Complex Numbers: Definitions, Operations, and Quadratic Equations

Complex Numbers

Introduction to Complex Numbers

The complex number system extends the real numbers to allow solutions to equations with negative radicands. This system is essential for solving equations that have no real solutions, such as those involving the square root of a negative number.

• Complex numbers are numbers of the form a + b i, where a and b are real numbers.

• The number a is called the real part, and b is the imaginary part.

• The imaginary unit i is defined by $i^2 = -1$.

Examples:

• -3 + 7i

• 4 - 5i

• 8 (which is 8 + 0i in standard form)

• 3i (which is 0 + 3i in standard form; called a pure imaginary number)

All real numbers are complex numbers (with b = 0), just as all integers are rational numbers.

Historical Context and Number Systems

Mathematicians have expanded number systems to solve equations that could not be solved within previous systems:

• No integer solution for $3x = 8$ → Rational numbers created ($x = \frac{8}{3}$).

• No rational solution for $x^2 = 5$ → Irrational numbers created ($x = \pm \sqrt{5}$).

• No real solution for $x^2 = -9$ → Complex numbers created ($x = \pm 3i$).

Operations with Complex Numbers

Addition and Subtraction

To add or subtract complex numbers, combine the real parts and the imaginary parts separately.

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

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

Example:

• $(3 + 5i) + (7 - 9i) = (3 + 7) + (5 - 9)i = 10 - 4i$

• $( -8 + 6i ) - (3 - 10i) = ( -8 - 3 ) + (6 - ( -10 ))i = -11 + 16i$

Multiplication

Multiply complex numbers using the distributive property (FOIL), remembering that $i^2 = -1$.

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

• Since $i^2 = -1$, $bdi^2 = -bd$

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

Example:

• $(4 + 11i)( -8 + 3i )$ $= 4 \times ( -8 ) + 4 \times 3i + 11i \times ( -8 ) + 11i \times 3i$ $= -32 + 12i - 88i + 33i^2$ $= -32 + 12i - 88i + 33( -1 )$ $= -32 + 12i - 88i - 33$ $= ( -32 - 33 ) + ( 12i - 88i )$ $= -65 - 76i$

Solving Quadratic Equations in the Complex Number System

Square Roots of Negative Numbers

When solving equations with negative radicands, use the imaginary unit:

• $\sqrt{ -25 } = \sqrt{ 25 \cdot ( -1 ) } = \sqrt{ 25 } \cdot \sqrt{ -1 } = 5i$

Examples of Quadratic Equations

Quadratic equations may have complex solutions if the discriminant is negative.

• Example 1: $x^2 + 49 = 0$ $x^2 = -49$ $x = \pm \sqrt{ -49 } = \pm 7i$

• Example 2: $x^2 = -24$ $x = \pm \sqrt{ -24 } = \pm 2 \sqrt{6} i$

• Example 3: $x^2 - 14x + 53 = 0$ Use the quadratic formula: $x = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a }$ Here, $a = 1$, $b = -14$, $c = 53$ $x = \frac{ 14 \pm \sqrt{ ( -14 )^2 - 4 \cdot 1 \cdot 53 } }{ 2 }$ $= \frac{ 14 \pm \sqrt{ 196 - 212 } }{ 2 }$ $= \frac{ 14 \pm \sqrt{ -16 } }{ 2 }$ $= \frac{ 14 \pm 4i }{ 2 }$ $= 7 \pm 2i$

• Example 4: $42x^2 + 1 = 8x$ Rearranged: $42x^2 - 8x + 1 = 0$ Use quadratic formula: $x = \frac{ 8 \pm \sqrt{ ( -8 )^2 - 4 \cdot 42 \cdot 1 } }{ 2 \cdot 42 }$ $= \frac{ 8 \pm \sqrt{ 64 - 168 } }{ 84 }$ $= \frac{ 8 \pm \sqrt{ -104 } }{ 84 }$ $= \frac{ 8 \pm 2 \sqrt{26} i }{ 84 }$ $= \frac{ 4 \pm \sqrt{26} i }{ 42 }$

Note: If directions specify "state all real solutions," complex solutions are not required.

Summary Table: Types of Numbers and Their Equations

Equation	Number System Needed	Example Solution
$3x = 8$	Rational Numbers	$x = \frac{8}{3}$
$x^2 = 5$	Irrational Numbers	$x = \pm \sqrt{5}$
$x^2 = -9$	Complex Numbers	$x = \pm 3i$

Additional info: The notes above expand on the original content by providing full explanations, formulas, and step-by-step examples for operations and solving quadratic equations in the complex number system.