Understand the Problem
The image contains notes regarding the representation of complex numbers, including operations like addition and subtraction, as well as formulas for their real part, imaginary part, argument, and modulus. It provides a visual depiction of these concepts.
Answer
Addition: $5 + 7i$, Subtraction: $-1 - 1i$, Multiplication: $-6 + 17i$, Modulus: $\sqrt{13}, 5$.
Answer for screen readers
- Addition: ( 5 + 7i )
- Subtraction: ( -1 - 1i )
- Multiplication: ( -6 + 17i )
- Modulus: ( \text{Mod}(x) = \sqrt{13}, \text{Mod}(y) = 5 )
Steps to Solve
- Identify the Complex Numbers
The given complex numbers are:
- ( x = 2 + 3i )
- ( y = 3 + 4i )
- Perform Addition
To add the two complex numbers: $$ x + y = (2 + 3i) + (3 + 4i) $$ Combine real parts and imaginary parts: $$ = (2 + 3) + (3 + 4)i = 5 + 7i $$
- Perform Subtraction
To subtract the two complex numbers: $$ x - y = (2 + 3i) - (3 + 4i) $$ Combine real parts and imaginary parts: $$ = (2 - 3) + (3 - 4)i = -1 - 1i $$
- Multiply the Complex Numbers
To multiply the two complex numbers: $$ x \cdot y = (2 + 3i)(3 + 4i) $$ Use the distributive property (FOIL): $$ = 2 \cdot 3 + 2 \cdot 4i + 3i \cdot 3 + 3i \cdot 4i = 6 + 8i + 9i - 12 = -6 + 17i $$
- 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 ): $$ \text{Mod}(x) = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} $$ $$ \text{Mod}(y) = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 $$
- 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 ): $$ \theta_y = \tan^{-1}\left(\frac{4}{3}\right) $$
- Addition: ( 5 + 7i )
- Subtraction: ( -1 - 1i )
- Multiplication: ( -6 + 17i )
- Modulus: ( \text{Mod}(x) = \sqrt{13}, \text{Mod}(y) = 5 )
More Information
Complex numbers combine real numbers with imaginary units, allowing for diverse mathematical applications, including engineering and physics. Operations like addition, subtraction, and multiplication showcase how complex numbers interact, revealing patterns and behaviors in multidimensional spaces.
Tips
- Forgetting to separate real and imaginary parts correctly during operations can lead to errors.
- Confusing the argument calculation; ensure both real and imaginary parts are used properly.
- Not simplifying the results for modulus and arguments can lead to incomplete answers.