Generate 5 questions on addition, subtraction, multiplication, and division of complex numbers each.

Steps to Solve

1. Define Complex Numbers A complex number is composed of a real part and an imaginary part. It is written in the form $a + bi$, where $a$ is the real part, $b$ is the imaginary part, and $i$ is the imaginary unit ($i^2 = -1$).

2. Create Problems for Addition For example, consider two complex numbers: $z_1 = 3 + 4i$ and $z_2 = 1 + 2i$. The addition of these numbers is calculated as follows: $$ z_1 + z_2 = (3 + 4i) + (1 + 2i) = (3 + 1) + (4 + 2)i = 4 + 6i $$

3. Create Problems for Subtraction Using the same complex numbers $z_1$ and $z_2$, the subtraction is: $$ z_1 - z_2 = (3 + 4i) - (1 + 2i) = (3 - 1) + (4 - 2)i = 2 + 2i $$

4. Create Problems for Multiplication For multiplication, use: $$ z_1 \times z_2 = (3 + 4i)(1 + 2i) $$ Using the distributive property: $$ = 3 \times 1 + 3 \times 2i + 4i \times 1 + 4i \times 2i = 3 + 6i + 4i + 8i^2 $$ Simplifying gives: $$ = 3 + 10i - 8 = -5 + 10i $$

5. Create Problems for Division For division, consider: $$ \frac{z_1}{z_2} = \frac{3 + 4i}{1 + 2i} $$ To simplify, multiply numerator and denominator by the conjugate of the denominator: $$ = \frac{(3 + 4i)(1 - 2i)}{(1 + 2i)(1 - 2i)} $$ Calculating gives: $$ = \frac{3 - 6i + 4i + 8}{1 + 4} = \frac{11 - 2i}{5} = \frac{11}{5} - \frac{2}{5}i $$