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

Link between fractions and rational numbers Q

All these fractions: 3/4, 6/8, ..., 27/36, ... are set by reducing (simplifying) or by multiplying both the numerator and denominator of the fraction by the same non-zero number, they 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 multiplying both their numerators and denominators by the same non-zero number are also contained in the rational numbers array, for example, 3/1 = 6/2 = ... = 18/6 = ... They may be substituted for each other. The integer 0 may be replaced by a number of counter fractions with numerator 0.

The denominator 0 is not allowed for any fraction.

Rational numbers ordering 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 out by their value we have to bring them to the same denominator and so the numerators will give the order of magnitude; 7/12 = (7 * 5) / (12 * 5) = 35/60 and 11/20 = (11 * 3) / (20 * 3) = 33/60, 35/60 > 33/60, so 7/12 > 11/20.

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

Sometimes it is easier to bring fractions to the same numerator: 3/15 = (3 * 5) / (15 * 5) = 15/75 and 5/28 = (5 * 3) / (28 * 3) = 15/84, 15/75 > 15/84, so 3/15 > 5/28.

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

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

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

More on ordinary math fractions theory:

Fractions operations that can be run automatically, with explanations: