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.


  • 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.

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