Complex numbers expand our numerical world, combining real and imaginary parts. They're essential for solving equations that stumped mathematicians for centuries. We use them in physics, engineering, and even signal processing.

Algebraic properties of complex numbers mirror those of real numbers. Addition, subtraction, multiplication, and division follow similar rules. But complex numbers introduce new concepts like conjugates and absolute values, crucial for advanced calculations.

Complex Numbers and their Components

Definition and Representation

• Complex numbers are numbers 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 ($i^2 = -1$)
• The real part of a complex number $a + bi$ is the real number $a$, representing the horizontal component on the complex plane
• The imaginary part of a complex number $a + bi$ is the real number $b$, representing the vertical component on the complex plane
• Complex numbers are represented as points or vectors on the complex plane, with the real part on the horizontal axis and the imaginary part on the vertical axis ($3 + 2i$ is plotted at the point (3, 2))

Properties and Operations

• The complex conjugate of a complex number $a + bi$ is $a - bi$, obtained by changing the sign of the imaginary part (the conjugate of $3 + 2i$ is $3 - 2i$)
• The absolute value (modulus) of a complex number $a + bi$ is the non-negative real number $\sqrt{a^2 + b^2}$, representing the distance from the origin on the complex plane (the absolute value of $3 + 2i$ is $\sqrt{3^2 + 2^2} = \sqrt{13}$)
• The sum and difference of complex conjugates result in real numbers: $(a + bi) + (a - bi) = 2a$ and $(a + bi) - (a - bi) = 2bi$
• The product of a complex number and its conjugate is a real number: $(a + bi)(a - bi) = a^2 + b^2$

Operations with Complex Numbers

Addition and Subtraction

• Addition of complex numbers is performed by adding the real and imaginary parts separately: $(a + bi) + (c + di) = (a + c) + (b + d)i$ ($(3 + 2i) + (1 - 4i) = (3 + 1) + (2 - 4)i = 4 - 2i$)
• Subtraction of complex numbers is performed by subtracting the real and imaginary parts separately: $(a + bi) - (c + di) = (a - c) + (b - d)i$ ($(3 + 2i) - (1 - 4i) = (3 - 1) + (2 - (-4))i = 2 + 6i$)
• The commutative and associative properties hold for addition and subtraction of complex numbers: $z1 + z2 = z2 + z1$ and $(z1 + z2) + z3 = z1 + (z2 + z3)$, where $z1$, $z2$, and $z3$ are complex numbers

Multiplication and Division

• Multiplication of complex numbers follows the distributive property and the rule $i^2 = -1$: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$ ($(3 + 2i)(1 - 4i) = (3 \cdot 1 - 2 \cdot (-4)) + (3 \cdot (-4) + 2 \cdot 1)i = 11 - 10i$)
• Division of complex numbers is performed by multiplying the numerator and denominator by the complex conjugate of the denominator to eliminate the imaginary part in the denominator: $(a + bi) \div (c + di) = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$ ($\frac{3 + 2i}{1 - 4i} = \frac{(3 + 2i)(1 + 4i)}{(1 - 4i)(1 + 4i)} = \frac{11 + 10i}{17} = \frac{11}{17} + \frac{10}{17}i$)
• The commutative, associative, and distributive properties hold for multiplication of complex numbers: $z1 \cdot z2 = z2 \cdot z1$, $(z1 \cdot z2) \cdot z3 = z1 \cdot (z2 \cdot z3)$, and $z1 \cdot (z2 + z3) = z1 \cdot z2 + z1 \cdot z3$

Simplifying Complex Expressions

Algebraic Properties

• Complex numbers follow the commutative, associative, and distributive properties for addition and multiplication, similar to real numbers
  • Commutative property: $z1 + z2 = z2 + z1$ and $z1 \cdot z2 = z2 \cdot z1$
  • Associative property: $(z1 + z2) + z3 = z1 + (z2 + z3)$ and $(z1 \cdot z2) \cdot z3 = z1 \cdot (z2 \cdot z3)$
  • Distributive property: $z1 \cdot (z2 + z3) = z1 \cdot z2 + z1 \cdot z3$
• The conjugate of a sum or difference of complex numbers is equal to the sum or difference of their conjugates: $(a + bi) \pm (c + di) = (a \pm c) - (b \pm d)i$ (the conjugate of $(3 + 2i) - (1 - 4i)$ is $(3 - 1) - (2 - (-4))i = 2 - 6i$)

Properties of Modulus and Polar Form

• The absolute value of a product or quotient of complex numbers is equal to the product or quotient of their absolute values: $|z1 \times z2| = |z1| \times |z2|$ and $|z1 \div z2| = |z1| \div |z2|$, where $z1$ and $z2$ are complex numbers (if $z1 = 3 + 2i$ and $z2 = 1 - 4i$, then $|z1 \cdot z2| = \sqrt{11^2 + 10^2} = \sqrt{221} = |z1| \cdot |z2| = \sqrt{13} \cdot \sqrt{17}$)
• De Moivre's formula relates complex numbers in polar form to trigonometric functions: $(cos \theta + i sin \theta)^n = cos(n\theta) + i sin(n\theta)$, where $n$ is an integer ($(cos \frac{\pi}{4} + i sin \frac{\pi}{4})^3 = cos(\frac{3\pi}{4}) + i sin(\frac{3\pi}{4})$)
• Euler's formula relates complex numbers in polar form to exponential functions: $e^{i\theta} = cos \theta + i sin \theta$, where $e$ is the mathematical constant ($e^{i\frac{\pi}{2}} = cos \frac{\pi}{2} + i sin \frac{\pi}{2} = i$)

Solving Equations with Complex Numbers

Linear and Quadratic Equations

• Linear equations with complex coefficients can be solved by isolating the variable on one side of the equation and performing arithmetic operations on complex numbers (solve $(2 + 3i)z = 4 - 5i$ by dividing both sides by $2 + 3i$: $z = \frac{4 - 5i}{2 + 3i} = \frac{(4 - 5i)(2 - 3i)}{(2 + 3i)(2 - 3i)} = \frac{7 + 22i}{13} = \frac{7}{13} + \frac{22}{13}i$)
• Quadratic equations with complex coefficients can be solved using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are complex numbers (solve $z^2 + (2 + i)z + 3i = 0$ using the quadratic formula with $a = 1$, $b = 2 + i$, and $c = 3i$)
• The discriminant ($b^2 - 4ac$) determines the nature of the roots: if it is positive, there are two distinct real roots; if it is zero, there is one real root; if it is negative, there are two distinct complex roots

Polynomial Equations and Systems of Equations

• Polynomial equations with complex coefficients can be solved by factoring, using the fundamental theorem of algebra, or by numerical methods such as Newton's method or the secant method
  • The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root
  • Newton's method is an iterative algorithm that approximates the roots of a polynomial equation by using the formula $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$, where $f(x)$ is the polynomial function and $f'(x)$ is its derivative
• Systems of linear equations with complex coefficients can be solved using methods such as substitution, elimination, or matrix methods
  • Cramer's rule uses determinants to solve systems of linear equations, where the solution for each variable is given by the ratio of two determinants (solve the system $\begin{cases} (1 + i)x + (2 - i)y = 5 + 3i \ (3 + 2i)x - (1 + 4i)y = 7 - i \end{cases}$ using Cramer's rule)
  • Gaussian elimination involves transforming the augmented matrix of the system into row echelon form and then solving for the variables by back-substitution