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Complex Numbers
The Foundation: Field Axioms
The set of real numbers $R$ is a Field, a mathematical structure defined by a set of rules for a Binary Operation like addition and multiplication.
$$a(b + c) = ab + ac$$
The Distributive Law links multiplication and addition across a field.
$a, b, c$=Any elements within the field(unitless)
$a = 0$
→$0(b + c) = 0$, illustrating the property of the additive identity.
Axiom Checklist: Includes Closure, Associative, Identity, Inverse, and Commutative properties.
Trichotomy Property: For any $a, b \in R$, exactly one of $a < b$, $a = b$, or $a > b$ must be true.
Additive Identity: The number 0, where $a + 0 = a$.
Introduction to Complex Numbers
Complex numbers extend the real number system by introducing the Iota ($i$), allowing for the square roots of negative numbers.
$$i = \sqrt{-1} \implies i^2 = -1$$
Definition of the imaginary unit $i$.
$i$=Imaginary unit (iota)(unitless)
$i^3$
→$-i$
$i^4$
→$1$
Standard Form: A complex number $z$ is written as $x + iy$, where $x$ is the Real Part and $y$ is the Imaginary Part.
Every Real Is Complex: Any real number $a$ is a complex number with imaginary part 0, i.e., $a = a + 0i$.
Purely Imaginary: If the real part $x = 0$, the number is called purely imaginary (e.g., $3i$, $-5i$).
Iota Cycle: Powers of $i$ repeat every four steps: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, then back to $i$.
Power Shortcut: To find $i^n$, divide $n$ by 4 and use the remainder: $i^{n \bmod 4}$. For example, $i^{13} = i^{13 \bmod 4} = i^1 = i$.
Consecutive Sum: Any four consecutive powers of $i$ always sum to zero: $i^k + i^{k+1} + i^{k+2} + i^{k+3} = 0$.
Algebraic Operations
Operations on complex numbers involve combining like terms (real with real, imaginary with imaginary) and applying the distributive property.
$$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$$
Standard multiplication formula for complex numbers.
$a, c$=Real components(unitless)
$b, d$=Imaginary components(unitless)
$(a + bi)(a - bi)$
→$a^2 + b^2$, a real non-negative number.
Equality: Two complex numbers are equal iff their real and imaginary parts match: $a + bi = c + di \Leftrightarrow a = c$ and $b = d$.
Addition: $(a + bi) + (c + di) = (a+c) + (b+d)i$ — add real parts, add imaginary parts.
Ordered Pair Form: Complex numbers can be expressed as $(a, b)$, where multiplication follows $(a,b)(c,d) = (ac-bd, ad+bc)$.
Scalar Multiplication: $k(a + bi) = ka + kbi$ for any real constant $k$.
Division Technique: To divide $\frac{z_1}{z_2}$, multiply numerator and denominator by the conjugate of $z_2$: $\frac{a+bi}{c+di} \times \frac{c-di}{c-di} = \frac{(ac+bd)+(bc-ad)i}{c^2+d^2}$.
Conjugate Properties and Identities
The Conjugate of a complex number satisfies several important algebraic identities used throughout complex number computations.
$$z\bar{z} = |z|^2 = a^2 + b^2$$
The product of a complex number with its conjugate always yields a real non-negative number equal to the square of its modulus.
$z$=Complex number $a + bi$(unitless)
$\bar{z}$=Conjugate $a - bi$(unitless)
$z$ is real ($b = 0$)
→$z\bar{z} = a^2$, and $\bar{z} = z$ (reals are self-conjugate).
Double Conjugate: $\bar{\bar{z}} = z$ — conjugating twice returns the original number.
Sum Property: $\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}$ — the conjugate distributes over addition.
Product Property: $\overline{z_1 z_2} = \bar{z_1} \cdot \bar{z_2}$ — the conjugate distributes over multiplication.
Quotient Property: $\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z_1}}{\bar{z_2}}$ for $z_2 \neq 0$.
Modulus Invariance: $|-z| = |z| = |\bar{z}| = |-\bar{z}|$ — negation and conjugation do not change the modulus.
Product of Moduli: $|z_1 z_2| = |z_1| \cdot |z_2|$ — the modulus of a product equals the product of moduli.
Multiplicative Inverse: $z^{-1} = \frac{\bar{z}}{|z|^2} = \left(\frac{a}{a^2+b^2}, \frac{-b}{a^2+b^2}\right)$ for $z \neq 0$.
The Complex Plane and Modulus
The Argand Diagram visualizes complex numbers as points or vectors, where the x-axis is the real axis and the y-axis is the imaginary axis.
$$|z| = \sqrt{x^2 + y^2}$$
The modulus represents the magnitude or distance from the origin.
$|z|$=Modulus of $z$(unitless)
$x$=Real component coordinate(unitless)
$y$=Imaginary component coordinate(unitless)
$y = 0$
→$|z| = \sqrt{x^2} = |x|$, reducing to the absolute value of a real number.
Pythagorean Link: The modulus formula is derived directly from the distance formula in Cartesian coordinates.
Triangle Inequality: $|z_1 + z_2| \le |z_1| + |z_2|$ — equality holds when $z_1$ and $z_2$ point in the same direction.
Reverse Triangle Inequality: $|z_1 + z_2| \ge ||z_1| - |z_2||$ — the minimum modulus of a sum.
Real Points: Points on the x-axis represent real numbers (imaginary part 0).
Imaginary Points: Points on the y-axis represent purely imaginary numbers (real part 0).
No Ordering: Unlike $R$, complex numbers cannot be ordered — statements like $z_1 > z_2$ are meaningless for non-real complex numbers.
Polar Form and De Moivre's Theorem
The Polar Form describes a complex number using its distance $r$ from the origin and the Argument $\theta$, facilitating complex exponentiation.
$$(\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta$$
De Moivre's Theorem for raising complex numbers to integer powers.
$\theta$=Argument (angle)(radians/degrees)
$n$=Integer exponent(unitless)
$n = 2$
→Double angle identities: $\cos 2\theta + i\sin 2\theta$.
Polar Form: $z = r(\cos\theta + i\sin\theta)$ where $r = \sqrt{x^2+y^2}$ is the modulus and $\theta = \tan^{-1}(y/x)$ is the argument.
Multiplication in Polar: $z_1 z_2 = r_1 r_2[\cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2)]$ — multiply moduli, add arguments.
De Moivre's Power: $[r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta)$ — raise modulus to the $n$th power, multiply argument by $n$.