The Main Idea 

Complex numbers are a type of number that expand the traditional notion of numbers by including imaginary numbers.

The imaginary number [latex]i[/latex] is defined to be [latex]i=\sqrt{-1}[/latex]. Any real multiple of [latex]i[/latex], like 5[latex]i[/latex], is also an imaginary number.

A complex number is composed of two parts: a real part and an imaginary part, often written in the form [latex]a + bi[/latex], where [latex]a[/latex] and [latex]b[/latex] are real numbers. This expanded number system allows for solutions to equations that cannot be solved using only real numbers.

The Main Idea 

In the complex plane, the horizontal axis is the real axis and the vertical axis is the imaginary axis.

Complex Plane Structure:

• Horizontal axis: Real part
• Vertical axis: Imaginary part
• [latex]z = a + bi[/latex] is plotted as point [latex](a, b)[/latex]

The Main Idea 

• Addition and Subtraction:
  • Add/subtract real and imaginary parts separately
  • [latex]\left(a+bi\right)+\left(c+di\right)=\left(a+c\right)+\left(b+d\right)i[/latex]
  • [latex]\left(a+bi\right)-\left(c+di\right)=\left(a-c\right)+\left(b-d\right)i[/latex]
• Multiplication:
  • With real number: Distribute
  • Between complex numbers: Use FOIL or distributive property
  • Key rule: [latex]i^2 = -1[/latex]
• Geometric Interpretation:
  • Addition/Subtraction: Translation in complex plane
  • Multiplication: Scaling and rotation
• Complex Number Division:
  • Cannot divide by imaginary numbers directly
  • Goal: Eliminate imaginary part in denominator
• Complex Conjugate:
  • Definition: For [latex]a+bi[/latex], conjugate is [latex]a-bi[/latex]
  • Key property: [latex]\begin{align}(a+bi)(a-bi)&=a^2+b^2\end{align}[/latex] (always real)
• Division Process:
  • Multiply numerator and denominator by denominator’s conjugate
  • Simplify and rationalize the denominator

The Main Idea

• Quadratic Equation:
  • General form: [latex]ax^2 + bx + c = 0[/latex]
  • Roots are x-intercepts of the parabola
• Quadratic Formula:
  • [latex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/latex]
  • Solves all quadratic equations
• Complex Roots:
  • Occur when [latex]b^2 - 4ac[/latex] is negative
  • Always come in conjugate pairs: [latex]a + bi[/latex] and [latex]a - bi[/latex]
  • No real [latex]x[/latex]-intercepts for the parabola
• Geometric Interpretation:
  • Complex roots: Parabola doesn’t cross x-axis
  • Real roots: Parabola touches or crosses x-axis

• Discriminant:
  • Expression under the square root: [latex]b^2 - 4ac[/latex]
  • Determines nature and number of solutions

  • Value of Discriminant	Results
    [latex]{b}^{2}-4ac=0[/latex]	One rational solution (double solution)
    [latex]{b}^{2}-4ac>0[/latex], perfect square	Two rational solutions
    [latex]{b}^{2}-4ac>0[/latex], not a perfect square	Two irrational solutions
    [latex]{b}^{2}-4ac<0[/latex]	Two complex solutions

• Types of Solutions:
  • [latex]b^2 - 4ac > 0[/latex]: Two real solutions
  • [latex]b^2 - 4ac = 0[/latex]: One real solution (double root)
  • [latex]b^2 - 4ac < 0[/latex]: Two complex solutions