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;
  • 7/12 = (7 * 5) / (12 * 5) = 35/60
  • 11/20 = (11 * 3) / (20 * 3) = 33/60
  • 35/60 > 33/60, so 7/12 > 11/20
  • If two positive fractions have the same denominator, the fraction with the larger numerator is the larger fraction.

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

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

More on ordinary math fractions theory:

Fractions operations that can be run automatically, with explanations: