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 }
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 }
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 Ready to create beautiful mathematical documents? You now have the foundation to typeset any mathematical expression in LaTeX! 