Ordinary fractions, theory and operations: adding, subtracting, multiplying, dividing, reducing (simplifying), comparing, sorting fractions

What fractions are?

If we evenly divide six apples to 3 children, then we divide 6 by 3 getting 2 each and we know that every child will get two apples. If we have to share 2 apples to 3 children, then partition of 2 by 3 has to be solved.

This operation, 2:3, has no solution in the set of natural numbers. However, we can still divide the apples with the knife. The amount of the apple will be defined by using the fraction 2/3. All similar cases lead to fractions.

Fractions are formed by dividing one or more integers:

Each fraction has the form p/q, where "p" is the numerator, and "q" is the denominator. Numerator, "p", term of the fraction, is written above the fraction line, and denominator, "q", written under the fraction line, shows in how many parts is the whole, "p", being divided.

These fractions in which both the numerator and denominator are integers are called simple fractions or common fractions or vulgar fractions or ordinary fractions.

Fractions numerators and denominators can also be negative, ie: -3/5, -2/-9 , 7/-4; due to sign rule: -3/5 = 3/-5 = -(3/5); -2/-9 = 2/9; 7/-4 = -7/4 = -(7/4)

Ordinary fractions types:

  • Subunitary or proper: 2/3, 1/7, 5/9, -11/13, 10/11, 15/-16; the absolute value (see below Note 1) of the numerator is smaller than the absolute value of the denominator, then the absolute value of the fraction is smaller than 1.
  • Unitary: 5/5, 11/11, -19/19; the absolute value of the numerator is equal to the absolute value of the denominator, then the absolute value of the fraction equals to 1.
  • Supraunitary (above unit), improper or top-heavy: 4/3, 16/3, 9/8, 123/-13; the absolute value of the numerator is greater than the absolute value of the denominator, then the absolute value of the fraction is greater than 1.
    • Improper fractions cand be written as mixed fractions:
    • 4/3 = 3/3 + 1/3 = 1 + 1/3, written as: 11/3
    • 16/3 = 15/3 + 1/3 = 5 + 1/3, written as: 51/3
    • 9/8 = 8/8 + 1/8 = 1 + 1/8, written as: 11/8
    • 123/-13 = -123/13 = - (117 + 6) / 13 = - 117/13 - 6/13 = - 9 - 6/13, written as: -96/13

(Note 1) Absolute value = The numerical value of a number without regard to its sign. For example, the absolute value of -7 (written │-7│) is 7. Also called numerical value. More exemples: |-17| = 17; |10| = 10; |-123| = 123; ...

  • If the numerator of a fraction is equal to the denominator of another fraction and vice versa, then fractions are called inverse or direct opposite:
  • 3/5 and 5/3
  • 17/6 and 6/17

How do fractions compare?

  • If two fractions have the same denominator, the fraction with the greater numerator is greater than the other: 2/7 < 6/7.
  • If two fractions have the same numerator, the fraction with higher denominator is smaller than the other: 5/9 < 5/7
  • If two fractions have both different numerators and denominators, first the fractions are brought to the same denominator, and the fraction with higher numerator is higher than the other: 8/9 vs. 5/7 => (8 * 7) / (9 * 7) vs. (5 * 9) / (7 * 9) => 56/63 > 45/63 => 8/9 > 5/7

More on ordinary math fractions theory:

Fractions operations that can be run automatically, with explanations: