Problem 4 - Advanced Engineering Mathematics

Question

In Problems 1-10, write the given complex number in polar form. $$ 6 i $$

Step-by-Step Solution

Answer

The polar form is \(6(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2})\).

Step 1: Identify the complex number components

The given complex number is purely imaginary and can be written as \( a + bi \), where \( a = 0 \) and \( b = 6 \). Thus, the complex number is in the form \( 0 + 6i \).

Step 2: Calculate the magnitude

To convert a complex number to polar form, we first calculate the magnitude \( r \). The magnitude is given by the formula:\[r = \sqrt{a^2 + b^2}\]Substitute \( a = 0 \) and \( b = 6 \):\[r = \sqrt{0^2 + 6^2} = \sqrt{36} = 6\]

Step 3: Determine the angle (argument)

The angle \( \theta \) is the angle made by the line representing the complex number with the positive real axis. Since this number is on the imaginary axis, its angle is \( \frac{\pi}{2} \) radians (90 degrees).

Step 4: Write the polar form

Now, we can write the complex number in polar form as:\[\r(\cos \theta + i \sin \theta) \]Using our computed values, the polar form of the complex number is:\[6(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2})\]

Key Concepts

Complex Numbers

Complex numbers are a blend of real and imaginary parts. They can be expressed in the form \( a + bi \), where \( a \) represents the real part and \( bi \) the imaginary part. In our exercise example, the complex number given is \( 6i \). Since there is no real part here, it can be rewritten as \( 0 + 6i \). This format shows that the number is purely imaginary.

• Real Part: The real component \( a = 0 \).
• Imaginary Part: The imaginary component \( b = 6 \), giving us the term \( 6i \).

You will often see complex numbers plotted on the complex plane, with the x-axis representing real numbers and the y-axis representing imaginary numbers. This visualization can help understand the distance and angle needed for the polar form.

Magnitude of a Complex Number

The magnitude, or absolute value, of a complex number is the distance from the origin to the point on the complex plane. For a complex number \( a + bi \), the magnitude is calculated using the formula:\[r = \sqrt{a^2 + b^2}\]In our situation with the complex number \( 0 + 6i \):

• Substitute \( a = 0 \) and \( b = 6 \) into the formula.
• Calculate: \( r = \sqrt{0^2 + 6^2} = \sqrt{36} = 6 \).

This result tells us that the complex number's magnitude is 6 units away from the origin. The magnitude is always a non-negative value, providing the "length" of the vector representing the complex number.

Argument of a Complex Number

The argument of a complex number is essentially the direction in which it points from the origin, measured as an angle \( \theta \) from the positive real axis. For the number \( 0 + 6i \), which lies on the positive imaginary axis, the argument is straightforward.

• Purely Imaginary Numbers: They are positioned vertically on the complex plane. In this case, the line is aligned parallel to the imaginary axis.
• The standard angle for purely imaginary numbers with positive \( b \) is \( \frac{\pi}{2} \) radians, or 90 degrees.

This angle shows the direction of the complex number relative to the positive x-axis (real axis), a crucial part when converting to the polar form. Understanding this angle helps in visualizing how a complex number's real and imaginary parts constructively contribute to its position on the plane.