Problem 17 - Precalculus

Question

Express each complex number in polar form. $$3+0 i$$

Step-by-Step Solution

Answer

The polar form of \( 3 + 0i \) is \( 3e^{i0} \) or \( 3(\cos 0 + i \sin 0) \).

Step 1: Identify Real and Imaginary Parts

The given complex number is \( 3+0i \). Here, the real part \( a = 3 \) and the imaginary part \( b = 0 \).

Step 2: Calculate the Magnitude

To convert the complex number to polar form, we first calculate the magnitude \( r \), which is given by the formula \( r = \sqrt{a^2 + b^2} \). Substitute \( a = 3 \) and \( b = 0 \) into the formula: \[ r = \sqrt{3^2 + 0^2} = \sqrt{9} = 3 \]

Step 3: Calculate the Argument

Next, calculate the argument \( \theta \), which is the angle the complex number makes with the positive real axis. Since \( b = 0 \) and \( a > 0 \), the argument \( \theta = 0 \) radians. Thus, in the complex plane, the angle with the positive real axis is 0.

Step 4: Write in Polar Form

The polar form of a complex number is \( r (\cos \theta + i \sin \theta) \) or simply \( r e^{i\theta} \). With \( r = 3 \) and \( \theta = 0 \), the polar form is:\[ 3(\cos 0 + i \sin 0) \quad \text{or} \quad 3 e^{i0} \]

Key Concepts

Complex Numbers

Complex numbers are a fundamental concept in mathematics that combine real and imaginary numbers. A complex number is usually expressed as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. The imaginary unit \( i \) is defined such that \( i^2 = -1 \).

Here's why complex numbers are useful:

• Facilitating Calculations: They simplify mathematical problems that involve oscillations, waves, and circles.
• Broadening Solutions: They allow solutions to equations that don't have real number solutions.

Understanding the separate roles of the real part \( a \) and the imaginary part \( bi \) is crucial before converting complex numbers into other forms.

Magnitude Calculation

The magnitude of a complex number, also known as the modulus, is a measure of its size or length in the complex plane. It is denoted by \( |z| \) and can be calculated using the Pythagorean theorem for the right triangle formed by the real and imaginary parts.

The formula for the magnitude is:\[| z | = \sqrt{a^2 + b^2}\]where \( a \) is the real part and \( b \) is the imaginary part. To illustrate, if you have a complex number \( 3 + 0i \), the magnitude is:

• \( a = 3 \)
• \( b = 0 \)
• Magnitude, \( r = \sqrt{3^2 + 0^2} = \sqrt{9} = 3 \)

This formula gives you the distance from the origin to the point \( (a, b) \) in the complex plane.

Argument of a Complex Number

The argument of a complex number is the angle \( \theta \) that the number, represented as a point in the complex plane, makes with the positive real axis. Calculating the argument helps in understanding the direction of the complex number.

To find \( \theta \), various rules apply depending on the coordinates:

• If \( b = 0 \) and \( a > 0 \), as in our example, \( \theta = 0 \) radians.
• If \( b = 0 \) and \( a < 0 \), then \( \theta = \pi \).

The argument is usually measured in radians and can be found using trigonometric functions. Being comfortable with these calculations is key to converting complex numbers to polar form accurately.

Conversion to Polar Form

Converting a complex number to its polar form involves expressing it in terms of magnitude and argument. The polar form of a complex number is represented as:\[r(\cos \theta + i \sin \theta) \quad \text{or} \quad r e^{i\theta}\]Here, \( r \) is the magnitude and \( \theta \) is the argument.

Why use polar form?

• Simplifies Multiplication/Division: Polar form makes these operations easier compared to standard form.
• Great for Calculus: Helpful when dealing with integrals and derivatives of complex functions.

For \( 3 + 0i \), simply calculate \( r = 3 \) and \( \theta = 0 \), leading to the polar representation:\[3(\cos 0 + i \sin 0) \quad \text{or} \quad 3 e^{i0}\]The polar form elegantly visualizes complex numbers as points in circular motion.