Basics Elementary Middle School High School University Advanced
How to Write Math Formulas

Math formulas are displayed by enclosing them in "$" (dollar signs).
Example: Writing $\sin^2 \theta + \cos^2 \theta = 1$ displays $\sin^2 \theta + \cos^2 \theta = 1$.

Table of Contents

Quadratic Functions & Equations

General Form of Quadratic Function
Code:
$y = ax^2 + bx + c$
Display:
$y = ax^2 + bx + c$
Vertex Form
Code:
$y = a(x - p)^2 + q$
Display:
$y = a(x - p)^2 + q$
Axis of Symmetry
Code:
$x = -\displaystyle\frac{b}{2a}$
Display:
$x = -\displaystyle\frac{b}{2a}$
Quadratic Formula
Code:
$x = \displaystyle\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Display:
$x = \displaystyle\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Discriminant
Code:
$D = b^2 - 4ac$
Display:
$D = b^2 - 4ac$
Vieta's Formulas (Sum of Roots)
Code:
$\alpha + \beta = -\displaystyle\frac{b}{a}$
Display:
$\alpha + \beta = -\displaystyle\frac{b}{a}$
Vieta's Formulas (Product of Roots)
Code:
$\alpha \beta = \displaystyle\frac{c}{a}$
Display:
$\alpha \beta = \displaystyle\frac{c}{a}$
Completing the Square
Code:
$ax^2 + bx + c = a\left(x + \displaystyle\frac{b}{2a}\right)^2 - \displaystyle\frac{b^2 - 4ac}{4a}$
Display:
$ax^2 + bx + c = a\left(x + \displaystyle\frac{b}{2a}\right)^2 - \displaystyle\frac{b^2 - 4ac}{4a}$

Permutations, Combinations & Probability

Factorial
Code:
$n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1$
Display:
$n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1$
Permutation
Code:
$_n P_r = \displaystyle\frac{n!}{(n-r)!}$
Display:
$_n P_r = \displaystyle\frac{n!}{(n-r)!}$
Combination
Code:
$_n C_r = \displaystyle\frac{n!}{r!(n-r)!}$
Display:
$_n C_r = \displaystyle\frac{n!}{r!(n-r)!}$
Combination (Alternative Notation)
Code:
$\binom{n}{r} = \displaystyle\frac{n!}{r!(n-r)!}$
Display:
$\binom{n}{r} = \displaystyle\frac{n!}{r!(n-r)!}$
Permutation with Repetition
Code:
$_n \Pi_r = n^r$
Display:
$_n \Pi_r = n^r$
Combination with Repetition
Code:
$_n H_r = _{n+r-1} C_r$
Display:
$_n H_r = _{n+r-1} C_r$
Basic Probability
Code:
$P(A) = \displaystyle\frac{n(A)}{n(U)}$
Display:
$P(A) = \displaystyle\frac{n(A)}{n(U)}$
Complement Rule
Code:
$P(\overline{A}) = 1 - P(A)$
Display:
$P(\overline{A}) = 1 - P(A)$
Addition Rule
Code:
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Display:
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Addition Rule for Mutually Exclusive Events
Code:
$P(A \cup B) = P(A) + P(B)$
Display:
$P(A \cup B) = P(A) + P(B)$
Multiplication Rule for Independent Events
Code:
$P(A \cap B) = P(A) \cdot P(B)$
Display:
$P(A \cap B) = P(A) \cdot P(B)$
Conditional Probability
Code:
$P(A|B) = \displaystyle\frac{P(A \cap B)}{P(B)}$
Display:
$P(A|B) = \displaystyle\frac{P(A \cap B)}{P(B)}$
Multiplication Rule
Code:
$P(A \cap B) = P(B) \cdot P(A|B)$
Display:
$P(A \cap B) = P(B) \cdot P(A|B)$
Repeated Trials (Binomial Probability)
Code:
$P(X = k) = {}_n C_k p^k (1-p)^{n-k}$
Display:
$P(X = k) = {}_n C_k p^k (1-p)^{n-k}$
Expected Value
Code:
$E(X) = \displaystyle\sum_{i=1}^{n} x_i p_i$
Display:
$E(X) = \displaystyle\sum_{i=1}^{n} x_i p_i$

Trigonometric Functions

Basic Trigonometric Functions
Code:
$\sin \theta, \cos \theta, \tan \theta$
Display:
$\sin \theta, \cos \theta, \tan \theta$
Pythagorean Identity
Code:
$\sin^2 \theta + \cos^2 \theta = 1$
Display:
$\sin^2 \theta + \cos^2 \theta = 1$
Addition Formulas
Code:
$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$
Display:
$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$
Inverse Trigonometric Functions (arc notation)
Code:
$\arcsin x, \arccos x, \arctan x$
Display:
$\arcsin x, \arccos x, \arctan x$
Inverse Trigonometric Functions (-1 notation)
Code:
$\sin^{-1} x, \cos^{-1} x, \tan^{-1} x$
Display:
$\sin^{-1} x, \cos^{-1} x, \tan^{-1} x$
About Inverse Trig Notation

Both notations are widely used: $\arcsin x$ and $\sin^{-1} x$ mean the same thing.
Note: $\sin^{-1} x$ means the inverse function, not $\frac{1}{\sin x}$ (which is $\csc x$).

