2.4 Complex Numbers

Complex numbers expand our number system, combining real and imaginary parts. They're written as a + bi, where i is the square root of -1. This powerful concept allows us to solve equations that were previously impossible.

We can add, subtract, multiply, and divide complex numbers using specific rules. They can be represented in different forms like algebraic and polar, and graphed on a complex plane. Understanding complex numbers opens up new mathematical possibilities.

Complex Numbers

Arithmetic with complex numbers

• Addition and subtraction
  • Add or subtract real parts and imaginary parts separately (like combining like terms)
  • $(2 + 3i) + (4 + 5i) = (2 + 4) + (3 + 5)i = 6 + 8i$
  • $(7 - 2i) - (3 + 4i) = (7 - 3) + (-2 - 4)i = 4 - 6i$
• Multiplication
  • Multiply terms using distributive property and $i^2 = -1$
  • $(2 + 3i)(4 + 5i) = 8 + 10i + 12i + 15i^2 = 8 + 22i - 15 = -7 + 22i$
  • $(1 - i)(2 + 3i) = 2 + 3i - 2i - 3i^2 = 2 + i + 3 = 5 + i$
• Division
  • Multiply numerator and denominator by complex conjugate of denominator to rationalize
  • Complex conjugate of $a + bi$ is $a - bi$
  • $\frac{2 + 3i}{4 + 5i} = \frac{(2 + 3i)(4 - 5i)}{(4 + 5i)(4 - 5i)} = \frac{8 - 10i + 12i - 15i^2}{16 + 20i - 20i - 25i^2} = \frac{23 + 2i}{41} = \frac{23}{41} + \frac{2}{41}i$

Powers of imaginary unit

• Cyclic pattern repeats every four powers of $i$
  • $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, then pattern repeats ($i^5 = i$, $i^6 = -1$, etc.)
• Simplify powers by breaking down into multiples of four
  • $i^7 = i^4 \cdot i^3 = 1 \cdot (-i) = -i$
  • $i^{15} = (i^4)^3 \cdot i^3 = 1^3 \cdot (-i) = -i$
  • $i^{-5} = \frac{1}{i^5} = \frac{1}{i} = -i$

Forms of complex numbers

• Algebraic form (rectangular form)
  • $a + bi$ where $a$ is real part and $b$ is imaginary part ($3 + 2i$, $-4 - 7i$)
• Polar form
  • $r(\cos\theta + i\sin\theta)$ or $re^{i\theta}$ where $r$ is modulus (magnitude) and $\theta$ is argument (angle)
  • Modulus $r = \sqrt{a^2 + b^2}$, argument $\theta = \tan^{-1}(\frac{b}{a})$
  • $2 + 2i$ in polar form: $r = \sqrt{2^2 + 2^2} = 2\sqrt{2}$, $\theta = \tan^{-1}(\frac{2}{2}) = 45^\circ$ or $\frac{\pi}{4}$ radians
  • Euler's formula relates exponential and trigonometric functions: $e^{i\theta} = \cos\theta + i\sin\theta$
• Converting between algebraic and polar forms
  • Algebraic to polar: use modulus and argument formulas
  • Polar to algebraic: $a = r\cos\theta$, $b = r\sin\theta$
  • $2(\cos 30^\circ + i \sin 30^\circ) = 2(\frac{\sqrt{3}}{2} + \frac{1}{2}i) = \sqrt{3} + i$
• Graphical representation on complex plane
  • Real part on horizontal axis, imaginary part on vertical axis
  • Modulus is distance from origin, argument is angle with positive real axis
  • $3 - 4i$ plotted 3 units right and 4 units down from origin
  • Complex plane coordinates (a, b) correspond to the complex number a + bi

Advanced Complex Number Concepts

• Real numbers are complex numbers with zero imaginary part (e.g., 3 + 0i)
• Imaginary numbers are complex numbers with zero real part (e.g., 0 + 2i)
• De Moivre's theorem: $(r(\cos\theta + i\sin\theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))$, useful for finding powers of complex numbers