Convert 1 - 3i and 5 + i into exponential form and hence calculate the modulus and argument of: (1 - 3i)/(5 + i)

Let's begin by converting the two complex numbers \( z_1 = 1 - 3i \) and \( z_2 = 5 + i \) into their exponential forms. The general form for expressing a complex number \( z = x + iy \) in exponential form is:

\[
z = r e^{i\theta}
\]
where:
- \( r = |z| \) is the modulus (magnitude) of the complex number, given by \( r = \sqrt{x^2 + y^2} \),
- \( \theta = \arg(z) \) is the argument (angle) of the complex number, given by \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).

### Step 1: Convert \( z_1 = 1 - 3i \) into exponential form.

- **Modulus**:
  \[
  r_1 = |z_1| = \sqrt{1^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}
  \]

- **Argument**:
  \[
  \theta_1 = \arg(z_1) = \tan^{-1}\left(\frac{-3}{1}\right) = \tan^{-1}(-3)
  \]
  Since the real part is positive and the imaginary part is negative, the complex number is in the fourth quadrant. The argument is:
  \[
  \theta_1 = -\tan^{-1}(3)
  \]

Thus, \( z_1 \) in exponential form is:
\[
z_1 = \sqrt{10} e^{-i\tan^{-1}(3)}
\]

### Step 2: Convert \( z_2 = 5 + i \) into exponential form.

- **Modulus**:
  \[
  r_2 = |z_2| = \sqrt{5^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26}
  \]

- **Argument**:
  \[
  \theta_2 = \arg(z_2) = \tan^{-1}\left(\frac{1}{5}\right)
  \]
  Since the real part is positive and the imaginary part is positive, the complex number is in the first quadrant. The argument is:
  \[
  \theta_2 = \tan^{-1}\left(\frac{1}{5}\right)
  \]

Thus, \( z_2 \) in exponential form is:
\[
z_2 = \sqrt{26} e^{i\tan^{-1}\left(\frac{1}{5}\right)}
\]

### Step 3: Calculate \( \frac{z_1}{z_2} \).

Now, divide \( z_1 \) by \( z_2 \):
\[
\frac{z_1}{z_2} = \frac{\sqrt{10} e^{-i\tan^{-1}(3)}}{\sqrt{26} e^{i\tan^{-1}\left(\frac{1}{5}\right)}}
\]
This simplifies to:
\[
\frac{z_1}{z_2} = \frac{\sqrt{10}}{\sqrt{26}} e^{i\left(-\tan^{-1}(3) - \tan^{-1}\left(\frac{1}{5}\right)\right)}
\]

### Step 4: Modulus of \( \frac{z_1}{z_2} \).

The modulus is simply the ratio of the moduli of \( z_1 \) and \( z_2 \):
\[
\text{Modulus} = \frac{|z_1|}{|z_2|} = \frac{\sqrt{10}}{\sqrt{26}} = \frac{\sqrt{10}}{\sqrt{26}} = \sqrt{\frac{10}{26}} = \frac{\sqrt{5}}{5}
\]

### Step 5: Argument of \( \frac{z_1}{z_2} \).

The argument is the difference of the arguments of \( z_1 \) and \( z_2 \):
\[
\text{Argument} = \arg\left(\frac{z_1}{z_2}\right) = -\tan^{-1}(3) - \tan^{-1}\left(\frac{1}{5}\right)
\]

Thus, the modulus and argument of \( \frac{z_1}{z_2} \) are:

- **Modulus**: \( \frac{\sqrt{5}}{5} \)
- **Argument**: \( -\tan^{-1}(3) - \tan^{-1}\left(\frac{1}{5}\right) \)

Would you like further clarification or details on any part of this process?

Here are some follow-up questions you might find interesting:

1. How do you calculate the real and imaginary parts from an exponential form?
2. What is the significance of the modulus and argument in complex numbers?
3. Can you express any complex number in polar form and vice versa?
4. How can you geometrically interpret the division of two complex numbers?
5. What are the applications of complex number operations in electrical engineering?

**Tip**: When working with arguments of complex numbers, make sure to adjust the angle depending on the quadrant where the number lies.

Learn how to convert complex numbers (1 - 3i) and (5 + i) into exponential form and calculate the modulus and argument of their division. This problem covers key concepts such as modulus, argument, and the use of exponential form. Suitable for students in Grades 10-12.

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