Double Angle Formula (sin)
Code:
$\sin 2\theta = 2\sin \theta \cos \theta$
Display:
$\sin 2\theta = 2\sin \theta \cos \theta$
Double Angle Formula (cos)
Code:
$\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta$
Display:
$\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta$
Double Angle Formula (tan)
Code:
$\tan 2\theta = \displaystyle\frac{2\tan \theta}{1 - \tan^2 \theta}$
Display:
$\tan 2\theta = \displaystyle\frac{2\tan \theta}{1 - \tan^2 \theta}$
Half Angle Formula (sin)
Code:
$\sin^2 \displaystyle\frac{\theta}{2} = \frac{1 - \cos \theta}{2}$
Display:
$\sin^2 \displaystyle\frac{\theta}{2} = \frac{1 - \cos \theta}{2}$
Half Angle Formula (cos)
Code:
$\cos^2 \displaystyle\frac{\theta}{2} = \frac{1 + \cos \theta}{2}$
Display:
$\cos^2 \displaystyle\frac{\theta}{2} = \frac{1 + \cos \theta}{2}$
Triple Angle Formula (sin)
Code:
$\sin 3\theta = 3\sin \theta - 4\sin^3 \theta$
Display:
$\sin 3\theta = 3\sin \theta - 4\sin^3 \theta$
Triple Angle Formula (cos)
Code:
$\cos 3\theta = 4\cos^3 \theta - 3\cos \theta$
Display:
$\cos 3\theta = 4\cos^3 \theta - 3\cos \theta$
Product-to-Sum Formula
Code:
$\sin \alpha \cos \beta = \displaystyle\frac{1}{2}\{\sin(\alpha + \beta) + \sin(\alpha - \beta)\}$
Display:
$\sin \alpha \cos \beta = \displaystyle\frac{1}{2}\{\sin(\alpha + \beta) + \sin(\alpha - \beta)\}$
Product-to-Sum Formula (cos cos)
Code:
$\cos \alpha \cos \beta = \displaystyle\frac{1}{2}\{\cos(\alpha + \beta) + \cos(\alpha - \beta)\}$
Display:
$\cos \alpha \cos \beta = \displaystyle\frac{1}{2}\{\cos(\alpha + \beta) + \cos(\alpha - \beta)\}$
Product-to-Sum Formula (sin sin)
Code:
$\sin \alpha \sin \beta = -\displaystyle\frac{1}{2}\{\cos(\alpha + \beta) - \cos(\alpha - \beta)\}$
Display:
$\sin \alpha \sin \beta = -\displaystyle\frac{1}{2}\{\cos(\alpha + \beta) - \cos(\alpha - \beta)\}$
Sum-to-Product Formula (sin + sin)
Code:
$\sin A + \sin B = 2\sin\displaystyle\frac{A+B}{2}\cos\frac{A-B}{2}$
Display:
$\sin A + \sin B = 2\sin\displaystyle\frac{A+B}{2}\cos\frac{A-B}{2}$
Sum-to-Product Formula (cos + cos)
Code:
$\cos A + \cos B = 2\cos\displaystyle\frac{A+B}{2}\cos\frac{A-B}{2}$
Display:
$\cos A + \cos B = 2\cos\displaystyle\frac{A+B}{2}\cos\frac{A-B}{2}$
Law of Sines
Code:
$\displaystyle\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$
Display:
$\displaystyle\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$
Law of Cosines
Code:
$a^2 = b^2 + c^2 - 2bc\cos A$
Display:
$a^2 = b^2 + c^2 - 2bc\cos A$
Area of Triangle (Sine Formula)
Code:
$S = \displaystyle\frac{1}{2}bc\sin A$
Display:
$S = \displaystyle\frac{1}{2}bc\sin A$
Heron's Formula
Code:
$S = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \displaystyle\frac{a+b+c}{2}$
Display:
$S = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \displaystyle\frac{a+b+c}{2}$
Trigonometric Synthesis
Code:
$a\sin\theta + b\cos\theta = \sqrt{a^2 + b^2}\sin(\theta + \phi)$
Display:
$a\sin\theta + b\cos\theta = \sqrt{a^2 + b^2}\sin(\theta + \phi)$

Exponentials and Logarithms

Exponential Function
Code:
$y = a^x$
Display:
$y = a^x$
Base of Natural Logarithm
Code:
$e^x$
Display:
$e^x$
Logarithm
Code:
$\log_a x$
Display:
$\log_a x$
Natural Logarithm
Code:
$\ln x$
Display:
$\ln x$
Properties of Logarithms
Code:
$\log_a (xy) = \log_a x + \log_a y$
Display:
$\log_a (xy) = \log_a x + \log_a y$

