# Fraction

## Fraction

Relationship between a part and the whole.

The whole is divided into a certain number of equivalent parts.
The numerator indicates the number of equivalent parts considered.
The denominator indicates how many equivalent parts the whole was divided into.

A fraction empirically represents a part of a whole expressed in the form of a ratio of two positive integers a and b. This fraction is represented by the symbol $$\dfrac{a}{b}$$, called fractional notation.

In general, a fraction is the representation of a rational number in the form $$\dfrac{a}{b}$$ where a and b are integers and b is not zero.

If a > b, the expression $$\dfrac{a}{b}$$ is called an improper fraction.

A fraction is not a number properly speaking, but rather a relationship between two numbers, just like a fractional expression can express a rational number in the form of a ratio between two integers.

### Examples

• In the fraction $$\dfrac{2}{3}$$, the number 2 is the numerator and the number 3 is the denominator.
• The expression $$\dfrac{11}{5}$$ is a fractional expression of the rational number 2.2 that we can also call an improper fraction.
• The expression $$5 \dfrac{1}{3}$$ is a mixed number, which is an expression formed by an integer and a fraction. This expression is equivalent to the fractional expression $$\dfrac{16}{3}$$.

### Historical Note

The first person to use a horizontal fraction bar to write a fraction (like $$\frac{3}{4}$$) was Leonardo of Pisa (1175-1250), better known as Fibonacci. He published a text in 1202 where he used the Hindu-Arabic numerical symbols. He was the first to do this. It seems that he was influenced by the Arabic mathematician Al-Hassar who lived in the 12th century.

The first time that the oblique line was used to write a fraction was by the Spanish mathematician Manuel Antonio Valdes around 1748 in his book Gazetasde Mexico. In fact, it was the Spanish mathematician Antonio y Oliveres who first used a straight oblique line (like 3/4). This made it possible to write a fraction on one line instead of three lines.