The connection between fractions and rational numbers Q

All these fractions: ^{3}/_{4}, ^{6}/_{8}, ^{9}/_{12}, ... ^{27}/_{36}, ... that are set by reducing (or by expanding), are equivalent fractions, ie they represent the same quantity, the unique rational number:

^{3}/_{4} = 3 ÷ 4 = 0.75.

^{3}/_{4} has a double meaning: it represents a fraction and a rational number, that is, it represents all fractions calculated out of ^{3}/_{4} by expanding it, but at the same time it is equal to the rational number 0.75.

The fractions with 1 as a denominator and those calculated by expanding them are also contained in the set of the rational numbers; for example:

^{3}1 = ^{6}/_{2} = ^{9}/_{3} = ... = ^{27}/_{9} = ... They can be substituted for each other, being equivalent.

The integer 0 can be substituted for by an infinite number of fractions having 0 as a numerator: