Fundamentals - Algebra, Number Systems, and Pre-Calculus

Table of Contents

1. Number Systems
2. Basic Algebra
3. Functions
4. Sequences and Series
5. Exponential and Logarithmic Functions
6. Trigonometric Functions
7. Polynomials
8. Rational Functions
9. Inequalities
10. Coordinate Geometry

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Number Systems

Real Numbers (ℝ)

The set of real numbers includes all rational and irrational numbers.

Properties:

• Commutativity: $a + b = b + a$, $ab = ba$
• Associativity: $(a + b) + c = a + (b + c)$, $(ab)c = a(bc)$
• Distributivity: $a(b + c) = ab + ac$
• Identity: $a + 0 = a$, $a \cdot 1 = a$
• Inverse: $a + (-a) = 0$, $a \cdot (1/a) = 1$ ($a \neq 0$)

Absolute Value

For any real number $x$: $|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$

Properties:

• $|xy| = |x||y|$
• $|x + y| \leq |x| + |y|$ (Triangle Inequality)
• $|x - y| \geq ||x| - |y||$
• $|x| < a \iff -a < x < a$

Complex Numbers (ℂ)

A complex number is of the form $z = a + bi$ where $a, b \in \mathbb{R}$ and $i^2 = -1$.

Operations:

• Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$
• Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$
• Conjugate: $\bar{z} = a - bi$, $z \cdot \bar{z} = a^2 + b^2 = |z|^2$
• Modulus: $|z| = \sqrt{a^2 + b^2}$
• Argument: $\arg(z) = \theta$ where $\tan \theta = b/a$

Polar Form: $z = r(\cos \theta + i \sin \theta) = re^{i\theta}$ (Euler’s formula)

De Moivre’s Theorem: $(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)$

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Basic Algebra

Binomial Theorem

$(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$

where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.

Pascal’s Triangle: Each entry is sum of two above it.

Factoring Formulas

• Difference of squares: $a^2 - b^2 = (a - b)(a + b)$
• Sum/difference of cubes:
  • $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
  • $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
• Perfect squares: $(a \pm b)^2 = a^2 \pm 2ab + b^2$

Quadratic Formula

For $ax^2 + bx + c = 0$ ($a \neq 0$): $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Discriminant: $\Delta = b^2 - 4ac$

• $\Delta > 0$: two distinct real roots
• $\Delta = 0$: one repeated real root
• $\Delta < 0$: two complex conjugate roots

Completing the Square

For $ax^2 + bx + c = 0$: $a\left(x + \frac{b}{2a}\right)^2 = ax^2 + bx + \frac{b^2}{4a}$

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Functions

Definition

A function $f: A \to B$ assigns to each element $x \in A$ exactly one element $f(x) \in B$.

Domain: Set A (inputs) Codomain: Set B (possible outputs) Range: {f(x) : x ∈ A} ⊆ B (actual outputs)

Types of Functions

Injection (One-to-One): $f(x_1) = f(x_2) \implies x_1 = x_2$

Surjection (Onto): For every $y \in B$, there exists $x \in A$ such that $f(x) = y$

Bijection (One-to-One and Onto): Both injective and surjective

Composition and Inverse

Composition: $(g \circ f)(x) = g(f(x))$

Inverse Function: $f^{-1}$ exists if and only if $f$ is bijective, and $f^{-1}(f(x)) = x$, $f(f^{-1}(y)) = y$

Even and Odd Functions

• Even: $f(-x) = f(x)$ for all $x$ in domain
• Odd: $f(-x) = -f(x)$ for all $x$ in domain

Periodic Functions

$f(x + T) = f(x)$ for all $x$, where $T$ is the period.

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Sequences and Series

Sequence

A sequence is a function from $\mathbb{N}$ to $\mathbb{R}$: {a_n} = {a₁, a₂, a₃, …}

Types of Sequences

Arithmetic: $a_n = a_1 + (n-1)d$ where $d$ is common difference $S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n-1)d]$

Geometric: $a_n = a_1r^{n-1}$ where $r$ is common ratio $S_n = a_1 \frac{1-r^n}{1-r} \quad (r \neq 1)$

Convergence of Sequences

A sequence {a_n} converges to L if for every ε > 0, there exists N such that |a_n - L| < ε for all n > N.

Series

A series is the sum of sequence terms: $\sum_{n=1}^{\infty} a_n = \lim_{N \to \infty} \sum_{n=1}^{N} a_n$

Important Series

Geometric Series: $S = \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \quad \text{for } |r| < 1$

Harmonic Series: $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges

p-Series: $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if $p > 1$, diverges if $p \leq 1$

Telescoping Series: Many terms cancel out in partial sums

See also: Calculus for convergence tests and power series.

