1}(\frac{b}{a})\), which is \(\tan^{-1}(\frac{0}{4}) = \tan^{-1}(0) = 0\). Since the real positive part is on the positive x-axis, \(\theta = 0\) is within the specified range of 0 to \(2\pi\).

Step 4: Write the Polar Form

With \(r = 4\) and \(\theta = 0\), the polar form of the complex number is \(4(\cos0 + i\sin0)\), which can also be expressed as \(4e^{i \cdot 0}\). Therefore, the polar form is \(4e^{i0}\).

Key Concepts

Complex numbers

Complex numbers are numbers that have both a real and an imaginary part. They can be written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, with \(i\) being the imaginary unit (\(i^2 = -1\)).
One interesting feature of complex numbers is how they can be expressed.

• A complex number like \(4 + 0i\) might seem like a regular number, but it is technically a complex number, with "\(a = 4\)" and "\(b = 0\)".
• The imaginary part is zero, illustrating that every real number can be seen as a special case of complex numbers.
• Remember, complex numbers are generally plotted on a plane, known as the complex plane, with a real axis and an imaginary axis.

In the original exercise, our focus was to convert such numbers into polar form, which offers a different way of representing these unique numbers.

Polar coordinates

Polar coordinates offer a different way to express complex numbers. Instead of using two Cartesian coordinates (\(a, b\)) which could be seen on a regular graph grid, the polar system uses a radius and an angle.
Complex numbers in polar form are represented as \(r(\cos\theta + i\sin\theta)\) or \(re^{i\theta}\), where:

• \(r\) is the distance from the origin to the point in the complex plane.
• \(\theta\) is the angle from the positive x-axis to the line connecting the origin to the point.
• It's essentially like using a laser pointer, where you set the distance \(r\) and the direction the laser points (given by \(\theta\)).

This method can sometimes simplify complicated calculations and shows another dimension of understanding numbers, beyond the traditional method.

Magnitude and argument

The concepts of magnitude and argument play crucial roles when converting complex numbers into polar form.
In this context:

• The magnitude \(r\) is calculated as \(\sqrt{a^2 + b^2}\). It represents the 'length' of the line segment from the origin to the point \(a + bi\) in the complex plane. Basically, it reveals how far the number is from the origin.
• The argument, \(\theta\), finds the angle between the positive real axis and the line connecting the origin to the complex number. It's found using \(\tan^{-1}(\frac{b}{a})\).
• In the exercise, for the number \(4 + 0i\), the magnitude \(r\) is 4 and the argument \(\theta\) is 0 because there's no imaginary component, positioning the number right on the real axis.

Together, these two values transform complex numbers, providing insights and alternative views which can be immensely useful for complex analysis.