Problem 35 - Precalculus Mathematics for Calculus

Question

Write the complex number in polar form with argument \(\theta\) between 0 and \(2 \pi\). $$-3 i$$

Step-by-Step Solution

Answer

The polar form of \(-3i\) is: \(3 (\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2})\).

Step 1: Identify the Real and Imaginary Parts

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

Step 2: Find the Magnitude of the Complex Number

The magnitude \(r\) of a complex number \(a + bi\) is found using the formula \(r = \sqrt{a^2 + b^2}\). Since \(a = 0\) and \(b = -3\), we have:\[ r = \sqrt{0^2 + (-3)^2} = \sqrt{9} = 3 \]

Step 3: Determine the Argument \(\theta\) for the Complex Number

For a complex number \(a + bi\), the argument is determined using the formula \(\theta = \tan^{-1}\left( \frac{b}{a} \right)\). Here, \(a = 0\) and \(b = -3\). Since \(a = 0\), the complex number lies on the negative imaginary axis, pointing directly downward, giving us \(\theta = \frac{3\pi}{2}\).

Step 4: Write the Polar Form

The polar form of a complex number is \(r(\cos \theta + i \sin \theta)\). Substituting \(r = 3\) and \(\theta = \frac{3\pi}{2}\), we get:\[ 3 \left( \cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2} \right) \]

Key Concepts

Complex Numbers

Complex numbers are a fascinating extension of our traditional number system. They are numbers that have two parts: a real part and an imaginary part. Typically, these numbers are written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, with \(i\) being the imaginary unit. Imaginary numbers are multiples of \(i\), where \(i\) is defined by the property \(i^2 = -1\). Understanding complex numbers is crucial because they allow us to solve equations that have no real solutions, such as those involving the square roots of negative numbers.

• The real part \(a\) represents the number on the horizontal axis of the complex plane.
• The imaginary part \(b\) is plotted on the vertical axis.
• When combined, they form a unique point on this plane.

By using complex numbers, mathematicians gain powerful tools to solve problems in engineering, physics, and many other fields.

Magnitude

The magnitude of a complex number is often compared to the length of the vector it forms in the complex plane. It is essentially the distance from the origin (0,0) to the point \((a, b)\) on the complex plane. This concept is deeply helpful when converting between rectangular (Cartesian) form and polar form. To compute the magnitude \(r\) of the complex number \(a + bi\), use the formula \(r = \sqrt{a^2 + b^2}\).

• Think of magnitude as measuring how far the complex number is from the origin.
• For example, in the complex number \(-3i\), the real part is 0, and the imaginary part is -3. Using our formula, the magnitude is \(\sqrt{0^2 + (-3)^2} = 3\).
• This tells us the point \(-3i\) is exactly 3 units away from the origin on the negative imaginary axis.

Knowing the magnitude is crucial when expressing complex numbers in polar form.

Argument of a Complex Number

The argument of a complex number is the angle that the line representing the complex number makes with the positive real axis. This angle, typically denoted as \(\theta\), is measured in radians. The argument helps describe the direction or orientation of the vector in the complex plane.

• To find the argument \(\theta\), you can use the formula \(\theta = \tan^{-1} \left( \frac{b}{a} \right)\), where \(a\) is the real part and \(b\) is the imaginary part.
• In situations where the real component \(a = 0\), like in \(-3i\), the complex number lies on the imaginary axis. The angle is not calculated by the arctangent function but directly based on its position.
• For \(-3i\), which lies directly downward on the negative imaginary axis, the argument \(\theta\) is \(\frac{3\pi}{2}\).

The argument is key in converting rectangular forms into polar forms, and this standardizes complex numbers into a form that is easy to work with in trigonometric functions, circuits, and oscillations.