Complex Numbers: Operations and Applications (Precalculus, Ch. 2.1)

Polynomial and Rational Functions

Complex Numbers

Complex numbers extend the real number system by including solutions to equations that have no real solutions, such as the square root of negative numbers. This section introduces the definition, operations, and applications of complex numbers in algebra.

Definition of Complex and Imaginary Numbers

• Imaginary Unit: The imaginary unit is defined as $i$, where $i^2 = -1$.

• Complex Number: Any number in the form $a + bi$, where $a$ and $b$ are real numbers, is a complex number. Here, $a$ is the real part and $b$ is the imaginary part.

• Pure Imaginary Number: A number of the form $bi$ (where $b \neq 0$) is called a pure imaginary number.

Operations on Complex Numbers

• Complex numbers are manipulated using the same algebraic rules as binomials.

Adding and Subtracting Complex Numbers

• To add complex numbers: Add the real parts and add the imaginary parts.

• To subtract complex numbers: Subtract the real parts and subtract the imaginary parts.

• Standard Form: The result is always written as $a + bi$.

• Example: $(3 + 2i) + (1 + 4i) = (3 + 1) + (2i + 4i) = 4 + 6i$

Multiplying Complex Numbers

• Multiply as you would binomials, using the distributive property (FOIL method).

• Remember that $i^2 = -1$.

• Example: $(2 + 3i)(1 - 4i) = 2 \times 1 + 2 \times (-4i) + 3i \times 1 + 3i \times (-4i) = 2 - 8i + 3i - 12i^2$ Since $i^2 = -1$, $-12i^2 = 12$, so the result is $2 - 8i + 3i + 12 = 14 - 5i$.

Complex Conjugate

• The conjugate of $a + bi$ is $a - bi$.

• Multiplying a complex number by its conjugate yields a real number: $(a + bi)(a - bi) = a^2 + b^2$.

Dividing Complex Numbers

• To divide by a complex number, multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator.

• Example: $\frac{3 + 2i}{1 - i}$ Multiply numerator and denominator by $1 + i$: $\frac{(3 + 2i)(1 + i)}{(1 - i)(1 + i)} = \frac{3 + 3i + 2i + 2i^2}{1^2 - i^2} = \frac{3 + 5i - 2}{1 + 1} = \frac{1 + 5i}{2} = 0.5 + 2.5i$

Principal Square Root of a Negative Number

• For any positive real number $b$, the principal square root of $-b$ is defined as $\sqrt{-b} = i\sqrt{b}$.

• Example: $\sqrt{-9} = i\sqrt{9} = 3i$

Solving Quadratic Equations with Imaginary Solutions

• Quadratic equations with negative discriminants have complex (imaginary) solutions.

• Example: Solve $x^2 + 4 = 0$ $x^2 = -4 \implies x = \pm\sqrt{-4} = \pm 2i$