Problem 40 - Precalculus Mathematics for Calculus

Question

Write the complex number in polar form with argument \(\theta\) between 0 and \(2 \pi\). $$8 i$$

Step-by-Step Solution

Answer

The polar form of the complex number is \(8(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2})\).

Step 1: Identify the Components

The complex number is given as \(8i\). This can be rewritten in standard complex form as \(0 + 8i\), which implies that the real part \(a = 0\) and the imaginary part \(b = 8\).

Step 2: Calculate the Magnitude

The magnitude \(r\) of a complex number \(a + bi\) is given by \(r = \sqrt{a^2 + b^2}\). Substituting \(a = 0\) and \(b = 8\), we have \(r = \sqrt{0^2 + 8^2} = \sqrt{64} = 8\).

Step 3: Determine the Argument

The argument \(\theta\) of the complex number is calculated using the formula \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). However, since \(a = 0\), we consider the position on the complex plane. Because \(b = 8\) (positive), the argument \(\theta = \frac{\pi}{2}\).

Step 4: Express in Polar Form

The polar form of a complex number is expressed as \(r(\cos\theta + i\sin\theta)\). Substituting \(r = 8\) and \(\theta = \frac{\pi}{2}\), we get \(8(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2})\). Simplifying, the polar form is \(8i\).

Key Concepts

polar form

The polar form of a complex number provides a unique way to represent complex numbers using a combination of a radius and an angle. Unlike the standard Cartesian form which uses real and imaginary parts, polar form utilizes the magnitude of the number and the angle it creates with the positive real axis in the complex plane. This can be especially useful for multiplying and dividing complex numbers.

• The polar form of a complex number is written as \( r(\cos\theta + i\sin\theta) \), where \( r \) is the magnitude and \( \theta \) is the argument.
• This form provides a geometric interpretation which can simplify complex number arithmetic.

To convert a number like \( 8i \) to polar form, first determine its magnitude and argument, and then apply them in the formula. The polar form reveals more about the number's "direction" in the complex plane, which can be particularly handy in practical applications like signal processing.

magnitude

In the context of complex numbers, the magnitude \( r \) represents the distance from the origin to the point defined by the complex number in the complex plane. This is analogous to understanding the length of a vector.

• For a complex number \( a + bi \), the magnitude \( r \) is calculated as \( r = \sqrt{a^2 + b^2} \).
• The magnitude is always a non-negative real number.

In our example with \( 8i \), since the real part \( a \) is 0, the calculation simplifies directly to \( r = \sqrt{0^2 + 8^2} = 8 \). This tells us that on the complex plane, the number \( 8i \) lies 8 units directly above the origin. The magnitude is a critical component in converting a complex number to its polar form, as it represents the overall size of the vector.

argument

The argument of a complex number is the angle it forms with the positive real axis in the complex plane. It is an essential aspect of the polar representation as it indicates the "direction" of the complex number.

• The argument \( \theta \) is usually given in radians and can range between 0 and \( 2\pi \). For negative numbers or those in different quadrants, adjustments are made accordingly.
• For \( a = 0 \), as seen in \( 8i \), determining \( \theta \) requires special attention since standard trigonometric rules may not directly apply.

In this exercise, since the imaginary part is positive 8 and the real part is 0, the number lies on the positive imaginary axis. Therefore, we determine the argument \( \theta \) as \( \frac{\pi}{2} \), representing a 90-degree rotation from the positive real axis. Understanding the argument helps in visualizing the complex number's orientation in the complex plane. This becomes a tool for solving many complex math problems and simplifies computations like powers and roots.