If you’re studying mathematics, you may have come across the term vinculum.

A vinculum is a horizontal line that is placed over an expression to indicate that it is to be considered as a single unit.

The fraction bar and the square root sign are expressions that you might already be familiar with but perhaps weren’t aware a vinculum was used.

For example, with the expression $\sqrt{25}$, the vinculum is the horizontal line that is placed over the number $25$. Or with the expression $\frac{1}{2}$ the vinculum is the horizontal line that separates the numerator and the denominator.

What Is a Vinculum?

A vinculum is a horizontal line used in mathematical notation for various purposes. It is also known as a fraction bar or a grouping symbol.

In mathematics, the vinculum is used to indicate that the expression below the line should be considered a single unit.

Definition

In mathematical notation, the vinculum is placed as an overline or underline over or under a mathematical expression to indicate that the expression is to be considered grouped together. It is most commonly used to denote:

  • A fraction: e.g., $\frac{a}{b}$ means “a over b”.
  • A radical: e.g., $\sqrt{a + b}$ means “the square root of a plus b”.
  • A complex number: e.g., $\overline{a + bi}$ denotes the complex conjugate “a minus bi”.
  • A repeating decimal: e.g., $0.\overline{3}$ is read as “0.3 recurring”.

Examples

  • $\frac{a + b}{c - d}$ — vinculum separates numerator and denominator.
  • $\sqrt{a^2 + b^2}$ — vinculum spans the expression under the radical.
  • $\overline{3 + 4i}$ — vinculum indicates complex conjugate.
  • $0.\overline{6}$ — vinculum over the repeating digit in a decimal.

Vinculum In Fractions

A vinculum is a horizontal line used to group the numerator and denominator in a fraction.

  • Numerator: top part (e.g., in $\frac{3}{4}$, the numerator is 3).
  • Denominator: bottom part (e.g., in $\frac{3}{4}$, the denominator is 4).

Note: A vinculum is not the same as a division symbol, even though both use a horizontal line.

Vinculum in Mathematical Notation

Vincula are supported in various mathematical systems.

Unicode

The overline is represented by U+203E. It is often used in statistical notation (e.g., $\overline{x}$ for the mean).

LaTeX

In LaTeX, use:

  • \overline{} for an overline: e.g., \overline{AB}.
  • \underline{} for an underline.
  • \frac{a}{b} automatically adds a vinculum.

These are useful in representing radicals, fractions, and complex numbers.

Vinculum Summary

A vinculum is a line used to indicate grouping. It is commonly found in:

  • Radicals: $\sqrt{4 + 9}$
  • Repeating decimals: $0.\overline{3}$
  • Fractions: $\frac{a + b}{a - b}$
  • Complex conjugates: $\overline{a + bi}$
  • Line segments: $\overline{AB}$
  • Logic: used for negation or complement operations

Understanding the vinculum helps you read and write mathematical expressions clearly and correctly.