LaTeX is renowned for its superior mathematical typesetting. This guide will take you from basic equations to advanced mathematical expressions.
Fun fact : LaTeX’s math rendering is so good that even Microsoft Word now uses a LaTeX-like syntax for its equation editor!
Why LaTeX for Math? Compare these approaches to writing the quadratic formula:
Plain text : x = (-b +/- sqrt(b^2 - 4ac)) / 2aLaTeX result : x = − b ± b 2 − 4 a c 2 a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} x = 2 a − b ± b 2 − 4 a c The difference is clear – LaTeX produces publication-quality mathematics.
Math Modes LaTeX has two math modes:
1. Inline Math Mode For math within text, use $...$ or \(...\):
The famous equation $ E = mc^ 2 $ was discovered by Einstein. We can also write \( a^ 2 + b^ 2 = c^ 2 \) for the Pythagorean theorem.
Rendered Output The famous equation E = m c 2 E = mc^2 E = m c 2 a 2 + b 2 = c 2 a^2 + b^2 = c^2 a 2 + b 2 = c 2 2. Display Math Mode For centered equations on their own line, use \[...\] or equation environment:
The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^ 2 - 4 ac}}{ 2 a} \] For numbered equations, use: \begin { equation } \int _ 0 ^ \infty e^{-x^ 2 } dx = \frac{\sqrt{ \pi }}{ 2 } \end { equation }
Rendered Output The quadratic formula is: x = − b ± b 2 − 4 a c 2 a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} x = 2 a − b ± b 2 − 4 a c For numbered equations: \int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2} \tag{1} Use \[...\] for important formulas you want to highlight. Use $...$ for variables and simple expressions within sentences.
Basic Math Elements Superscripts and Subscripts % Superscripts with ^ $ x^ 2 $ , $ x^{ 10 } $ , $ x^{n+ 1 } $ % Subscripts with _ $ x_ 1 $ , $ x_{ 10 } $ , $ x_{i,j} $ % Combined $ x_ 1 ^ 2 $ , $ a_n^{k+ 1 } $ % Chemical formulas $ \text{H}_ 2 \text{O} $ , $ \text{CO}_ 2 $
Rendered Output Superscripts: x 2 x^2 x 2 x 10 x^{10} x 10 x n + 1 x^{n+1} x n + 1 Subscripts: x 1 x_1 x 1 x 10 x_{10} x 10 x i , j x_{i,j} x i , j Combined: x 1 2 x_1^2 x 1 2 a n k + 1 a_n^{k+1} a n k + 1 Chemical formulas: H 2 O \text{H}_2\text{O} H 2 O CO 2 \text{CO}_2 CO 2 Fractions % Simple fractions $ \frac{ 1 }{ 2 } $ , $ \frac{a}{b} $ % Nested fractions $ \frac{ 1 }{ 1 + \frac{ 1 }{ 2 }} $ % Display style in inline math $ \displaystyle\frac{a+b}{c+d} $ % Alternative notation $ a/b $ or $ ^a/_b $
Rendered Output Simple fractions: 1 2 \frac{1}{2} 2 1 a b \frac{a}{b} b a Nested fractions: 1 1 + 1 2 \frac{1}{1 + \frac{1}{2}} 1 + 2 1 1 Display style: a + b c + d \displaystyle\frac{a+b}{c+d} c + d a + b Alternative notation: a / b a/b a / b a / b ^a/_b a / b Roots % Square root $ \sqrt{ 2 } $ , $ \sqrt{x^ 2 + y^ 2 } $ % nth root $ \sqrt[ 3 ]{ 8 } $ , $ \sqrt[n]{x} $ % Nested roots $ \sqrt{ 2 + \sqrt{ 3 }} $
