Problem 39 - Advanced Engineering Mathematics

Question

(a) If \(z_{1}=-1\) and \(z_{2}=5 i\), verify that $$ \operatorname{Arg}\left(z_{1} z_{2}\right) \neq \operatorname{Arg}\left(z_{1}\right)+\operatorname{Arg}\left(z_{2}\right) $$ (b) If \(z_{1}=-1\) and \(z_{2}=-5 i\), verify that $$ \operatorname{Arg}\left(z_{1} / z_{2}\right) \neq \operatorname{Arg}\left(z_{1}\right)-\operatorname{Arg}\left(z_{2}\right) $$

Step-by-Step Solution

Answer

For both parts, inequalities are verified: \(\operatorname{Arg}(z₁z₂) \neq \operatorname{Arg}(z₁) + \operatorname{Arg}(z₂)\) and \(\operatorname{Arg}(z₁/z₂) \neq \operatorname{Arg}(z₁) - \operatorname{Arg}(z₂)\).

Step 1: Find Arg(z₁) and Arg(z₂) for Part (a)

For the given complex numbers, find the arguments \(\operatorname{Arg}(z_1)\) and \(\operatorname{Arg}(z_2)\). \(z_1 = -1\) lies on the negative real axis, so \(\operatorname{Arg}(z_1) = \pi\). For \(z_2 = 5i\), this lies on the positive imaginary axis, hence \(\operatorname{Arg}(z_2) = \frac{\pi}{2}\).

Step 2: Find Arg(z₁z₂) for Part (a)

Calculate the product \(z_1z_2 = (-1)(5i) = -5i\). The complex number \(-5i\) lies on the negative imaginary axis, thus \(\operatorname{Arg}(z_1z_2) = -\frac{\pi}{2}\).

Step 3: Verify the inequality for Part (a)

Sum the arguments from Step 1: \(\operatorname{Arg}(z_1) + \operatorname{Arg}(z_2) = \pi + \frac{\pi}{2} = \frac{3\pi}{2}\). Compare this with \(\operatorname{Arg}(z_1z_2) = -\frac{\pi}{2}\). Since \(-\frac{\pi}{2} eq \frac{3\pi}{2}\), the inequality is verified.

Step 4: Find Arg(z₁) and Arg(z₂) for Part (b)

Reuse \(\operatorname{Arg}(z_1) = \pi\) from Part (a). For \(z_2 = -5i\), which lies on the negative imaginary axis, \(\operatorname{Arg}(z_2) = -\frac{\pi}{2}\).

Step 5: Find Arg(z₁/z₂) for Part (b)

Calculate the division \(z_1/z_2 = \frac{-1}{-5i} = \frac{i}{5}\), which is equivalent to the complex number \(\frac{i}{5}\), lying on the positive imaginary axis, hence \(\operatorname{Arg}(z_1/z_2) = \frac{\pi}{2}\).

Step 6: Verify the inequality for Part (b)

Calculate the difference of the arguments from Step 4: \(\operatorname{Arg}(z_1) - \operatorname{Arg}(z_2) = \pi - \left(-\frac{\pi}{2}\right) = \pi + \frac{\pi}{2} = \frac{3\pi}{2}\). Compare this with \(\operatorname{Arg}(z_1/z_2) = \frac{\pi}{2}\). Since \(\frac{\pi}{2} eq \frac{3\pi}{2}\), the inequality is verified.

Key Concepts

Argument of a complex number

The argument of a complex number, often denoted as \( \operatorname{Arg}(z) \), is the angle formed by the complex number with the positive real axis in the complex plane. Consider a complex number \( z = a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part.

• If \( z \) lies on the positive real axis, then \( \operatorname{Arg}(z) = 0 \).
• If \( z \) lies on the negative real axis, then \( \operatorname{Arg}(z) = \pi \).
• For positive imaginary axis, \( \operatorname{Arg}(z) = \frac{\pi}{2} \).
• For negative imaginary axis, \( \operatorname{Arg}(z) = -\frac{\pi}{2} \).

The argument takes into account the quadrant in which the complex number lies, and it ranges generally from \(-\pi \) to \( \pi \).
It's essential when calculating the argument to consider the location of the complex number on the plane for accuracy. In this exercise, we observe that for given numbers, distinct axes result in differing angles.

Multiplication of complex numbers

Multiplying complex numbers involves combining not only their magnitudes but also their arguments. To multiply two complex numbers \( z_1 = r_1 (\cos \theta_1 + i \sin \theta_1) \) and \( z_2 = r_2 (\cos \theta_2 + i \sin \theta_2) \) using polar form:

• The magnitude of the product \( |z_1 z_2| = r_1 \cdot r_2 \)
• The argument of the product \( \operatorname{Arg}(z_1 z_2) = \operatorname{Arg}(z_1) + \operatorname{Arg}(z_2) \)

However, as demonstrated in Part (a) of the exercise, these sums might be affected by the periodic nature of angles. The product \(-5i\) discussed in the exercise appears on the negative imaginary axis, making \( \operatorname{Arg}(-5i) = -\frac{\pi}{2} \), not simply a straightforward sum like \( \frac{3\pi}{2} \).
Always ensure that the resultant angle is adjusted within the standard range for comparison and calculations.

Division of complex numbers

Dividing complex numbers is somewhat like the reverse of multiplication. When dividing \( z_1 = r_1 (\cos \theta_1 + i \sin \theta_1) \) by \( z_2 = r_2 (\cos \theta_2 + i \sin \theta_2) \):

• The magnitude of the quotient is \( \left| \frac{z_1}{z_2} \right| = \frac{r_1}{r_2} \)
• The argument of the quotient is \( \operatorname{Arg}\left(\frac{z_1}{z_2}\right) = \operatorname{Arg}(z_1) - \operatorname{Arg}(z_2) \)

But as Part (b) of the exercise reveals, there is potential for discrepancies due to the circular nature of angles.
Although the calculation suggests a subtraction of arguments, \( \frac{i}{5} \) lies on the positive imaginary axis, resulting in \( \operatorname{Arg}\left(\frac{z_1}{z_2}\right) = \frac{\pi}{2} \). Be sure to normalize the result within the proper angle range \([-\pi, \pi]\) to avoid any misinterpretations.