Section 4.2: Operations with Complex Numbers

This section covers addition, subtraction, multiplication, and division of complex numbers in standard form.

Example 1: Adding Complex Numbers

Simplify \( (4 + 3i) + (2 - 5i) \).

Step 1: Add real parts: 4 + 2 = 6

Step 2: Add imaginary parts: 3i - 5i = -2i

Result: \( 6 - 2i \)

Simplify \( (1 + 2i)(3 - i) \).

Step 1: Apply distributive property: \( 1\cdot3 + 1\cdot(-i) + 2i\cdot3 + 2i\cdot(-i) \)

Step 2: Simplify: \( 3 - i + 6i - 2i^2 = 3 + 5i + 2 \) (since \( i^2 = -1 \))

Result: \( 5 + 5i \)

Simplify \( \frac{3 + 4i}{1 - 2i} \).

Step 1: Multiply numerator and denominator by the conjugate: \( \frac{3+4i}{1-2i} \cdot \frac{1+2i}{1+2i} \)

Step 2: Expand numerator: \( (3+4i)(1+2i) = 3 + 6i + 4i + 8i^2 = 3 + 10i - 8 = -5 + 10i \)

Step 3: Expand denominator: \( (1-2i)(1+2i) = 1 - 4i^2 = 1 + 4 = 5 \)

Step 4: Divide: \( \frac{-5+10i}{5} = -1 + 2i \)

Simplify \( (2 + 5i) + (7 - 3i) \)
Simplify \( (4 - i) - (2 + 6i) \)
Multiply \( (3 + i)(2 - 4i) \)
Divide \( \frac{5 + 2i}{1 + i} \)
Simplify \( (1 + 3i)^2 \)