Higher-Order Equations

Factor Theorem
Code:
$P(a) = 0 \Leftrightarrow P(x)$ is divisible by $(x - a)$
Display:
$P(a) = 0 \Leftrightarrow P(x)$ is divisible by $(x - a)$
Remainder Theorem
Code:
When $P(x)$ is divided by $(x - a)$, the remainder is $P(a)$
Display:
When $P(x)$ is divided by $(x - a)$, the remainder is $P(a)$
Cubic Equation Vieta's Formulas
Code:
For $ax^3 + bx^2 + cx + d = 0$ with roots $\alpha, \beta, \gamma$: $\alpha + \beta + \gamma = -\displaystyle\frac{b}{a}$
Display:
For $ax^3 + bx^2 + cx + d = 0$ with roots $\alpha, \beta, \gamma$: $\alpha + \beta + \gamma = -\displaystyle\frac{b}{a}$
Sum of Cubes
Code:
$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
Display:
$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
Difference of Cubes
Code:
$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
Display:
$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Analytic Geometry

Distance Between Two Points
Code:
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Display:
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Internal Division Point
Code:
$\left(\displaystyle\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)$
Display:
$\left(\displaystyle\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)$
Midpoint Formula
Code:
$\left(\displaystyle\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
Display:
$\left(\displaystyle\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
Equation of a Circle (Center-Radius Form)
Code:
$(x - a)^2 + (y - b)^2 = r^2$
Display:
$(x - a)^2 + (y - b)^2 = r^2$
Equation of a Circle (General Form)
Code:
$x^2 + y^2 + Dx + Ey + F = 0$
Display:
$x^2 + y^2 + Dx + Ey + F = 0$
Distance from a Point to a Line
Code:
$d = \displaystyle\frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}$
Display:
$d = \displaystyle\frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}$
Tangent Line to a Circle
Code:
Tangent at $(x_1, y_1)$ on $x^2 + y^2 = r^2$: $x_1 x + y_1 y = r^2$
Display:
Tangent at $(x_1, y_1)$ on $x^2 + y^2 = r^2$: $x_1 x + y_1 y = r^2$
Equation of an Ellipse
Code:
$\displaystyle\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
Display:
$\displaystyle\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
Equation of a Hyperbola
Code:
$\displaystyle\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
Display:
$\displaystyle\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
Equation of a Parabola
Code:
$y^2 = 4px$
Display:
$y^2 = 4px$
Asymptotes of a Hyperbola
Code:
$y = \pm\displaystyle\frac{b}{a}x$
Display:
$y = \pm\displaystyle\frac{b}{a}x$
Focus of an Ellipse
Code:
$c = \sqrt{a^2 - b^2}$ (when $a > b$)
Display:
$c = \sqrt{a^2 - b^2}$ (when $a > b$)
Eccentricity of an Ellipse
Code:
$e = \displaystyle\frac{c}{a} = \frac{\sqrt{a^2 - b^2}}{a}$
Display:
$e = \displaystyle\frac{c}{a} = \frac{\sqrt{a^2 - b^2}}{a}$
Parametric Form of an Ellipse
Code:
$x = a\cos\theta, \quad y = b\sin\theta$
Display:
$x = a\cos\theta, \quad y = b\sin\theta$
Directrix of a Parabola
Code:
For $y^2 = 4px$: Focus $(p, 0)$, Directrix $x = -p$
Display:
For $y^2 = 4px$: Focus $(p, 0)$, Directrix $x = -p$
Polar Equation of a Conic
Code:
$r = \displaystyle\frac{l}{1 + e\cos\theta}$
Display:
$r = \displaystyle\frac{l}{1 + e\cos\theta}$

Differentiation

Derivative
Code:
$f'(x)$ or $\displaystyle\frac{dy}{dx}$
Display:
$f'(x)$ or $\displaystyle\frac{dy}{dx}$
Power Rule
Code:
$(x^n)' = nx^{n-1}$
Display:
$(x^n)' = nx^{n-1}$
Derivative of Trigonometric Functions
Code:
$(\sin x)' = \cos x$
Display:
$(\sin x)' = \cos x$
Derivative of Exponential Function
Code:
$(e^x)' = e^x$
Display:
$(e^x)' = e^x$
Derivative of Logarithmic Function
Code:
$(\ln x)' = \displaystyle\frac{1}{x}$
Display:
$(\ln x)' = \displaystyle\frac{1}{x}$
Product Rule
Code:
$(fg)' = f'g + fg'$
Display:
$(fg)' = f'g + fg'$
Quotient Rule
Code:
$\left(\displaystyle\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$
Display:
$\left(\displaystyle\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$
Chain Rule
Code:
$\{f(g(x))\}' = f'(g(x)) \cdot g'(x)$
Display:
$\{f(g(x))\}' = f'(g(x)) \cdot g'(x)$
Derivative of Cosine
Code:
$(\cos x)' = -\sin x$
Display:
$(\cos x)' = -\sin x$
Derivative of Tangent
Code:
$(\tan x)' = \displaystyle\frac{1}{\cos^2 x} = \sec^2 x$
Display:
$(\tan x)' = \displaystyle\frac{1}{\cos^2 x} = \sec^2 x$
Derivative of General Exponential
Code:
$(a^x)' = a^x \ln a$
Display:
$(a^x)' = a^x \ln a$
Derivative of Logarithm (any base)
Code:
$(\log_a x)' = \displaystyle\frac{1}{x \ln a}$
Display:
$(\log_a x)' = \displaystyle\frac{1}{x \ln a}$
Definition of Derivative
Code:
$f'(a) = \displaystyle\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$
Display:
$f'(a) = \displaystyle\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

