Problem 24 - Precalculus

Question

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

Step-by-Step Solution

Answer

The polar form of the complex number is \(3\sqrt{2}(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})\).

Step 1: Identify Components of the Complex Number

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

Step 2: Calculate the Magnitude

The magnitude \(r\) of the complex number can be calculated using the formula \(r = \sqrt{a^2 + b^2}\). For \(a = 3\) and \(b = 3\), compute \(r = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}\).

Step 3: Calculate the Argument

The argument \(\theta\) of the complex number is given by \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). So, compute \(\theta = \tan^{-1}\left(\frac{3}{3}\right) = \tan^{-1}(1)\). Hence, \(\theta = \frac{\pi}{4}\) radians since this angle corresponds to \(\tan^{-1}(1)\) in the first quadrant.

Step 4: Express in Polar Form

The polar form of a complex number is given by \(r(\cos \theta + i\sin \theta)\). Substitute \(r = 3\sqrt{2}\) and \(\theta = \frac{\pi}{4}\) to get the polar form: \(3\sqrt{2}(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})\).

Key Concepts

Complex Numbers

Complex numbers are fascinating entities in mathematics, consisting of two parts: a real part and an imaginary part. Generally expressed in the form \(a + bi\), a complex number includes:

\(a\), which is the real part, and \(b\), the imaginary part, multiplied by \(i\), where \(i\) is the imaginary unit satisfying \(i^2 = -1\).
This discovery extends the real number system to solve equations that do not have real solutions.
For example, the complex number \(3 + 3i\) means the real part is 3, and the imaginary part is also 3.

Understanding complex numbers is crucial for solving advanced mathematics problems involving roots of equations, vectors, and transformations.

Magnitude of a Complex Number

The magnitude, or modulus, of a complex number is a measure of its 'size' or distance from the origin in the complex plane.
It is denoted by \(|z|\) for a complex number \(z = a + bi\) and calculated using the formula:

\[r = \sqrt{a^2 + b^2}\]

For the complex number \(3 + 3i\), we substitute \(a = 3\) and \(b = 3\) to get:
\[r = \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2}\]
This tells us that the complex number \(3 + 3i\) is exactly \(3\sqrt{2}\) units away from the origin \((0,0)\) on the complex plane.
Understanding the magnitude is essential for comparing complex numbers or converting them into different forms, such as polar form.

Argument of a Complex Number

The argument of a complex number is the direction of the number from the origin, measured as the angle in the complex plane from the positive real axis.
This angle is commonly denoted by \(\theta\) and calculated using:

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

For our example, \(3 + 3i\):
- Real part \(a = 3\)
- Imaginary part \(b = 3\)

So, \(\theta = \tan^{-1}(1) = \frac{\pi}{4}\) radians.
This means the complex number \(3 + 3i\) forms an angle of \(45^\circ\) with the positive real axis.
The argument helps us translate complex numbers into the polar coordinate system, making it easier to visualize and manipulate them.

Trigonometric Form

When converting complex numbers into other forms, the trigonometric (or polar) form is extremely useful.
This form represents a complex number using its magnitude and argument:

\[r(\cos \theta + i\sin \theta)\]

Here, \(r\) is the magnitude, and \(\theta\) the argument.
For the complex number \(3 + 3i\), the polar form becomes:
\[3\sqrt{2}(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})\]
This notation efficiently captures the number's size and direction, proving particularly useful in complex multiplication and division.
Expressing complex numbers this way simplifies many mathematical problems by leveraging trigonometric identities.