Complex numbers expand our number system, letting us solve equations that were once impossible. They're like a secret weapon in math, opening up new possibilities in algebra and beyond.

In this part of the chapter, we'll learn how to work with complex numbers. We'll cover their different forms, basic operations, and even see how they connect to quadratic equations. It's gonna be eye-opening!

Complex Number Representation

Definition and Components

• A complex number is a number expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit defined as the square root of -1
• The real part of a complex number is the $a$ value, and the imaginary part is the $b$ value
• Examples of complex numbers: $3 + 2i$, $-4 - 7i$, $6i$

Rectangular and Polar Forms

• In rectangular form $(a + bi)$, $a$ represents the point's horizontal position and $b$ represents the vertical position on the complex plane
• The absolute value (modulus) of a complex number is the distance from the origin to the point representing the complex number on the complex plane, calculated using the Pythagorean theorem: $|a + bi| = \sqrt{a^2 + b^2}$
• In polar form, a complex number is defined as $r(\cos(\theta) + i \sin(\theta))$, where $r$ is the absolute value (modulus) and $\theta$ is the angle (argument) formed with the positive real axis on the complex plane
• The argument $(\theta)$ of a complex number can be calculated using the arctangent function: $\theta = \arctan(b/a)$, with special consideration given to the signs of $a$ and $b$ to determine the quadrant
• Euler's formula, $e^{i\theta} = \cos(\theta) + i \sin(\theta)$, provides a connection between the polar and exponential forms of complex numbers

Complex Number Arithmetic

Addition, Subtraction, and Multiplication

• Addition and subtraction of complex numbers are performed by adding or subtracting the real and imaginary parts separately: $(a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i$
  • Example: $(3 + 2i) + (4 - 5i) = (3 + 4) + (2 - 5)i = 7 - 3i$
• Multiplication of complex numbers follows the distributive property and the rule $i^2 = -1$: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$
  • Example: $(2 + 3i)(4 - i) = (2 \cdot 4 - 3 \cdot (-1)) + (2 \cdot (-1) + 3 \cdot 4)i = 11 + 10i$
• The conjugate of a complex number $a + bi$ is $a - bi$. The product of a complex number and its conjugate is always a real number: $(a + bi)(a - bi) = a^2 + b^2$

Division and De Moivre's Theorem

• Division of complex numbers is performed by multiplying the numerator and denominator by the conjugate of the denominator, then simplifying: $(a + bi) \div (c + di) = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i$
  • Example: $(3 + 4i) \div (1 - 2i) = \frac{(3 + 4i)(1 + 2i)}{(1 - 2i)(1 + 2i)} = \frac{11 + 2i}{5} = \frac{11}{5} + \frac{2}{5}i$
• De Moivre's Theorem states that for any complex number $(\cos(\theta) + i \sin(\theta))$ and any integer $n$, $(\cos(\theta) + i \sin(\theta))^n = \cos(n\theta) + i \sin(n\theta)$. This theorem is useful for finding roots and powers of complex numbers

Complex Solutions to Quadratics

Quadratic Formula and Discriminant

• The quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, can be used to solve quadratic equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$
• When the discriminant $(b^2 - 4ac)$ is negative, the quadratic equation has two complex solutions, which are complex conjugates of each other
• To find the complex solutions, substitute the negative discriminant into the quadratic formula and simplify the expression, expressing the result in the form $a + bi$
  • Example: For the quadratic equation $x^2 + 2x + 5 = 0$, the discriminant is $2^2 - 4 \cdot 1 \cdot 5 = -16$. The complex solutions are $x = \frac{-2 \pm \sqrt{-16}}{2} = -1 \pm 2i$

Sum and Product of Complex Solutions

• The sum of the complex solutions of a quadratic equation is equal to $-b/a$, and the product of the solutions is equal to $c/a$, which are the negatives of the coefficients of the quadratic equation in standard form
  • Example: For the quadratic equation $x^2 + 2x + 5 = 0$, the sum of the solutions is $-2/1 = -2$, and the product of the solutions is $5/1 = 5$

Geometric Interpretation of Complex Numbers

Complex Plane (Argand Plane)

• The complex plane (Argand plane) is a two-dimensional representation of complex numbers, where the horizontal axis represents the real part and the vertical axis represents the imaginary part
• Each complex number $a + bi$ corresponds to a unique point $(a, b)$ on the complex plane, with $a$ being the x-coordinate and $b$ being the y-coordinate
• The absolute value (modulus) of a complex number is the distance from the origin $(0, 0)$ to the point representing the complex number
• The argument (angle) of a complex number is the angle formed between the positive real axis and the line segment connecting the origin to the point representing the complex number, measured in radians or degrees

Geometric Transformations

• Geometric transformations such as rotation, reflection, and dilation can be performed on complex numbers in the complex plane
• Multiplication by $i$ corresponds to a 90-degree counterclockwise rotation in the complex plane, while multiplication by $-i$ corresponds to a 90-degree clockwise rotation
  • Example: Multiplying $2 + 3i$ by $i$ results in $-3 + 2i$, which is a 90-degree counterclockwise rotation of the point $(2, 3)$ in the complex plane
• Reflection across the real axis is achieved by taking the complex conjugate of a complex number
  • Example: The complex conjugate of $2 + 3i$ is $2 - 3i$, which is the reflection of the point $(2, 3)$ across the real axis in the complex plane