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Exponential and Logarithmic Functions

Exponential Function

$e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n$

Properties:

• $e^0 = 1$
• $e^{x+y} = e^x \cdot e^y$
• $e^{-x} = 1/e^x$
• $e^{xy} = (e^x)^y$
• $d/dx(e^x) = e^x$

Logarithmic Function

$y = \ln x \quad \text{if and only if} \quad x = e^y$

Properties:

• $\ln(1) = 0$
• $\ln(e) = 1$
• $\ln(xy) = \ln x + \ln y$
• $\ln(x/y) = \ln x - \ln y$
• $\ln(x^r) = r \ln x$
• $d/dx(\ln x) = 1/x$

Change of Base: $\log_a x = \frac{\ln x}{\ln a}$

Applications

• Exponential growth/decay: $y = Ce^{kt}$
• Half-life: $t_{1/2} = (\ln 2)/k$
• Continuous compounding: $A = Pe^{rt}$

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Trigonometric Functions

Basic Definitions

• $\sin \theta = \text{opposite}/\text{hypotenuse}$
• $\cos \theta = \text{adjacent}/\text{hypotenuse}$
• $\tan \theta = \sin \theta/\cos \theta$
• $\csc \theta = 1/\sin \theta$
• $\sec \theta = 1/\cos \theta$
• $\cot \theta = 1/\tan \theta$

Unit Circle

On unit circle, point at angle $\theta$ is $(\cos \theta, \sin \theta)$

Fundamental Identities

Pythagorean: $\\sin^2 \theta + \cos^2 \theta = 1$ $1 + \tan^2 \theta = \sec^2 \theta$ $1 + \cot^2 \theta = \csc^2 \theta$

Angle Sum/Difference:

• $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
• $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$
• $\tan(A \pm B) = (\tan A \pm \tan B)/(1 \mp \tan A \tan B)$

Double Angle:

• $\sin 2\theta = 2 \sin \theta \cos \theta$
• $\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2 \cos^2 \theta - 1 = 1 - 2 \sin^2 \theta$
• $\tan 2\theta = 2 \tan \theta/(1 - \tan^2 \theta)$

Half Angle:

• $\sin^2(\theta/2) = (1 - \cos \theta)/2$
• $\cos^2(\theta/2) = (1 + \cos \theta)/2$
• $\tan(\theta/2) = \sin \theta/(1 + \cos \theta) = (1 - \cos \theta)/\sin \theta$

Product to Sum:

• $\sin A \sin B = [\cos(A-B) - \cos(A+B)]/2$
• $\cos A \cos B = [\cos(A-B) + \cos(A+B)]/2$
• $\sin A \cos B = [\sin(A+B) + \sin(A-B)]/2$

Law of Sines: $a/\sin A = b/\sin B = c/\sin C$

Law of Cosines: $a^2 = b^2 + c^2 - 2bc \cos A$

Inverse Trigonometric Functions

• $\arcsin x$: $[-\pi/2, \pi/2]$
• $\arccos x$: $[0, \pi]$
• $\arctan x$: $(-\pi/2, \pi/2)$
• $\text{arcsec } x, \text{arccsc } x, \text{arccot } x$

Derivatives:

• $d/dx(\arcsin x) = 1/\sqrt{1-x^2}$
• $d/dx(\arccos x) = -1/\sqrt{1-x^2}$
• $d/dx(\arctan x) = 1/(1+x^2)$

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Polynomials

Polynomial Degree

$P(x) = a_n x^n + a_{n-1}x^{n-1} + \dots + a_1 x + a_0$ Degree is $n$ if $a_n \neq 0$.

Fundamental Theorem of Algebra

Every polynomial of degree $n$ has exactly $n$ roots (counting multiplicity) over $\mathbb{C}$.

Synthetic Division

Method for dividing polynomials by linear factors.

Rational Root Theorem

If $p/q$ (in lowest terms) is a root of $P(x) = a_n x^n + \dots + a_0$, then $p$ divides $a_0$ and $q$ divides $a_n$.

Vieta’s Formulas

For $P(x) = x^2 + px + q = (x-r)(x-s)$:

• Sum of roots: $r + s = -p$
• Product of roots: $rs = q$

For cubic $ax^3 + bx^2 + cx + d = a(x-r)(x-s)(x-t)$:

• $r + s + t = -b/a$
• $rs + rt + st = c/a$
• $rst = -d/a$

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Rational Functions

Definition

$R(x) = P(x)/Q(x)$ where $P$ and $Q$ are polynomials.

Partial Fractions

Decompose rational functions for integration.

