Welcome back to Writing Math in LaTeX! In the first two lessons of this course, you learned how to enter math mode and how to write subscripts and superscripts. With those tools in hand, you are already capable of expressing a wide range of notation. Now, in this third lesson — the halfway point of the course — we turn to one of the most recognizable structures in all of mathematics: fractions. By the end of this lesson, you will know how to typeset fractions with the \frac command, nest fractions within fractions, mix fractions into larger expressions, and understand how their appearance changes between inline and display modes.

Fractions appear everywhere in mathematics and science. From the slope formula $\frac{\Delta y}{\Delta x}$ in algebra to dosage ratios in medicine, the familiar "one thing over another" structure is one of the most common patterns in technical writing. In plain text, we often resort to a forward slash, writing something like 1/2 or (a+b)/(c+d). This works for quick notes, but it becomes hard to read as expressions grow more complex.

LaTeX solves this with a clean, stacked layout where the numerator sits above a horizontal bar and the denominator sits below it. This is the polished format you see in textbooks and published papers, and it is exactly what the \frac command produces.

The \frac command takes two required arguments enclosed in curly braces: the numerator first, then the denominator.

As you know from working with subscripts and superscripts, curly braces group content so that LaTeX treats it as a single unit. Here, the first pair of braces holds whatever goes on top of the fraction bar, and the second pair holds whatever goes below it. Let's see a simple example in display math mode:

This produces:

The numerator a appears above the fraction bar and the denominator b appears below it. We can place any valid math expression inside those braces — for instance, \frac{x + 1}{2} renders as:

Fractions behave quite differently depending on which math mode you use, and it is important to know what to expect. In display mode, fractions appear full-sized with generous spacing:

In inline mode, the same code produces a noticeably smaller result. Writing $\frac{1}{2}$ gives us $\frac{1}{2}$, where LaTeX compresses the numerator and denominator so the fraction does not push the surrounding text lines apart.

Because each argument of \frac can contain any math expression, we can place one fraction inside another. This is called nesting. For example, suppose we want to write "one divided by one plus one over x." We simply place a \frac inside the denominator of an outer \frac:

This renders as:

Fractions rarely appear in isolation. In real formulas, they are mixed with other terms, variables, and operators. You can freely combine \frac with addition, subtraction, subscripts, superscripts, and anything else you have learned so far. Consider this expression:

This gives us:

As with subscripts and superscripts, a few small errors can cause big headaches when working with fractions. Here are the most frequent pitfalls:

• Missing braces: Writing \frac x y instead of \frac{x}{y}. Without braces, LaTeX may grab only the first character for each argument, producing unexpected output.
• Unbalanced braces in nested fractions: Every opening { must have a matching }. Count your braces carefully when you have two or more levels of nesting.
• Overusing inline fractions: A complex fraction like $\frac{x^2 + 1}{2x - 3}$ can feel cramped inside a paragraph. When a fraction holds anything beyond a simple ratio, consider switching to display mode instead.

In this lesson, we explored the \frac command and its two required arguments for building fractions in LaTeX. We compared how fractions render in inline versus display mode, practiced nesting one fraction inside another, and saw how fractions combine naturally with subscripts, superscripts, and other terms in larger expressions. Careful brace management is the key skill here, especially as nesting depth increases.

Now it is time to put these ideas into practice. Head over to the exercises, where you will build fractions from the ground up — starting with simple ratios, comparing inline and display rendering, and working your way to nested, compound expressions. In the next lesson, we will move on to square roots, adding yet another essential piece to your math typesetting toolkit.