Steps to Solve

1. Identify the Complex Numbers

The given complex numbers are:

• ( x = 2 + 3i )
• ( y = 3 + 4i )

2. Perform Addition

To add the two complex numbers: $$ x + y = (2 + 3i) + (3 + 4i) $$ Combine real parts and imaginary parts: $$ = (2 + 3) + (3 + 4)i = 5 + 7i $$

3. Perform Subtraction

To subtract the two complex numbers: $$ x - y = (2 + 3i) - (3 + 4i) $$ Combine real parts and imaginary parts: $$ = (2 - 3) + (3 - 4)i = -1 - 1i $$

4. Multiply the Complex Numbers

To multiply the two complex numbers: $$ x \cdot y = (2 + 3i)(3 + 4i) $$ Use the distributive property (FOIL): $$ = 2 \cdot 3 + 2 \cdot 4i + 3i \cdot 3 + 3i \cdot 4i = 6 + 8i + 9i - 12 = -6 + 17i $$

5. Find the Real Part, Imaginary Part, and Modulus

The real and imaginary parts of ( x ): $$ \text{Re}(x) = 2, \quad \text{Im}(x) = 3 $$ The real and imaginary parts of ( y ): $$ \text{Re}(y) = 3, \quad \text{Im}(y) = 4 $$ The modulus of ( x ) and ( y ): $$ \text{Mod}(x) = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} $$ $$ \text{Mod}(y) = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 $$

6. Calculate the Argument

The argument ( \theta ) can be calculated using: $$ \theta = \tan^{-1}\left(\frac{\text{Im}}{\text{Re}}\right) $$ For ( x ): $$ \theta_x = \tan^{-1}\left(\frac{3}{2}\right) $$ For ( y ): $$ \theta_y = \tan^{-1}\left(\frac{4}{3}\right) $$