Fractions, theory: rational numbers, sorting fractions by bringing them to the same denominator or common numerator

Fractions and rational numbers Q

Link between fractions and rational numbers Q

All these fractions: ^{3}/_{4}; ^{6}/_{8}; ...; ^{27}/_{36}; ... that are set by reducing (or by expanding) represent the same amount, a unique rational number: ^{3}/_{4}

So ^{3}/_{4} has a double meaning: as a fraction and as a rational number.

Fractions with denominator 1 and those set up by expanding (multiplying both their numerators and denominators by the same non-zero number) are also contained in the set of rational numbers, for example, ^{3}/_{1} = ^{6}/_{2} = ... = ^{18}/_{6} = ... They may be substituted for each other.

The integer 0 may be replaced by an infinite number of counter fractions with numerator 0.

The denominator 0 is not allowed in any fraction.

Rational numbers sorting by their value:

As in the case of natural or whole numbers, we have for rational numbers: r1 < r2, r1 = r2 or r1 > r2.

To sort fractions by their value we bring them to the same denominator and so the numerators will give the order of magnitude;

If two positive fractions have the same numerator, the fraction with the smaller denominator is the larger fraction.

If two negative fractions have the same numerator, the fraction with the smaller denominator is the smaller fraction.

A rational number does not have a unique successor nor a unique predecessor.

In the set of rational numbers there is not the lowest nor there is the highest number. A rational number does not have a unique successor nor a unique predecessor.

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