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 is placed over the denominator, being the number 2 .

## 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: In a fraction, the vinculum separates the numerator and denominator. For example, \frac{a}{b} is read as “a over b”.
- A radical: In a radical, the vinculum is placed over the expression under the radical sign. For example, \sqrt{a+b} is read as “the square root of a plus b”.
- A complex number: In a complex number, the vinculum is placed over the imaginary part of the number. For example, \overline{a+bi} is read as “a minus bi”, where the bar over the expression indicates the complex conjugate.
- A repeating decimal: In a repeating decimal, the vinculum is placed over the repeating digits. For example, 0.\overline{3} is read as “0.3 recurring”.

### Examples

Let’s take a look at some examples of how the vinculum is used in mathematical notation:

- \frac{a+b}{c-d} : In this expression, the vinculum separates the numerator and denominator of the fraction. The expression can be read as “a plus b over c minus d”.
- \sqrt{a^2+b^2} : In this expression, the vinculum is placed over the expression under the radical sign. The expression can be read as “the square root of a squared plus b squared”.
- \overline{3+4i} : In this expression, the vinculum is placed over the imaginary part of the complex number. The expression can be read as “3 minus 4i”.
- 0.\overline{6} : In this expression, the vinculum is placed over the repeating digits of the decimal. The expression can be read as “0.666 recurring”.

Overall, the vinculum is an important symbol in mathematical notation, used to indicate grouping, fractions, radicals, complex numbers, and repeating decimals.

## Vinculum In Fractions

A vinculum is a horizontal line used in mathematical notation to group together expressions. In fractions, the vinculum separates the numerator and denominator.

### Numerator

The numerator is the top part of a fraction, above the vinculum. It represents the number of parts being considered.

For example, in the fraction \frac{3}{4} , the numerator is 3 .

### Denominator

The denominator is the bottom part of a fraction,
**
below the vinculum
**
. It represents the total number of parts in the whole.

For example, in the fraction \frac{3}{4} , the denominator is 4 .

When working with fractions, it is important to understand the role of the vinculum in separating the numerator and denominator. Without the vinculum, the fraction would not be properly defined and could not be used in calculations.

In some countries, the vinculum is also called a “fraction bar”. It is important to note that the vinculum is not the same as a division symbol. While both use a horizontal line, the vinculum groups together expressions, while the division symbol indicates the operation of division.

## Vinculum in Mathematical Notation

In mathematical notation, a vinculum is a horizontal line that is placed over or under a mathematical expression to indicate that the expression is grouped together.

The vinculum is used in several mathematical notations, including Unicode and LaTeX.

### Unicode

In Unicode, the vinculum is represented by the character U+203E, which is called “overline”. The overline character can be used to create a vinculum over a single character or a group of characters. The overline character is often used to represent a bar in statistics, such as the mean or standard deviation.

### LaTeX

In LaTeX, the vinculum is represented by the
```
\overline
```

and
```
\underline
```

commands. The
```
\overline
```

command is used to create a vinculum over a single character or a group of characters, while the
```
\underline
```

command is used to create a vinculum under a single character or a group of characters. The
```
\overline
```

and
```
\underline
```

commands can be used to create mathematical expressions such as radicals, and complex numbers.

To create a fraction in LaTeX where the vinculum is automatically inserted between the numerator and the denominator, the
```
\frac{numerator}{denominator}
```

is used.

Overall, the vinculum is an important tool in mathematical notation that is used to group together expressions and represent repeating decimals. Whether you are using Unicode or LaTeX, the vinculum is a versatile and powerful tool that can help you express complex mathematical ideas.

## Vinculum Summary

A vinculum is a horizontal line used in mathematical notation for various purposes. It is a symbol that is placed as an overline or underline over (or under) a mathematical expression to indicate that the expression is to be considered grouped together.

The vinculum is most commonly used to denote:

- A radical expression: A radical is a mathematical symbol that is used to denote the square root of a number. The vinculum is used to indicate which values are part of the square root. For example, the square root of 4+9 is denoted as \sqrt{4+9} .
- Repeating decimals: The vinculum is used to indicate that a decimal number repeats infinitely. For example, the repeating decimal 0.333... is denoted as 0.\overline{3} .
- A unit: The vinculum is used to indicate that multiple quantities form a unit. For example, \frac{a+b}{a-b} can be written as a+b\overline{a-b} .

The vinculum is also used in other contexts, such as to denote the line segment joining two points (for example line \overline{AB} , to indicate the complex conjugate, or to negate a logical expression.