Complex Numbers: Operations and Standard Form

Complex Numbers

Operations with Complex Numbers

Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined by $i^2 = -1$. In Precalculus, understanding how to perform operations with complex numbers and express results in standard form is essential.

• Addition and Subtraction: To add or subtract complex numbers, combine the real parts and the imaginary parts separately.

• Multiplication: Use the distributive property (FOIL method) and apply $i^2 = -1$ to simplify.

• Standard Form: After performing operations, always write the result as a + bi.

Key Formulas

• Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$

• Subtraction: $(a + bi) - (c + di) = (a - c) + (b - d)i$

• Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

Example

• Addition Example: $(2 + 3i) + (4 - 5i) = (2 + 4) + (3 - 5)i = 6 - 2i$

• Multiplication Example: $(1 + 2i)(3 + 4i) = (1 \times 3 - 2 \times 4) + (1 \times 4 + 2 \times 3)i = (3 - 8) + (4 + 6)i = -5 + 10i$

Additional info: The standard form a + bi is used to clearly separate the real and imaginary components, which is important for further operations such as division and finding complex conjugates.