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.