Math Modes, Fractions, and Symbols

Mathematics is the reason most people learn LaTeX. This chapter covers everything you need for undergraduate homework and exams: inline and display math, fractions, roots, Greek letters, operators, delimiters, and the spacing rules that separate amateur typesetting from professional.

LaTeX math mode is the boundary between text and mathematics: inline math flows with your sentence, display math gets its own line and proper spacing. This lesson covers both modes plus the fractions, roots, scripts, and Greek letters that handle most undergraduate work.

Math Modes

LaTeX has two math modes:

Inline Math

Inline math appears within a paragraph. Surround it with dollar signs or <!--M5-->:

The equation $E = mc^2$
relates energy and mass.

Alternatively: <!--M6-->.

The equation \( E = mc^2 \) relates energy and mass.

Alternatively: \( E = mc^2 \).

Tip: Both $...$ and <!--M9--> produce identical output. The <!--M10--> form is recommended by the LaTeX project because it’s easier to match paired delimiters. In practice, most people use dollar signs because they’re faster to type. Pick one and be consistent.

Display Math

Display math is centered on its own line. Use <!--M86--> or the equation environment:

The quadratic formula:
<!--M87-->

Numbered version:
\begin{equation}
  x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
\end{equation}

The quadratic formula:

$$ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} $$

The numbered version produces the same formula with an equation number in the right margin.

Warning: Never use <!--M1--> for display math. It’s plain TeX syntax that produces inconsistent spacing and breaks several LaTeX features. Always use <!--M88--> or equation.

Superscripts and Subscripts

$x^2$, $x_i$, $x^2_i$

$x^{n+1}$, $a_{ij}$

$x^{2^n}$ (nested)

${}_nC_r = \binom{n}{r}$

\( x^2 \), \( x_i \), \( x^2_i \) \( x^{n+1} \), \( a_{ij} \) \( x^{2^n} \) (nested) \( {}_nC_r = \binom{n}{r} \)

Warning: For multi-character superscripts or subscripts, always use braces: x^{n+1} not x^n+1. Without braces, only the first character is raised: \( x^n+1 \) vs. \( x^{n+1} \).

Fractions

LaTeX provides three fraction commands:

% Standard (auto-sizing)
$\frac{a}{b}$ inline,
<!--M89--> display

% Display-style (always large)
$\dfrac{a}{b}$ forces
full size inline

% Text-style (always small)
<!--M90--> forces
small size in display

\( \frac{a}{b} \) inline, $$ \frac{a}{b} $$

display

\( \dfrac{a}{b} \) forces full size inline $$ \tfrac{a}{b} $$

forces small size in display

Tip: Use \dfrac when an inline fraction is too small to read (common with nested fractions). Use \tfrac in display math when you want a compact fraction that doesn’t dominate the line.

For continued fractions, use \cfrac:

<!--M91-->

$$ \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cdots}}} $$

Roots

$\sqrt{2}$

$\sqrt{x^2 + y^2}$

$\sqrt[3]{27} = 3$

$\sqrt[n]{a}$

\( \sqrt{2} \) \( \sqrt{x^2 + y^2} \) \( \sqrt[3]{27} = 3 \) \( \sqrt[n]{a} \)

Greek Letters

Greek letters are commands preceded by a backslash:

• Lower: \alpha | Output: \( \alpha \) | Upper: \Alpha | Output: \( A \)

• Lower: \beta | Output: \( \beta \) | Upper: \Beta | Output: \( B \)

• Lower: \gamma | Output: \( \gamma \) | Upper: \Gamma | Output: \( \Gamma \)

• Lower: \delta | Output: \( \delta \) | Upper: \Delta | Output: \( \Delta \)

• Lower: \epsilon | Output: \( \epsilon \) | Upper: \Epsilon | Output: \( E \)

• Lower: \zeta | Output: \( \zeta \) | Upper: \Zeta | Output: \( Z \)

• Lower: \eta | Output: \( \eta \) | Upper: \Eta | Output: \( H \)

• Lower: \theta | Output: \( \theta \) | Upper: \Theta | Output: \( \Theta \)

• Lower: \iota | Output: \( \iota \) | Upper: \Iota | Output: \( I \)

• Lower: \kappa | Output: \( \kappa \) | Upper: \Kappa | Output: \( K \)

• Lower: \lambda | Output: \( \lambda \) | Upper: \Lambda | Output: \( \Lambda \)

• Lower: \mu | Output: \( \mu \) | Upper: \Mu | Output: \( M \)

• Lower: \nu | Output: \( \nu \) | Upper: \Nu | Output: \( N \)

• Lower: \xi | Output: \( \xi \) | Upper: \Xi | Output: \( \Xi \)

• Lower: \pi | Output: \( \pi \) | Upper: \Pi | Output: \( \Pi \)

• Lower: \rho | Output: \( \rho \) | Upper: \Rho | Output: \( P \)

• Lower: \sigma | Output: \( \sigma \) | Upper: \Sigma | Output: \( \Sigma \)

• Lower: \tau | Output: \( \tau \) | Upper: \Tau | Output: \( T \)

• Lower: \phi | Output: \( \phi \) | Upper: \Phi | Output: \( \Phi \)

• Lower: \chi | Output: \( \chi \) | Upper: \Chi | Output: \( X \)

• Lower: \psi | Output: \( \psi \) | Upper: \Psi | Output: \( \Psi \)

• Lower: \omega | Output: \( \omega \) | Upper: \Omega | Output: \( \Omega \)
  Several letters have variant forms:

• Standard: \epsilon | Output: \( \epsilon \) | Variant: \varepsilon | Output: \( \varepsilon \)

• Standard: \theta | Output: \( \theta \) | Variant: \vartheta | Output: \( \vartheta \)

• Standard: \pi | Output: \( \pi \) | Variant: \varpi | Output: \( \varpi \)

• Standard: \rho | Output: \( \rho \) | Variant: \varrho | Output: \( \varrho \)

• Standard: \sigma | Output: \( \sigma \) | Variant: \varsigma | Output: \( \varsigma \)

• Standard: \phi | Output: \( \phi \) | Variant: \varphi | Output: \( \varphi \)

  Tip: Most mathematicians and physicists prefer \varepsilon (\( \varepsilon \)) over \epsilon (\( \epsilon \)), and \varphi (\( \varphi \)) over \phi (\( \phi \)). Check your field’s conventions.

Once these basics feel natural, operators and delimiters completes the everyday toolkit, and multi-line equations and matrices handles the bigger structures.

Quick answers to common questions: