# 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.

### 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 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

### 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.