Complex numbers expand our understanding of mathematics beyond real numbers. They introduce the imaginary unit i, where i² = -1, allowing us to work with square roots of negative numbers. This concept opens up new possibilities in algebra and problem-solving.

On the complex plane, these numbers are represented as points with real and imaginary components. We can perform arithmetic operations with complex numbers, including addition, subtraction, multiplication, and division. They're crucial for solving polynomial equations and have applications in various fields of mathematics and science.

Complex Numbers

Square roots of negative numbers

• Imaginary unit $i$ defined as $i = \sqrt{-1}$ allows expressing square roots of negative numbers
  • $i^2 = -1$ is a fundamental property of the imaginary unit
• Square roots of negative numbers expressed using $i$ by multiplying the square root of the positive number by $i$
  • $\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i$ demonstrates this process
  • $\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i$ is another example
• Complex numbers have the general form $a + bi$, where $a$ is the real part and $b$ is the imaginary part
  • Real numbers $a$ and $b$ can be any values, including zero ($0 + 3i$ or $2 + 0i$)

Graphical representation on complex plane

• Complex plane is a 2D coordinate system with real numbers on the horizontal axis (x-axis) and imaginary numbers on the vertical axis (y-axis)
• Complex numbers plotted as points on the complex plane using their real part as the x-coordinate and imaginary part as the y-coordinate
  • $3 + 2i$ is plotted at the point $(3, 2)$ on the complex plane
• Modulus or absolute value of a complex number is the distance from the origin to the point representing the complex number
  • For a complex number $z = a + bi$, its modulus is calculated using the formula $|z| = \sqrt{a^2 + b^2}$
  • Modulus of $3 + 4i$ is $\sqrt{3^2 + 4^2} = 5$
• Polar form represents complex numbers using magnitude and angle (e.g., $r(\cos\theta + i\sin\theta)$)

Arithmetic operations with complex numbers

• Addition and subtraction of complex numbers performed by adding or subtracting the real parts and imaginary parts separately
  • $(3 + 2i) + (1 - 4i) = (3 + 1) + (2 - 4)i = 4 - 2i$ shows the addition process
  • $(5 - 3i) - (2 + i) = (5 - 2) + (-3 - 1)i = 3 - 4i$ demonstrates subtraction
• Multiplication of complex numbers uses the distributive property and the fact that $i^2 = -1$
  • $(3 + 2i)(1 - 4i) = 3 - 12i + 2i - 8i^2 = 3 - 10i + 8 = 11 - 10i$ is an example of multiplication
  • $(2 - i)(3 + 4i) = 6 + 8i - 3i + i^2 = 6 + 5i - 1 = 5 + 5i$ is another example
• Division of complex numbers involves multiplying the numerator and denominator by the complex conjugate of the denominator to rationalize it
  • Complex conjugate of $a + bi$ is $a - bi$, obtained by changing the sign of the imaginary part
  • $\frac{3 + 2i}{1 - 4i} = \frac{(3 + 2i)(1 + 4i)}{(1 - 4i)(1 + 4i)} = \frac{3 + 14i - 8}{1 + 16} = \frac{-5 + 14i}{17} = -\frac{5}{17} + \frac{14}{17}i$ demonstrates the division process
  • $\frac{2 - 3i}{2 + i} = \frac{(2 - 3i)(2 - i)}{(2 + i)(2 - i)} = \frac{4 - 5i - 3i^2}{4 - i^2} = \frac{4 - 5i + 3}{4 + 1} = \frac{7 - 5i}{5} = \frac{7}{5} - i$ is another example

Complex numbers for polynomial equations

• Polynomial equations may have non-real roots, which are complex numbers
• Quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ used to find roots of quadratic equations
  • If $b^2 - 4ac < 0$, the roots will be complex numbers
• Solving $x^2 + 2x + 5 = 0$ using the quadratic formula:
  1. Identify $a = 1, b = 2, c = 5$
  2. Substitute values into the quadratic formula: $x = \frac{-2 \pm \sqrt{2^2 - 4(1)(5)}}{2(1)} = \frac{-2 \pm \sqrt{-16}}{2} = \frac{-2 \pm 4i}{2} = -1 \pm 2i$
  3. The roots are $-1 + 2i$ and $-1 - 2i$
• Solving $2x^2 - 3x + 4 = 0$ using the quadratic formula:
  1. Identify $a = 2, b = -3, c = 4$
  2. Substitute values into the quadratic formula: $x = \frac{3 \pm \sqrt{(-3)^2 - 4(2)(4)}}{2(2)} = \frac{3 \pm \sqrt{9 - 32}}{4} = \frac{3 \pm \sqrt{-23}}{4} = \frac{3 \pm i\sqrt{23}}{4}$
  3. The roots are $\frac{3 + i\sqrt{23}}{4}$ and $\frac{3 - i\sqrt{23}}{4}$

Advanced Complex Number Concepts

• Euler's formula ($e^{ix} = \cos x + i\sin x$) connects complex exponentials to trigonometric functions
• Complex roots of unity are solutions to $z^n = 1$, forming a regular polygon in the complex plane
• De Moivre's theorem relates complex number exponentiation to trigonometry: $(r(\cos\theta + i\sin\theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))$