LaTeX Fractions — \frac, \dfrac, and \tfrac

The core command is \frac{numerator}{denominator}. You wrap it in math mode — dollar signs for inline math ($\frac{a}{b}$) or display math for a centered, standalone equation (\[\frac{a}{b}\]).

LaTeX automatically picks a style: in inline math the fraction is compact so it does not stretch the line height, while in display math it renders at full size. When you need to override that behavior, \dfrac(display style) and \tfrac (text style) force the size explicitly. Both come from the amsmath package.

Fractions nest naturally. You can place a fraction in the numerator or denominator of another fraction, and LaTeX shrinks each level so the expression stays readable. For continued fractions where you want every level at full size, use \cfrac instead of \frac.

The single most confusing thing about fractions is why the same \frac{1}{2}sometimes renders small and sometimes large. The answer is math style, not the command. Inside $...$LaTeX enters text style: the fraction bar sits at the current line height and the numerator and denominator shrink to about 70% size. This keeps a sentence like "the probability is $\frac{1}{2}$" from blowing up the line spacing.

Inside display math (\[...\], or the \begin{equation} and\begin{align} environments) LaTeX uses display style: the fraction renders at full size with the bar centered and generous spacing. You can think of \frac as "the right size for where I am,"\dfrac as "always full size," and \tfrac as "always compact." When you see an unexpectedly small fraction, you are almost always looking at a \frac that LaTeX decided should be text-style.

\dfrac{a}{b} forces the display fraction everywhere — inside inline math, inside exponents, inside matrices. This is what you reach for when a textbook-style fraction is required mid-sentence. Conversely \tfrac{a}{b} forces the compact form even inside display math, which is useful when a fraction appears in a limit, a table cell, or a heading where full display size looks top-heavy.

If you prefer not to change the command, wrap a group in a style declaration:\displaystyle \frac{a}{b} makes every math command after it display-style, and\textstyle \frac{a}{b} forces text style. Because these apply to an entire group, they are handy when a fraction sits inside an \int limit or a \sum where you want the surrounding operator to stay display-sized but the fraction compact.

You will meet fractions in nearly every branch of math. In calculus, derivatives are written as fractions: the chain rule \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}chains three fractions together, and second derivatives use \frac{d^2 y}{dx^2}. In combinatorics the binomial \binom{n}{k} counts combinations, while the probability\frac{n!}{k!(n-k)!} answers "how many ways can this happen."

In statistics an average is a sum over a count: \frac{1}{n}\sum_{i=1}^n x_i. In physics, rates and densities are fractions of differentials, such as velocity\frac{dx}{dt} or the electric field \frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}. Fractions also appear inside matrices: a transition matrix often stores probabilities like\frac{1}{2} in each cell. Whatever the field, the same two-argument\frac command — with \dfrac whenever you need it full-size — is all you need.