Problem 12 - College Trigonometry

Question

Find a polar representation for the complex number \(z\) and then identify \(\operatorname{Re}(z)\), \(\operatorname{Im}(z),|z|, \arg (z)\) and \(\operatorname{Arg}(z)\). $$ z=i \sqrt[3]{7} $$

Step-by-Step Solution

Answer

\(\operatorname{Re}(z) = 0\), \(\operatorname{Im}(z) = \sqrt[3]{7}\), \(|z| = 7^{1/3}\), \(\arg(z) = \frac{\pi}{2}\), and \(\operatorname{Arg}(z) = \frac{\pi}{2}\).

Step 1: Identify Real and Imaginary Parts

The given complex number is \(z = i \sqrt[3]{7}\). Here the real part is 0, since there is no real number part, and the imaginary part is \(\sqrt[3]{7}\). Thus, \(\operatorname{Re}(z) = 0\) and \(\operatorname{Im}(z) = \sqrt[3]{7}\).

Step 2: Calculate the Magnitude of \(z\)

The magnitude (or modulus) \(|z|\) of a complex number \(z = a + bi\) is given by \(|z| = \sqrt{a^2 + b^2}\). Here, \(a = 0\) and \(b = \sqrt[3]{7}\). Thus, \(|z| = \sqrt{0^2 + (\sqrt[3]{7})^2} = \sqrt{7^{2/3}} = 7^{1/3}\).

Step 3: Determine the Argument of \(z\)

The argument \(\arg(z)\) of a complex number given by \(z = a + bi\) is the angle \(\theta\) such that \(\tan(\theta) = \frac{b}{a}\). In this case, since \(a = 0\) and \(b = \sqrt[3]{7}\), the angle \(\theta\) that meets this criterion is \(\frac{\pi}{2}\), because it places our vector on the positive imaginary axis.

Step 4: Calculate Principal Argument

The principal argument (denoted \(\operatorname{Arg}(z)\)) is the value of the argument \(\arg(z)\) that lies in the interval \( (-\pi, \pi] \). In our case, since \(\arg(z) = \frac{\pi}{2}\), \(\operatorname{Arg}(z) = \frac{\pi}{2}\), which naturally falls in the required interval.

Key Concepts

Real and Imaginary Parts

In complex numbers, we often encounter terms like "real part" and "imaginary part". A complex number is expressed as \( z = a + bi \), where \( a \) and \( b \) are real numbers. Here, \( a \) is the real part, and \( b \) is the imaginary part which is multiplied by the imaginary unit \( i \). In the given complex number \( z = i \sqrt[3]{7} \), we do not see any real component because there is no standalone number outside the imaginary part.

Thus, for \( z = i \sqrt[3]{7} \):

• The real part, \( \operatorname{Re}(z) = 0 \).
• The imaginary part, \( \operatorname{Im}(z) = \sqrt[3]{7} \).

It's essential to note that having an imaginary part does not imply it is "unreal", but instead is a different dimension in the complex plane.

Magnitude of Complex Numbers

The magnitude of a complex number, also known as the modulus, is like a distance in mathematics. It tells us how far the number is from the origin on the complex plane. Mathematically, if a complex number is expressed as \( z = a + bi \), the magnitude \( |z| \) is calculated using the Pythagorean theorem as \( |z| = \sqrt{a^2 + b^2} \).

For our complex number \( z = i \sqrt[3]{7} \), we establish:

• Real part \( a = 0 \).
• Imaginary part \( b = \sqrt[3]{7} \).
• Hence, \( |z| = \sqrt{0^2 + (\sqrt[3]{7})^2} = \sqrt{7^{2/3}} = 7^{1/3} \).

Think of it as how long the "vector" is that represents this complex number on a graph, originating from the point (0,0).

Argument of Complex Numbers

The argument of a complex number is an angle which indicates its direction with respect to the positive real axis on the complex plane. Given a complex number \( z = a + bi \), its argument \( \arg(z) \) is calculated using the formula \( \tan(\theta) = \frac{b}{a} \), where \( \theta \) represents the angle.

In the complex number \( z = i \sqrt[3]{7} \), we find:

• \( a = 0 \)
• \( b = \sqrt[3]{7} \)

The angle that gives us a ratio where the tangent function is undefined can be more directly observed. Here, the angle placing us on the positive imaginary axis is \( \theta = \frac{\pi}{2} \). This result aligns with the fact that the imaginary part lies on the vertical axis.

Principal Argument

The principal argument is a specialized kind of argument. It refers to the value of the angle \( \arg(z) \) that lies within the interval \((-\pi, \pi]\). It helps ensure that every complex number has a single, unique polar representation.

In our example, since the argument \( \arg(z) = \frac{\pi}{2} \) already lies within the set range \((-\pi, \pi]\), it also serves as the principal argument:

• \( \operatorname{Arg}(z) = \frac{\pi}{2} \)

Having a specified range for the principal argument helps avoid ambiguity when expressing the angle  associated with a complex number, so two identical points do not have different representations.