Section 4.1: Complex Numbers Introduction
This section introduces complex numbers, including imaginary units, standard form, and basic arithmetic.
Example 1: Identifying Complex Numbers
Write each number in standard form a + bi:
- \( \sqrt{-9} \)
- \( 5 - \sqrt{-16} \)
\( \sqrt{-9} = 3i \)
\( 5 - \sqrt{-16} = 5 - 4i \)
Example 2: Adding Complex Numbers
Simplify \( (3 + 2i) + (1 - 5i) \).
Step 1: Add real parts: 3 + 1 = 4
Step 2: Add imaginary parts: 2i - 5i = -3i
Result: \( 4 - 3i \)
Practice Problems
- Express \( \sqrt{-25} \) in standard form
- Simplify \( (2 + 3i) + (4 - i) \)
- Simplify \( (5 - 2i) - (3 + 6i) \)
- Multiply \( i \cdot (3 + 4i) \)
- Simplify \( (2i)^2 + 3i \)