Complex Numbers: Definitions, Operations, and Conjugates

Complex Numbers

Introduction to Complex Numbers

Complex numbers extend the real number system by including imaginary numbers. A complex number is written in the standard form:

• Standard Form: $a + bi$, where a is the real part and b is the imaginary part.

• Imaginary Unit: $i$ is defined such that $i^2 = -1$.

Example: $3 + 2i$ has real part 3 and imaginary part 2.

Identifying Real and Imaginary Parts

Each complex number can be separated into its real and imaginary components.

• Real Part: The coefficient of 1 ($a$ in $a + bi$).

• Imaginary Part: The coefficient of $i$ ($b$ in $a + bi$).

Example: For $4 - 2i$, real part is 4, imaginary part is -2.

Adding and Subtracting Complex Numbers

To add or subtract complex numbers, combine like terms (real with real, imaginary with imaginary).

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

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

Example: $(2 + 4i) - (6 - i) = (2 - 6) + (4 - (-1))i = -4 + 5i$

Multiplying Complex Numbers

Multiplication uses the distributive property (FOIL method), remembering that $i^2 = -1$.

• Formula: $(a + bi)(c + di) = ac + adi + bci + bdi^2$

• Simplify using $i^2 = -1$.

Example: $(3i)(2 - i) = 3i imes 2 + 3i imes (-i) = 6i - 3i^2 = 6i - 3(-1) = 6i + 3$

Complex Conjugates

The conjugate of a complex number $a + bi$ is $a - bi$. Conjugates are useful for rationalizing denominators and always yield a real number when multiplied together.

• Conjugate: If $z = a + bi$, then $ar{z} = a - bi$.

• Product: $(a + bi)(a - bi) = a^2 + b^2$

Example: The conjugate of $1 - 2i$ is $1 + 2i$.

Dividing Complex Numbers

To divide by a complex number, multiply numerator and denominator by the conjugate of the denominator to eliminate $i$ from the denominator.

• Formula: $\frac{c}{a + bi} = \frac{c(a - bi)}{(a + bi)(a - bi)}$

• Simplify denominator to $a^2 + b^2$.

Example: $\frac{3}{1 + 2i}$: Multiply numerator and denominator by $1 - 2i$ to get $\frac{3(1 - 2i)}{(1 + 2i)(1 - 2i)} = \frac{3 - 6i}{1 + 4} = \frac{3 - 6i}{5}$

Summary Table: Complex Number Operations

Operation	Formula	Example
Addition	$(a + bi) + (c + di) = (a + c) + (b + d)i$	$(2 + 3i) + (4 + 5i) = 6 + 8i$
Subtraction	$(a + bi) - (c + di) = (a - c) + (b - d)i$	$(5 + 2i) - (3 + i) = 2 + i$
Multiplication	$(a + bi)(c + di) = ac + adi + bci + bdi^2$	$(1 + 2i)(3 - i) = 3 + 6i - i - 2i^2 = 3 + 5i + 2 = 5 + 5i$
Conjugate	$a + bi \rightarrow a - bi$	Conjugate of $4 - 3i$ is $4 + 3i$
Division	$\frac{c}{a + bi} = \frac{c(a - bi)}{a^2 + b^2}$	$\frac{2}{3 + 4i} = \frac{2(3 - 4i)}{9 + 16} = \frac{6 - 8i}{25}$

Additional info: These notes cover the foundational operations with complex numbers, which are essential for Precalculus topics such as polynomial equations, rational functions, and further algebraic manipulations.