Integration

Indefinite Integral
Code:
$\displaystyle\int f(x) dx$
Display:
$\displaystyle\int f(x) dx$
Definite Integral
Code:
$\displaystyle\int_a^b f(x) dx$
Display:
$\displaystyle\int_a^b f(x) dx$
Power Rule for Integration
Code:
$\displaystyle\int x^n dx = \frac{x^{n+1}}{n+1} + C$
Display:
$\displaystyle\int x^n dx = \frac{x^{n+1}}{n+1} + C$
Integral of Trigonometric Functions
Code:
$\displaystyle\int \cos x dx = \sin x + C$
Display:
$\displaystyle\int \cos x dx = \sin x + C$
Integral of Exponential Function
Code:
$\displaystyle\int e^x dx = e^x + C$
Display:
$\displaystyle\int e^x dx = e^x + C$
Integral of Sine
Code:
$\displaystyle\int \sin x dx = -\cos x + C$
Display:
$\displaystyle\int \sin x dx = -\cos x + C$
Integral of 1/x
Code:
$\displaystyle\int \frac{1}{x} dx = \ln |x| + C$
Display:
$\displaystyle\int \frac{1}{x} dx = \ln |x| + C$
Fundamental Theorem of Calculus
Code:
$\displaystyle\int_a^b f(x) dx = F(b) - F(a)$ where $F'(x) = f(x)$
Display:
$\displaystyle\int_a^b f(x) dx = F(b) - F(a)$ where $F'(x) = f(x)$
Area Under a Curve
Code:
$S = \displaystyle\int_a^b |f(x)| dx$
Display:
$S = \displaystyle\int_a^b |f(x)| dx$
Area Between Two Curves
Code:
$S = \displaystyle\int_a^b |f(x) - g(x)| dx$
Display:
$S = \displaystyle\int_a^b |f(x) - g(x)| dx$
Volume of Revolution (x-axis)
Code:
$V = \pi\displaystyle\int_a^b \{f(x)\}^2 dx$
Display:
$V = \pi\displaystyle\int_a^b \{f(x)\}^2 dx$
Volume of Revolution (y-axis)
Code:
$V = \pi\displaystyle\int_c^d \{g(y)\}^2 dy$
Display:
$V = \pi\displaystyle\int_c^d \{g(y)\}^2 dy$
Arc Length
Code:
$L = \displaystyle\int_a^b \sqrt{1 + \{f'(x)\}^2} dx$
Display:
$L = \displaystyle\int_a^b \sqrt{1 + \{f'(x)\}^2} dx$

Vectors

Vector Notation
Code:
$\vec{a}$ or $\boldsymbol{a}$
Display:
$\vec{a}$ or $\boldsymbol{a}$
Component Form
Code:
$\vec{a} = (a_1, a_2, a_3)$
Display:
$\vec{a} = (a_1, a_2, a_3)$
Dot Product
Code:
$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$
Display:
$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$
Magnitude of Vector
Code:
$|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$
Display:
$|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$
Dot Product (Component Form)
Code:
$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$
Display:
$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$
Perpendicularity Condition
Code:
$\vec{a} \perp \vec{b} \Leftrightarrow \vec{a} \cdot \vec{b} = 0$
Display:
$\vec{a} \perp \vec{b} \Leftrightarrow \vec{a} \cdot \vec{b} = 0$
Position Vector of Internal Division Point
Code:
$\vec{p} = \displaystyle\frac{n\vec{a} + m\vec{b}}{m + n}$
Display:
$\vec{p} = \displaystyle\frac{n\vec{a} + m\vec{b}}{m + n}$
Vector Equation of a Line (2D)
Code:
$\vec{p} = \vec{a} + t\vec{d}$
Display:
$\vec{p} = \vec{a} + t\vec{d}$
Vector Equation of a Plane
Code:
$\vec{p} = \vec{a} + s\vec{b} + t\vec{c}$
Display:
$\vec{p} = \vec{a} + s\vec{b} + t\vec{c}$
Position Vector of Centroid
Code:
$\vec{g} = \displaystyle\frac{\vec{a} + \vec{b} + \vec{c}}{3}$
Display:
$\vec{g} = \displaystyle\frac{\vec{a} + \vec{b} + \vec{c}}{3}$
Condition for Collinearity
Code:
$\vec{AB} = k\vec{AC}$ (for some real $k$)
Display:
$\vec{AB} = k\vec{AC}$ (for some real $k$)
Projection of a Vector
Code:
$\text{proj}_{\vec{b}}\vec{a} = \displaystyle\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\vec{b}$
Display:
$\text{proj}_{\vec{b}}\vec{a} = \displaystyle\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\vec{b}$
Unit Vector
Code:
$\vec{e} = \displaystyle\frac{\vec{a}}{|\vec{a}|}$
Display:
$\vec{e} = \displaystyle\frac{\vec{a}}{|\vec{a}|}$

