Given the complex number $z = -1 + i$, we need to find its modulus $r$ and argument $\theta$ in exact form.

Step 1: Find the Modulus $r$

The modulus $r$ of a complex number $z = a + bi$ is given by: $r = \sqrt{a^2 + b^2}$ For $z = -1 + i$:

• $a = -1$
• $b = 1$

Thus: $r = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2}$

So, the modulus $r$ is: $r = \sqrt{2}$

Step 2: Find the Argument $\theta$

The argument $\theta$ of a complex number is given by: $\theta = \tan^{-1}\left(\frac{b}{a}\right)$ For $z = -1 + i$: $\theta = \tan^{-1}\left(\frac{1}{-1}\right) = \tan^{-1}(-1)$

Since the complex number is in the second quadrant (negative real part and positive imaginary part), the correct angle is: $\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$

So, the argument $\theta$ is: $\theta = \frac{3\pi}{4}$

Final Answers:

$r = \sqrt{2}$ $\theta = \frac{3\pi}{4}$

Would you like further details or explanations?

Here are some related questions you might find useful:

1. How do you convert from rectangular to polar form for any complex number?
2. What is the significance of the modulus in the complex plane?
3. How do you determine the correct quadrant for the argument of a complex number?
4. What is the geometric interpretation of the argument $\theta$?
5. Can you find the argument for a purely imaginary or purely real number?

Tip: Always consider the quadrant of the complex number to determine the correct value of the argument $\theta$.