Find the modulus and amplitude of the following complex numbers and hence express them into polar form
`-1-i`

Answer

To find the modulus and amplitude of the complex number \( z = -1 - i \), and express it in polar form, we will follow these steps:

### Step 1: Identify the real and imaginary parts
The complex number can be expressed in the form \( z = x + iy \). Here, we have:
- \( x = -1 \) (real part)
- \( y = -1 \) (imaginary part)

### Step 2: Calculate the modulus
The modulus of a complex number \( z \) is given by the formula:
\[ |z| = \sqrt{x^2 + y^2} \]
Substituting the values of \( x \) and \( y \):
\[ |z| = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 3: Calculate the amplitude (argument)
The amplitude (or argument) \( \theta \) of a complex number is given by:
\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]
Substituting the values of \( y \) and \( x \):
\[ \theta = \tan^{-1}\left(\frac{-1}{-1}\right) = \tan^{-1}(1) \]
The value of \( \tan^{-1}(1) \) is \( \frac{\pi}{4} \) (or 45 degrees).

However, since both \( x \) and \( y \) are negative, the complex number lies in the third quadrant. To find the correct angle in the third quadrant, we add \( \pi \) to the acute angle:
\[ \theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4} \]

### Step 4: Express in polar form
The polar form of a complex number is given by:
\[ z = r(\cos \theta + i \sin \theta) \]
Substituting the modulus \( r = \sqrt{2} \) and the amplitude \( \theta = \frac{5\pi}{4} \):
\[ z = \sqrt{2} \left( \cos\left(\frac{5\pi}{4}\right) + i \sin\left(\frac{5\pi}{4}\right) \right) \]

### Final Result
Thus, the modulus of \( z \) is \( \sqrt{2} \), the amplitude is \( \frac{5\pi}{4} \), and the polar form of the complex number \( -1 - i \) is:
\[ z = \sqrt{2} \left( \cos\left(\frac{5\pi}{4}\right) + i \sin\left(\frac{5\pi}{4}\right) \right) \]