Sequences

General Term of Sequence
Code:
$\{a_n\}$
Display:
$\{a_n\}$
Arithmetic Sequence
Code:
$a_n = a_1 + (n-1)d$
Display:
$a_n = a_1 + (n-1)d$
Geometric Sequence
Code:
$a_n = a_1 \cdot r^{n-1}$
Display:
$a_n = a_1 \cdot r^{n-1}$
Summation (Sigma)
Code:
$\displaystyle\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$
Display:
$\displaystyle\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$
Product Notation (Pi)
Code:
$\displaystyle\prod_{k=1}^{n} k = n!$
Display:
$\displaystyle\prod_{k=1}^{n} k = n!$

Recurrence Relations

Arithmetic Type
Code:
$a_{n+1} = a_n + d$
Display:
$a_{n+1} = a_n + d$
Geometric Type
Code:
$a_{n+1} = r \cdot a_n$
Display:
$a_{n+1} = r \cdot a_n$
Linear Type
Code:
$a_{n+1} = pa_n + q$
Display:
$a_{n+1} = pa_n + q$
Difference Sequence Type
Code:
$a_{n+1} - a_n = f(n)$
Display:
$a_{n+1} - a_n = f(n)$
Three-Term Recurrence
Code:
$a_{n+2} = pa_{n+1} + qa_n$
Display:
$a_{n+2} = pa_{n+1} + qa_n$
Fibonacci Sequence
Code:
$F_{n+2} = F_{n+1} + F_n$ ($F_1 = F_2 = 1$)
Display:
$F_{n+2} = F_{n+1} + F_n$ ($F_1 = F_2 = 1$)
Characteristic Equation
Code:
For $a_{n+2} = pa_{n+1} + qa_n$: $x^2 = px + q$
Display:
For $a_{n+2} = pa_{n+1} + qa_n$: $x^2 = px + q$
General Solution (Distinct Roots)
Code:
$a_n = A\alpha^n + B\beta^n$
Display:
$a_n = A\alpha^n + B\beta^n$

Limits

Limit Notation
Code:
$\displaystyle\lim_{x \to a} f(x)$
Display:
$\displaystyle\lim_{x \to a} f(x)$
Limit as x Approaches Infinity
Code:
$\displaystyle\lim_{x \to \infty} \frac{1}{x} = 0$
Display:
$\displaystyle\lim_{x \to \infty} \frac{1}{x} = 0$
Right and Left Limits
Code:
$\displaystyle\lim_{x \to a^+} f(x)$, $\displaystyle\lim_{x \to a^-} f(x)$
Display:
$\displaystyle\lim_{x \to a^+} f(x)$, $\displaystyle\lim_{x \to a^-} f(x)$
Limit of Sequence
Code:
$\displaystyle\lim_{n \to \infty} a_n = L$
Display:
$\displaystyle\lim_{n \to \infty} a_n = L$

Matrices

2×2 Matrix
Code:
$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$
Display:
$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$
3×3 Matrix
Code:
$\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$
Display:
$\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$
Determinant
Code:
$\det A$ or $|A|$
Display:
$\det A$ or $|A|$
Inverse Matrix
Code:
$A^{-1}$
Display:
$A^{-1}$
Transpose Matrix
Code:
$A^T$
Display:
$A^T$