Rendered Output Square root: 2 \sqrt{2} 2 x 2 + y 2 \sqrt{x^2 + y^2} x 2 + y 2 nth root: 8 3 \sqrt[3]{8} 3 8 x n \sqrt[n]{x} n x Nested roots: 2 + 3 \sqrt{2 + \sqrt{3}} 2 + 3 Common Math Symbols Greek Letters % Lowercase $ \alpha , \beta , \gamma , \delta , \epsilon $ $ \theta , \lambda , \mu , \pi , \sigma , \phi $ % Uppercase $ \Gamma , \Delta , \Theta , \Lambda , \Sigma , \Phi $ % Variants $ \epsilon $ vs $ \varepsilon $ $ \phi $ vs $ \varphi $
Rendered Output Lowercase: α , β , γ , δ , ϵ \alpha, \beta, \gamma, \delta, \epsilon α , β , γ , δ , ϵ θ , λ , μ , π , σ , ϕ \theta, \lambda, \mu, \pi, \sigma, \phi θ , λ , μ , π , σ , ϕ Uppercase: Γ , Δ , Θ , Λ , Σ , Φ \Gamma, \Delta, \Theta, \Lambda, \Sigma, \Phi Γ , Δ , Θ , Λ , Σ , Φ Variants: ϵ \epsilon ϵ ε \varepsilon ε ϕ \phi ϕ φ \varphi φ Operators and Relations % Basic operators $ a + b - c \times d \div e $ % Comparison $ a < b \leq c = d \geq e > f $ $ a \neq b \approx c \equiv d $ % Set operations $ A \cup B \cap C \subset D $ $ x \in A, y \notin B $ % Logic $ p \land q \lor r \implies s $ $ \forall x \exists y $
Rendered Output Basic operators: a + b − c × d ÷ e a + b - c \times d \div e a + b − c × d ÷ e Comparison: a < b ≤ c = d ≥ e > f a < b \leq c = d \geq e > f a < b ≤ c = d ≥ e > f a ≠ b ≈ c ≡ d a \neq b \approx c \equiv d a = b ≈ c ≡ d Set operations: A ∪ B ∩ C ⊂ D A \cup B \cap C \subset D A ∪ B ∩ C ⊂ D x ∈ A , y ∉ B x \in A, y \notin B x ∈ A , y ∈ / B Logic: p ∧ q ∨ r ⟹ s p \land q \lor r \implies s p ∧ q ∨ r ⟹ s ∀ x ∃ y \forall x \exists y ∀ x ∃ y Arrows % Basic arrows $ \rightarrow , \leftarrow , \leftrightarrow $ $ \Rightarrow , \Leftarrow , \Leftrightarrow $ % Long arrows $ \longrightarrow , \longleftarrow $ % Special arrows $ \uparrow , \downarrow , \updownarrow $ $ \nearrow , \searrow , \swarrow , \nwarrow $
Rendered Output Basic arrows: → , ← , ↔ \rightarrow, \leftarrow, \leftrightarrow → , ← , ↔ ⇒ , ⇐ , ⇔ \Rightarrow, \Leftarrow, \Leftrightarrow ⇒ , ⇐ , ⇔ Long arrows: ⟶ , ⟵ \longrightarrow, \longleftarrow ⟶ , ⟵ Special arrows: ↑ , ↓ , ↕ \uparrow, \downarrow, \updownarrow ↑ , ↓ , ↕ ↗ , ↘ , ↙ , ↖ \nearrow, \searrow, \swarrow, \nwarrow ↗ , ↘ , ↙ , ↖ Functions and Operators Standard Functions % Trigonometric $ \sin \theta , \cos \theta , \tan \theta $ % Logarithms $ \log x, \ln x, \log_ 2 x $ % Limits $ \lim_{x \to 0 } \frac{\sin x}{x} = 1 $ % Min/Max $ \min(a,b), \max(a,b) $
Rendered Output Trigonometric: sin θ , cos θ , tan θ \sin\theta, \cos\theta, \tan\theta sin θ , cos θ , tan θ Logarithms: log x , ln x , log 2 x \log x, \ln x, \log_2 x log x , ln x , log 2 x Limits: lim x → 0 sin x x = 1 \displaystyle\lim_{x \to 0} \frac{\sin x}{x} = 1 x → 0 lim x sin x = 1 Min/Max: min ( a , b ) , max ( a , b ) \min(a,b), \max(a,b) min ( a , b ) , max ( a , b ) Sums and Products % Summation $ \sum _{i= 1 }^{n} i = \frac{n(n+ 1 )}{ 2 } $ % Product $ \prod _{i= 1 }^{n} i = n! $ % Multiple lines $ \sum _{\substack{i= 1 \\ i \neq j}}^{n} a_i $
