Common Mistakes to Avoid When Working with Modulus and Argument:

1. Getting the Modulus Wrong:

   • Many students forget the formula for modulus. If you have a complex number like $z = a + bi$, the modulus $r$ is found using this formula: $r = \sqrt{a^2 + b^2}$.

2. Misunderstanding the Argument:

   • The argument, or angle $\theta$, must be found carefully using the right quadrant. You use the formula $\tan^{-1}\left(\frac{b}{a}\right)$. If you make mistakes here, you might get the wrong angle.

3. Ignoring Periodicity:

   • Remember that the argument is periodic! This means that $\theta + 2k\pi$ (where $k$ is any whole number) represents the same complex number. If you forget this, you might get confused.

4. Mixing Up Polar and Cartesian Forms:

   • Some students have trouble changing between polar form and Cartesian form. This can lead to mistakes in how the numbers are represented.

5. Forgetting About Signs:

   • It's important to pay attention to whether the numbers are positive or negative. Overlooking this can cause mistakes in both the modulus and the argument calculations.