write polar form of this complex number
`1-i`

Explanation

To convert the complex number \(1 - i\) into its polar form, we will follow these steps:

### Step 1: Identify the complex number
The complex number given is \( z = 1 - i \).

### Step 2: Calculate the modulus \( r \)
The modulus \( r \) of a complex number \( z = a + bi \) is given by the formula:
\[ r = \sqrt{a^2 + b^2} \]
For our complex number:
- \( a = 1 \)
- \( b = -1 \)

Now, substituting the values:
\[ r = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 3: Calculate the argument \( \theta \)
The argument \( \theta \) is found using the formula:
\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \]
Substituting the values:
\[ \theta = \tan^{-1}\left(\frac{-1}{1}\right) = \tan^{-1}(-1) \]
The value of \( \tan^{-1}(-1) \) is \( -\frac{\pi}{4} \).

Since the complex number \( 1 - i \) lies in the fourth quadrant (where the real part is positive and the imaginary part is negative), we confirm that:
\[ \theta = -\frac{\pi}{4} \]

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

### Step 5: Simplify using trigonometric identities
Using the identities:
- \( \cos(-\theta) = \cos(\theta) \)
- \( \sin(-\theta) = -\sin(\theta) \)

We have:
\[ \cos\left(-\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \]
\[ \sin\left(-\frac{\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} \]

Thus, substituting these values back:
\[ z = \sqrt{2} \left( \frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2} \right) \]

### Final Polar Form
This simplifies to:
\[ z = 1 - i = \sqrt{2} \left( \cos\left(-\frac{\pi}{4}\right) + i \sin\left(-\frac{\pi}{4}\right) \right) \]

### Conclusion
The polar form of the complex number \( 1 - i \) is:
\[ z = \sqrt{2} \left( \cos\left(-\frac{\pi}{4}\right) + i \sin\left(-\frac{\pi}{4}\right) \right) \]