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:
- ^{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:
- ^{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;
More on ordinary (common) fractions / theory:
(9) Fractions as rational numbers
Fractions operations that can be run automatically, with explanations:
Mathematical operations with fractions, plus the theory behind: addition, subtraction, multiplication, division, fractions reducing (simplifying), comparing, sorting in ascending order, converting numbers to fractions and percentages.