Rendered Output Summation: ∑ i = 1 n i = n ( n + 1 ) 2 \displaystyle\sum_{i=1}^{n} i = \frac{n(n+1)}{2} i = 1 ∑ n i = 2 n ( n + 1 ) Product: ∏ i = 1 n i = n ! \displaystyle\prod_{i=1}^{n} i = n! i = 1 ∏ n i = n ! Multiple lines: ∑ i = 1 i ≠ j n a i \displaystyle\sum_{\substack{i=1 \\ i \neq j}}^{n} a_i i = 1 i = j ∑ n a i Integrals and Derivatives % Derivatives $ f'(x), f''(x), f^{(n)}(x) $ $ \frac{df}{dx}, \frac{d^ 2 f}{dx^ 2 } $ $ \frac{ \partial f}{ \partial x} $ % Integrals $ \int f(x) \, dx $ $ \int _a^b f(x) \, dx $ $ \iint_D f(x,y) \, dx \, dy $ % Special notation $ \oint _C F \cdot dr $
Rendered Output Derivatives: f ′ ( x ) , f ′ ′ ( x ) , f ( n ) ( x ) f'(x), f''(x), f^{(n)}(x) f ′ ( x ) , f ′′ ( x ) , f ( n ) ( x ) d f d x , d 2 f d x 2 \frac{df}{dx}, \frac{d^2f}{dx^2} d x df , d x 2 d 2 f ∂ f ∂ x \frac{\partial f}{\partial x} ∂ x ∂ f Integrals: ∫ f ( x ) d x \displaystyle\int f(x)\,dx ∫ f ( x ) d x ∫ a b f ( x ) d x \displaystyle\int_a^b f(x)\,dx ∫ a b f ( x ) d x ∬ D f ( x , y ) d x d y \displaystyle\iint_D f(x,y)\,dx\,dy ∬ D f ( x , y ) d x d y Special notation: ∮ C F ⋅ d r \displaystyle\oint_C F \cdot dr ∮ C F ⋅ d r Matrices and Arrays Basic Matrices % Using pmatrix (parentheses) $ \begin { pmatrix } a & b \\ c & d \end { pmatrix } $ % Using bmatrix (brackets) $ \begin { bmatrix } 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end { bmatrix } $ % Using vmatrix (determinant) $ \begin { vmatrix } a & b \\ c & d \end { vmatrix } = ad - bc $
Rendered Output pmatrix (parentheses): ( a b c d ) \begin{pmatrix} a & b \\ c & d \end{pmatrix} ( a c b d ) bmatrix (brackets): [ 1 2 3 4 5 6 7 8 9 ] \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} 1 4 7 2 5 8 3 6 9 vmatrix (determinant): ∣ a b c d ∣ = a d − b c \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc a c b d = a d − b c Advanced Arrays % Custom arrays $ \left[ \begin { array }{cc|c} 1 & 2 & 3 \\ 4 & 5 & 6 \end { array } \right] $ % Cases (piecewise functions) $ f(x) = \begin { cases } x^ 2 & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end { cases } $
Rendered Output Custom arrays: [ 1 2 3 4 5 6 ] \left[ \begin{array}{cc|c} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array} \right] [ 1 4 2 5 3 6 ] Cases (piecewise functions): f ( x ) = { x 2 if x ≥ 0 − x if x < 0 f(x) = \begin{cases} x^2 & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} f ( x ) = { x 2 − x if x ≥ 0 if x < 0 Spacing in Math Mode % Default spacing $ a b $ vs $ ab $ % Manual spacing $ a \, b $ % thin space $ a \: b $ % medium space $ a \; b $ % thick space $ a\quad b $ % quad space $ a\qquad b $ % double quad % Negative space $ a \! b $ % negative thin space
Rendered Output Default: a b a b ab a b ab ab Thin space: a b a\,b a b Medium space: a b a\:b a b Thick space: a b a\;b a b Quad space: a b a\quad b a b Double quad: a b a\qquad b a b Negative space: a b a\!b a b Use \, before differentials in integrals: \int f(x)\,dx looks better than \int f(x)dx.
