Let's solve this step by step for the given complex numbers $a = -1 + 3i$ and $b = -1 - i$.

Questions to solve:

1. $\text{Re}(a)$: The real part of $a$.
2. $\text{Im}(a)$: The imaginary part of $a$.
3. $\overline{a}$: The conjugate of $a$.
4. $|a|$: The modulus of $a$.
5. $\arg(a)$: The argument of $a$.

Solution:

1. Real part of $a$: $\text{Re}(a) = -1$

2. Imaginary part of $a$: $\text{Im}(a) = 3$

3. Conjugate of $a$: The conjugate of a complex number $a = x + yi$ is $\overline{a} = x - yi$. $\overline{a} = -1 - 3i$

4. Modulus of $a$: The modulus of a complex number $a = x + yi$ is given by: $|a| = \sqrt{x^2 + y^2}$ Substituting $x = -1$ and $y = 3$: $|a| = \sqrt{(-1)^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}$

5. Argument of $a$: The argument $\arg(a)$ of a complex number $a = x + yi$ is given by: $\arg(a) = \tan^{-1}\left(\frac{y}{x}\right)$ For $a = -1 + 3i$, $x = -1$ and $y = 3$: $\arg(a) = \tan^{-1}\left(\frac{3}{-1}\right)$ This simplifies to: $\arg(a) = \tan^{-1}(-3)$ Since the complex number lies in the second quadrant (negative real, positive imaginary), we adjust the angle to: $\arg(a) = \pi - |\tan^{-1}(3)|$ Numerically, $\arg(a) \approx 2.82 \, \text{radians}.$

Summary:

• $\text{Re}(a) = -1$
• $\text{Im}(a) = 3$
• $\overline{a} = -1 - 3i$
• $|a| = \sqrt{10}$
• $\arg(a) \approx 2.82 \, \text{radians}$

Would you like to confirm these calculations or solve for $b$ as well?

Follow-Up Questions:

1. How would the modulus of $b$ differ from $a$?
2. What is the graphical interpretation of $\arg(a)$ on the complex plane?
3. Can you verify the conjugate property $a \cdot \overline{a} = |a|^2$?
4. How does changing the sign of the imaginary part affect $\arg(a)$?
5. What happens to the modulus if both real and imaginary parts are doubled?

Tip:

Always remember that arguments of complex numbers are calculated with respect to their quadrant position!