# Fractions, theory: rational numbers

## Fractions and rational numbers Q

### 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:
• 31 = 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:
• 0/1 = 0/2 = 0/3 = ... 0/125 = ...
• The denominator 0 is excluded. There cannot be a fraction such us:
• 0/0 or 9/0 or 200/0...

### A rational number has no predecessor and no unique successor.

• Between two rational numbers r1 and r2 there is an infinite number of rational numbers r:
• r1 < r < r2 or r1 > r > r2;