Problem 40 - Algebra and Trigonometry

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 is \(8 (\cos \frac{\pi}{2} + i \sin \frac{\pi}{2})\) or \(8 e^{i \frac{\pi}{2}}\).

Step 1: Identify the components of the complex number

The given complex number is purely imaginary and can be written as \(0 + 8i\). Here, the real part \(a = 0\) and the imaginary part \(b = 8\).

Step 2: Calculate the modulus (magnitude)

The modulus \(r\) of a complex number \(a + bi\) is given by \(r = \sqrt{a^2 + b^2}\). In this case, \(r = \sqrt{0^2 + 8^2} = \sqrt{64} = 8\).

Step 3: Find the argument \(\theta\)

The argument \(\theta\) is the angle the line that represents the number makes with the positive real axis. For \(0 + 8i\), since it lies on the positive imaginary axis, \(\theta = \frac{\pi}{2}\).

Step 4: Write the complex number in polar form

In polar form, the complex number \(z = a + bi\) is represented as \(r(\cos \theta + i \sin \theta)\) or \(r e^{i\theta}\). Substituting the values for \(r\) and \(\theta\), we have: $$8 (\cos \frac{\pi}{2} + i\sin \frac{\pi}{2})$$ or \(8 e^{i \frac{\pi}{2}}\).

Key Concepts

Modulus of a Complex Number

Understanding the modulus of a complex number is fundamental when working with complex numbers, especially in polar form. The modulus, often referred to as the magnitude, represents how "long" the complex number is from the origin on the complex plane. It is similar to finding the length of the hypotenuse of a right triangle when given the other two sides.
For a complex number, written as \(a + bi\), the modulus \(r\) is calculated using the formula:

• \(r = \sqrt{a^2 + b^2}\).

The modulus is always a non-negative number. For example, if we have a complex number \(0 + 8i\), the modulus is \(8\). This is because:

• \(r = \sqrt{0^2 + 8^2} = 8\).

The modulus helps in transforming a complex number from its rectangular form \(a + bi\) to a polar form, where it is represented as a combination of an angle and distance.

Argument of a Complex Number

The argument of a complex number provides the direction of the complex number on the complex plane. It is the angle formed by the line representing the complex number with the positive real axis, measured counterclockwise.
Calculating the argument \(\theta\) involves identifying where the complex number is situated on the complex plane. The common formula for finding the argument when the number is not on an axis is:

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

but special attention is needed for numbers that lie directly on the axes, like pure imaginaries or reals.
For the complex number \(0 + 8i\), it is situated on the positive imaginary axis. Therefore, the angle made with the real axis is \(\pi/2\) radians, because it is 90 degrees counterclockwise from the real axis.
Understanding the argument is crucial for converting complex numbers into polar form, as it defines the angle of rotation from the positive real axis.

Imaginary Numbers

Imaginary numbers are the part of a complex number that include the imaginary unit \(i\), which is defined as \(i = \sqrt{-1}\).
This concept might initially seem abstract but has practical applications in various fields like engineering and physics. An imaginary number can be expressed as \(bi\), where \(b\) is a real number.
In the complex number form \(a + bi\):

• \(a\) is the real part.
• \(bi\) is the imaginary part.

In the case of \(0 + 8i\), the real part is 0, and the imaginary part is 8. Imaginary numbers enable the expansion of the number line into the complex plane, allowing the representation of rotations and oscillations in mathematical problems.
By pairing real numbers with imaginary numbers, we get a robust system that can describe wave functions, electrical currents, and much more.