Complex Numbers

Imaginary Unit
Code:
$i$ or $i^2 = -1$
Display:
$i$ or $i^2 = -1$
Complex Number Notation
Code:
$z = a + bi$
Display:
$z = a + bi$
Complex Conjugate
Code:
$\overline{z} = a - bi$
Display:
$\overline{z} = a - bi$
Absolute Value
Code:
$|z| = \sqrt{a^2 + b^2}$
Display:
$|z| = \sqrt{a^2 + b^2}$
Euler's Formula
Code:
$e^{i\theta} = \cos\theta + i\sin\theta$
Display:
$e^{i\theta} = \cos\theta + i\sin\theta$
Polar Form
Code:
$z = r(\cos\theta + i\sin\theta)$
Display:
$z = r(\cos\theta + i\sin\theta)$
De Moivre's Theorem
Code:
$(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$
Display:
$(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$
Product of Complex Numbers (Polar Form)
Code:
$z_1 z_2 = r_1 r_2 \{\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)\}$
Display:
$z_1 z_2 = r_1 r_2 \{\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)\}$
Quotient of Complex Numbers (Polar Form)
Code:
$\displaystyle\frac{z_1}{z_2} = \frac{r_1}{r_2} \{\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)\}$
Display:
$\displaystyle\frac{z_1}{z_2} = \frac{r_1}{r_2} \{\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)\}$
n-th Roots of Unity
Code:
$z^n = 1 \Rightarrow z = \cos\displaystyle\frac{2k\pi}{n} + i\sin\frac{2k\pi}{n}$
Display:
$z^n = 1 \Rightarrow z = \cos\displaystyle\frac{2k\pi}{n} + i\sin\frac{2k\pi}{n}$
Rotation on the Complex Plane
Code:
Rotation of $z$ by $\theta$: $w = z \cdot (\cos\theta + i\sin\theta)$
Display:
Rotation of $z$ by $\theta$: $w = z \cdot (\cos\theta + i\sin\theta)$
Product of Conjugates
Code:
$z \cdot \overline{z} = |z|^2$
Display:
$z \cdot \overline{z} = |z|^2$
Real and Imaginary Parts
Code:
$\text{Re}(z) = \displaystyle\frac{z + \overline{z}}{2}, \quad \text{Im}(z) = \frac{z - \overline{z}}{2i}$
Display:
$\text{Re}(z) = \displaystyle\frac{z + \overline{z}}{2}, \quad \text{Im}(z) = \frac{z - \overline{z}}{2i}$
Argument (Angle)
Code:
$\arg(z) = \theta$ where $z = r(\cos\theta + i\sin\theta)$
Display:
$\arg(z) = \theta$ where $z = r(\cos\theta + i\sin\theta)$
Argument of Product/Quotient
Code:
$\arg(z_1 z_2) = \arg(z_1) + \arg(z_2), \quad \arg\left(\displaystyle\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)$
Display:
$\arg(z_1 z_2) = \arg(z_1) + \arg(z_2), \quad \arg\left(\displaystyle\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)$
Distance on the Complex Plane
Code:
Distance between $z_1$ and $z_2$: $|z_1 - z_2|$
Display:
Distance between $z_1$ and $z_2$: $|z_1 - z_2|$

Common Greek Letters

Lowercase
Code:
$\alpha, \beta, \gamma, \delta, \epsilon, \theta, \lambda, \mu, \sigma, \phi, \omega$
Display:
$\alpha, \beta, \gamma, \delta, \epsilon, \theta, \lambda, \mu, \sigma, \phi, \omega$
Uppercase
Code:
$\Gamma, \Delta, \Theta, \Lambda, \Sigma, \Phi, \Omega$
Display:
$\Gamma, \Delta, \Theta, \Lambda, \Sigma, \Phi, \Omega$

Summation and Integration Formulas

Binomial Theorem
Code:
$(a+b)^n = \displaystyle\sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k$
Display:
$(a+b)^n = \displaystyle\sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k$
Integration by Parts
Code:
$\displaystyle\int u \, dv = uv - \int v \, du$
Display:
$\displaystyle\int u \, dv = uv - \int v \, du$
Integration by Substitution
Code:
$\displaystyle\int f(g(x))g'(x)dx = \int f(u)du$ (where $u=g(x)$)
Display:
$\displaystyle\int f(g(x))g'(x)dx = \int f(u)du$ (where $u=g(x)$)
Effect of \displaystyle

Adding \displaystyle changes the size and position of limits and fractions:

Without displaystyle: $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$

Result: $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$

With displaystyle: $\displaystyle\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$

Result: $\displaystyle\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$

With \displaystyle, fractions and limits are displayed in full size for better readability.

Vectors and Coordinate Systems

Cross Product
Code:
$\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \vec{n}$
Display:
$\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \vec{n}$
Parametric Equations
Code:
$\begin{cases} x = f(t) \\ y = g(t) \end{cases}$
Display:
$\begin{cases} x = f(t) \\ y = g(t) \end{cases}$
Polar Coordinates
Code:
$\begin{cases} x = r\cos\theta \\ y = r\sin\theta \end{cases}$
Display:
$\begin{cases} x = r\cos\theta \\ y = r\sin\theta \end{cases}$

Piecewise Notation

Piecewise Function Notation
Code:
$f(x) = \begin{cases} x^2 & (x \geq 0) \\ -x^2 & (x < 0) \end{cases}$
Display:
$f(x) = \begin{cases} x^2 & (x \geq 0) \\ -x^2 & (x < 0) \end{cases}$

Multi-line Equations

Multi-line Equation Tips

When writing step-by-step equation transformations or proofs, use \begin{align}...\end{align}.
- Use & to specify alignment position (usually placed before =)
- Use \\ for line breaks