Advanced Features Theorem Environments \documentclass { article } \usepackage { amsthm } \newtheorem {theorem}{Theorem} \newtheorem {lemma}{Lemma} \newtheorem {proof}{Proof} \begin { document } \begin { theorem }[Pythagoras] For a right triangle with legs $ a $ and $ b $ and hypotenuse $ c $ , we have $ a^ 2 + b^ 2 = c^ 2 $ . \end { theorem } \begin { proof } Consider a square with side length $ a + b $ ... \end { proof } \end { document }
Theorem 1 (Pythagoras). For a right triangle with legs
a and b and hypotenuse c, we have a² + b² = c².
Proof. Consider a square with side length a + b…
Aligning Equations \begin { align } 2 x + 3 y & = 7 \\ x - y & = 1 \end { align } % Multi-line derivation \begin { align } (x + y)^ 2 & = (x + y)(x + y) \\ & = x^ 2 + xy + yx + y^ 2 \\ & = x^ 2 + 2 xy + y^ 2 \end { align }
2x + 3y = 7 (1)
(x + y)² = (x + y)(x + y) (3)
Common Mistakes to Avoid 1. Forgetting braces for multi-character super/subscripts
Wrong: $x^10$ → x¹0
Right: $x^{10}$ → x¹⁰
2. Using text in math mode
Wrong: $x = speed * time$
Right: $x = \text{speed} \times \text{time}$
3. Incorrect fraction syntax
Wrong: $\frac{1/2}$
Right: $\frac{1}{2}$
Math Packages Essential packages for advanced mathematics:
\usepackage { amsmath } % Advanced math environments \usepackage { amssymb } % Additional symbols \usepackage { mathtools } % Enhanced amsmath \usepackage { physics } % Physics notation \usepackage { siunitx } % SI units
Practice Exercises Try typesetting these formulas:
Euler’s Identity : e i π + 1 = 0 e^{i\pi} + 1 = 0 e iπ + 1 = 0 Gaussian Integral : ∫ − ∞ ∞ e − x 2 d x = π \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} ∫ − ∞ ∞ e − x 2 d x = π Binomial Theorem : ( x + y ) n = ∑ k = 0 n ( n k ) x n − k y k (x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k ( x + y ) n = ∑ k = 0 n ( k n ) x n − k y k Maxwell’s Equation : ∇ × E = − ∂ B ∂ t \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} ∇ × E = − ∂ t ∂ B
Quick Reference Feature Syntax Example Inline math $...$$x^2$Display math \[...\]\[x^2\]Fraction \frac{num}{den}$\frac{a}{b}$Square root \sqrt{x}$\sqrt{2}$Subscript _$x_1$Superscript ^$x^2$Greek letter \alpha$\alpha$Sum \sum$\sum_{i=1}^n$Integral \int$\int_a^b$
Next Steps
Math Symbols Reference Complete list of mathematical symbols
Advanced Equations Multi-line equations and advanced layouts
Matrices & Arrays Complex matrix operations and layouts
Scientific Notation Physics, chemistry, and scientific formatting
Ready to create beautiful mathematical documents? You now have the foundation to typeset any mathematical expression in LaTeX! 