For non-repeated linear factors: $\frac{A}{(x-a)(x-b)} = \frac{C_1}{x-a} + \frac{C_2}{x-b}$

For repeated factors: $\frac{A}{(x-a)^k} = \frac{C_1}{x-a} + \frac{C_2}{(x-a)^2} + \cdots + \frac{C_k}{(x-a)^k}$

For quadratic factors: $\frac{Ax+B}{x^2+px+q} \text{ (if irreducible)}$

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Inequalities

Basic Rules

• If $a < b$ and $c > 0$, then $ac < bc$
• If $a < b$ and $c < 0$, then $ac > bc$
• If $a < b$ and $b < c$, then $a < c$
• If $a < b$ and $c < d$, then $a + c < b + d$

Important Inequalities

Arithmetic-Geometric Mean (AM-GM): For positive numbers: $(x_1 + x_2 + \dots + x_n)/n \geq (x_1x_2\dots x_n)^{1/n}$

Cauchy-Schwarz: $(a_1b_1 + \cdots + a_nb_n)^2 \leq (a_1^2 + \cdots + a_n^2)(b_1^2 + \cdots + b_n^2)$

Bernoulli’s Inequality: $(1 + x)^n \geq 1 + nx$ for $x \geq -1, n \in \mathbb{N}$

Triangle Inequality: $|a + b| \le |a| + |b|$

Solving Polynomial Inequalities

• Find critical points (zeros and discontinuities)
• Test intervals between critical points
• Determine sign of expression in each interval

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Coordinate Geometry

Distance Formula

Distance between $(x_1, y_1)$ and $(x_2, y_2)$: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

Midpoint Formula

Midpoint of $(x_1, y_1)$ and $(x_2, y_2)$: $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$

Slope

Slope of line through $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2-y_1}{x_2-x_1} = \tan \theta$

Equations of Lines

• Point-slope: $y - y_1 = m(x - x_1)$
• Slope-intercept: $y = mx + b$
• Standard: $Ax + By + C = 0$
• Intercept form: $x/a + y/b = 1$
• Parametric: $x = x_0 + at, y = y_0 + bt$

Angle Between Lines

For lines with slopes $m_1$ and $m_2$: $\tan \theta = \left|\frac{m_2-m_1}{1+m_1m_2}\right|$

Lines are perpendicular if $m_1m_2 = -1$.

Conic Sections

Circle: $(x-h)^2 + (y-k)^2 = r^2$

• Center: $(h, k)$, Radius: $r$

Ellipse: $(x-h)^2/a^2 + (y-k)^2/b^2 = 1$

• Foci at distance $c$ from center: $c^2 = a^2 - b^2$

Parabola: $(y-k)^2 = 4p(x-h)$ or $(x-h)^2 = 4p(y-k)$

• Focus distance $p$ from vertex

Hyperbola: $(x-h)^2/a^2 - (y-k)^2/b^2 = 1$

• Asymptotes: $y-k = \pm(b/a)(x-h)$
• Foci: $c^2 = a^2 + b^2$

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Extra Algebra Reference (merged from mathematics_GPT)

Powers and Roots

• $x^m \cdot x^n = x^{m+n}$
• $(x^m)^n = x^{mn}$
• $(xy)^n = x^n y^n$
• $(x/y)^n = x^n / y^n$
• $x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$

Basic Factorings

• $a^2-b^2 = (a+b)(a-b)$
• $a^2+2ab+b^2=(a+b)^2$
• $a^2-2ab+b^2=(a-b)^2$
• $a^3+b^3=(a+b)(a^2-ab+b^2)$
• $a^3-b^3=(a-b)(a^2+ab+b^2)$

Logarithmic Identities

• $\log_a(xy) = \log_a x + \log_a y$
• $\log_a(x/y) = \log_a x - \log_a y$
• $\log_a(x^n) = n \log_a x$
• $\log_a x = \frac{\log_b x}{\log_b a}$
• $\log_a a = 1$, $\log_a 1 = 0$

Quadratic Equations and Vieta

(Virtually redundant but preserve Vieta formulas for completeness):

• $r_1+r_2=-b/a$
• $r_1r_2=c/a$

Rational Root Theorem

If $p/q$ in lowest terms is a root of $a_nx^n+...+a_0$, then $p$ divides $a_0$, $q$ divides $a_n$.

Remainder Theorem

$P(x)$ divided by $(x-c)$: remainder is $P(c)$.

Other exponent rules/inequalities

• $a^x \cdot a^y = a^{x+y}$, $(a^x)^y = a^{xy}$, $a^{-x}=1/a^x$, $a^{1/n}=\sqrt[n]{a}$
• If $a<b$, $c>0$, then $ac<bc$. If $a<b$ and $c<0$, then $ac>bc$. If $0<a<b$, $0<1/b<1/a$

Common Algebraic Mistakes

• $\sqrt{a+b} \neq \sqrt{a} + \sqrt{b}$
• $(x+y)^2 \neq x^2+y^2$
• $1/(a+b) \neq 1/a + 1/b$
• $\sin^{-1}(x)$ is not $1/\sin x$

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Next: Calculus

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Last updated: Comprehensive fundamentals reference covering undergraduate through basic graduate material.