Find the modulus and argument of the complex number `(1+2i)/(1-3i)`.

Explanation

To find the modulus and argument of the complex number \(\frac{1+2i}{1-3i}\), we will follow these steps:

### Step 1: Simplify the Complex Number
We need to simplify the complex number by multiplying the numerator and the denominator by the conjugate of the denominator.

\[ \text{Conjugate of } (1 - 3i) = 1 + 3i \]

Now, we multiply both the numerator and the denominator by \(1 + 3i\):

\[ \frac{1 + 2i}{1 - 3i} \cdot \frac{1 + 3i}{1 + 3i} = \frac{(1 + 2i)(1 + 3i)}{(1 - 3i)(1 + 3i)} \]

### Step 2: Calculate the Denominator
Using the formula \(a^2 - b^2\) for the denominator:

\[ (1 - 3i)(1 + 3i) = 1^2 - (3i)^2 = 1 - 9(-1) = 1 + 9 = 10 \]

### Step 3: Calculate the Numerator
Now, we expand the numerator:

\[ (1 + 2i)(1 + 3i) = 1 \cdot 1 + 1 \cdot 3i + 2i \cdot 1 + 2i \cdot 3i = 1 + 3i + 2i + 6i^2 \]

Since \(i^2 = -1\):

\[ = 1 + 5i - 6 = -5 + 5i \]

### Step 4: Combine the Results
Now we can combine the results from the numerator and denominator:

\[ \frac{-5 + 5i}{10} = -\frac{1}{2} + \frac{1}{2}i \]

### Step 5: Find the Modulus
The modulus of a complex number \(z = a + bi\) is given by:

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

Here, \(a = -\frac{1}{2}\) and \(b = \frac{1}{2}\):

\[ |z| = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{1}{4}} = \sqrt{\frac{2}{4}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

### Step 6: Find the Argument
The argument \(\theta\) of a complex number is given by:

\[ \tan(\theta) = \frac{b}{a} \]

In our case:

\[ \tan(\theta) = \frac{\frac{1}{2}}{-\frac{1}{2}} = -1 \]

The angle whose tangent is \(-1\) is \(-\frac{\pi}{4}\). However, since the real part is negative and the imaginary part is positive, the complex number lies in the second quadrant. Therefore, we adjust the angle:

\[ \theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \]

### Final Answer
Thus, the modulus and argument of the complex number \(\frac{1+2i}{1-3i}\) are:

- Modulus: \(\frac{1}{\sqrt{2}}\)
- Argument: \(\frac{3\pi}{4}\)