Solving Linear Equations (align environment)
Code:
$\begin{align} 2x + 3 &= 11 \\ 2x &= 8 \\ x &= 4 \end{align}$
Display:
$\begin{align} 2x + 3 &= 11 \\ 2x &= 8 \\ x &= 4 \end{align}$
Solving Quadratic Equations
Code:
$\begin{align} x^2 - 5x + 6 &= 0 \\ (x - 2)(x - 3) &= 0 \\ x &= 2, 3 \end{align}$
Display:
$\begin{align} x^2 - 5x + 6 &= 0 \\ (x - 2)(x - 3) &= 0 \\ x &= 2, 3 \end{align}$
Integration Step by Step
Code:
$\begin{align} \int_0^2 x^2 dx &= \left[ \frac{x^3}{3} \right]_0^2 \\ &= \frac{8}{3} - 0 \\ &= \frac{8}{3} \end{align}$
Display:
$\begin{align} \int_0^2 x^2 dx &= \left[ \frac{x^3}{3} \right]_0^2 \\ &= \frac{8}{3} - 0 \\ &= \frac{8}{3} \end{align}$
Trigonometric Proof
Code:
$\begin{align} \sin 2\theta &= \sin(\theta + \theta) \\ &= \sin\theta\cos\theta + \cos\theta\sin\theta \\ &= 2\sin\theta\cos\theta \end{align}$
Display:
$\begin{align} \sin 2\theta &= \sin(\theta + \theta) \\ &= \sin\theta\cos\theta + \cos\theta\sin\theta \\ &= 2\sin\theta\cos\theta \end{align}$
Logarithm Calculation
Code:
$\begin{align} \log_2 8 + \log_2 4 &= \log_2 (8 \times 4) \\ &= \log_2 32 \\ &= 5 \end{align}$
Display:
$\begin{align} \log_2 8 + \log_2 4 &= \log_2 (8 \times 4) \\ &= \log_2 32 \\ &= 5 \end{align}$
Equation Transformation
Code:
$\begin{align} \frac{x+1}{2} &= \frac{x-1}{3} \\ 3(x+1) &= 2(x-1) \\ 3x + 3 &= 2x - 2 \\ x &= -5 \end{align}$
Display:
$\begin{align} \frac{x+1}{2} &= \frac{x-1}{3} \\ 3(x+1) &= 2(x-1) \\ 3x + 3 &= 2x - 2 \\ x &= -5 \end{align}$

Statistical Inference

Expected Value (Mean)
Code:
$E(X) = \displaystyle\sum_{i=1}^{n} x_i p_i$
Display:
$E(X) = \displaystyle\sum_{i=1}^{n} x_i p_i$
Variance and Standard Deviation
Code:
$V(X) = E(X^2) - \{E(X)\}^2, \quad \sigma(X) = \sqrt{V(X)}$
Display:
$V(X) = E(X^2) - \{E(X)\}^2, \quad \sigma(X) = \sqrt{V(X)}$
Binomial Distribution
Code:
$P(X = k) = {}_n C_k p^k (1-p)^{n-k}$
Display:
$P(X = k) = {}_n C_k p^k (1-p)^{n-k}$
Binomial Distribution: Expected Value and Variance
Code:
$X \sim B(n, p) \Rightarrow E(X) = np, \quad V(X) = np(1-p)$
Display:
$X \sim B(n, p) \Rightarrow E(X) = np, \quad V(X) = np(1-p)$
Normal Distribution
Code:
$X \sim N(\mu, \sigma^2)$
Display:
$X \sim N(\mu, \sigma^2)$
Standardization
Code:
$Z = \frac{X - \mu}{\sigma} \sim N(0, 1)$
Display:
$Z = \frac{X - \mu}{\sigma} \sim N(0, 1)$
Confidence Interval for Population Mean
Code:
$\bar{X} - 1.96 \cdot \frac{\sigma}{\sqrt{n}} \leq \mu \leq \bar{X} + 1.96 \cdot \frac{\sigma}{\sqrt{n}}$
Display:
$\bar{X} - 1.96 \cdot \frac{\sigma}{\sqrt{n}} \leq \mu \leq \bar{X} + 1.96 \cdot \frac{\sigma}{\sqrt{n}}$
Hypothesis Testing (Null and Alternative Hypotheses)
Code:
$H_0: \mu = \mu_0, \quad H_1: \mu \neq \mu_0$
Display:
$H_0: \mu = \mu_0, \quad H_1: \mu \neq \mu_0$
Test Statistic
Code:
$Z_0 = \frac{\bar{X} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$
Display:
$Z_0 = \frac{\bar{X} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$

Usage Examples in Reading Notes

Example 1: Differentiation Problem
Code:
p.124 problem: Differentiating $f(x) = x^3 - 3x^2 + 2$ gives $f'(x) = 3x^2 - 6x$. Setting $f'(x) = 0$ to find extrema gives $x = 0, 2$
Display:
p.124 problem: Differentiating $f(x) = x^3 - 3x^2 + 2$ gives $f'(x) = 3x^2 - 6x$. Setting $f'(x) = 0$ to find extrema gives $x = 0, 2$
Example 2: Integration Calculation
Code:
Definite integral $\displaystyle\int_0^1 (2x + 1) dx = [x^2 + x]_0^1 = (1 + 1) - 0 = 2$
Display:
Definite integral $\displaystyle\int_0^1 (2x + 1) dx = [x^2 + x]_0^1 = (1 + 1) - 0 = 2$
Example 3: Vector Dot Product
Code:
For $\vec{a} = (1, 2, 3)$ and $\vec{b} = (4, 5, 6)$, $\vec{a} \cdot \vec{b} = 1 \times 4 + 2 \times 5 + 3 \times 6 = 32$
Display:
For $\vec{a} = (1, 2, 3)$ and $\vec{b} = (4, 5, 6)$, $\vec{a} \cdot \vec{b} = 1 \times 4 + 2 \times 5 + 3 \times 6 = 32$
Common Symbol Reference
  • \sin, \cos, \tan → Trigonometric functions
  • \log, \ln → Logarithm
  • \int → Integral symbol ∫
  • \sum → Summation symbol Σ
  • \prod → Product symbol Π
  • \lim → Limit symbol lim
  • \frac{numerator}{denominator} → Fraction
  • \sqrt{} → Square root √
  • \vec{} → Vector arrow
  • \theta, \alpha, \beta → Greek letters
  • \infty → Infinity ∞
  • \cdot → Dot (for dot product, etc.)
  • \times → Multiplication
  • \pm → Plus-minus ±
  • ^ → Superscript (exponents, etc.)
  • _ → Subscript (indices, etc.)

Chemical Formulas & Equations (High School Chemistry)

Ammonia Synthesis (Haber-Bosch Process)
Code:
$\text{N}_2 + 3\text{H}_2 \rightleftharpoons 2\text{NH}_3$
Display:
$\text{N}_2 + 3\text{H}_2 \rightleftharpoons 2\text{NH}_3$
Combustion of Ethanol
Code:
$\text{C}_2\text{H}_5\text{OH} + 3\text{O}_2 \rightarrow 2\text{CO}_2 + 3\text{H}_2\text{O}$
Display:
$\text{C}_2\text{H}_5\text{OH} + 3\text{O}_2 \rightarrow 2\text{CO}_2 + 3\text{H}_2\text{O}$
Neutralization of HCl and NaOH
Code:
$\text{HCl} + \text{NaOH} \rightarrow \text{NaCl} + \text{H}_2\text{O}$
Display:
$\text{HCl} + \text{NaOH} \rightarrow \text{NaCl} + \text{H}_2\text{O}$
Reaction of Sulfuric Acid and Barium Hydroxide
Code:
$\text{H}_2\text{SO}_4 + \text{Ba(OH)}_2 \rightarrow \text{BaSO}_4 \downarrow + 2\text{H}_2\text{O}$
Display:
$\text{H}_2\text{SO}_4 + \text{Ba(OH)}_2 \rightarrow \text{BaSO}_4 \downarrow + 2\text{H}_2\text{O}$
Acetic Acid Dissociation Equilibrium
Code:
$\text{CH}_3\text{COOH} \rightleftharpoons \text{CH}_3\text{COO}^- + \text{H}^+$
Display:
$\text{CH}_3\text{COOH} \rightleftharpoons \text{CH}_3\text{COO}^- + \text{H}^+$
Electrolysis of Copper Chloride Solution
Code:
Cathode: $\text{Cu}^{2+} + 2e^- \rightarrow \text{Cu}$ Anode: $2\text{Cl}^- \rightarrow \text{Cl}_2 + 2e^-$
Display:
Cathode: $\text{Cu}^{2+} + 2e^- \rightarrow \text{Cu}$ Anode: $2\text{Cl}^- \rightarrow \text{Cl}_2 + 2e^-$
Esterification Reaction
Code:
$\text{CH}_3\text{COOH} + \text{C}_2\text{H}_5\text{OH} \rightleftharpoons \text{CH}_3\text{COOC}_2\text{H}_5 + \text{H}_2\text{O}$
Display:
$\text{CH}_3\text{COOH} + \text{C}_2\text{H}_5\text{OH} \rightleftharpoons \text{CH}_3\text{COOC}_2\text{H}_5 + \text{H}_2\text{O}$
Thermochemical Equation
Code:
$\text{C(graphite)} + \text{O}_2\text{(g)} = \text{CO}_2\text{(g)} + 394\text{ kJ}$
Display:
$\text{C(graphite)} + \text{O}_2\text{(g)} = \text{CO}_2\text{(g)} + 394\text{ kJ}$
Tips for High School Chemistry Notation

• Reversible reaction arrow: \rightleftharpoons → $\rightleftharpoons$
• Precipitate notation: \downarrow → $\downarrow$
• Superscript (ion charges): ^{2+} → $^{2+}$
• Electron: e^- → $e^-$