Fractions: what are they, value and sign of a fraction, fractions types, how do positive and negative fractions compare

Ordinary fractions: what are they?

If we have to divide evenly 6 apples to 3 children, then we perform the operation:

If we have to divide evenly 2 apples to 3 children, then we perform the operation:

2 ÷ 3 = ?
this operation has no solution in the set of natural numbers;
however we will be able to divide the apples with the help of a knife: the quantity of apple for each child will be defined using the fraction ²/₃
all similar cases lead to fractions.

each fraction has the form ^a/_b
"a" is the numerator, written above the fraction bar;
"b" is the denominator, written below the fraction bar; "b" may not be zero;
"b" shows us in how many equal parts "a" has been divided;
the value of the fraction is calculated by dividing the numerator, "a", by the denominator, "b":
"a" ÷ "b"
these fractions, in which both the numerator and the denominator are integers, are called ordinary fractions (also called normal, or simple).

A fraction's numerator and denominator can be positive or negative integers.
Example of fractions with positive numerators and denominators: ⁷/₆, ³/₄, ¹³/₂₀
Example of fractions with negative numerators and denominators: ^-7/_-6, ^-3/_-4, ^-13/_-20
Example of fractions with positive and / or negative numerators and denominators: ^-7/₆, ³/_-4, ^-13/_-20

The signs of the numerator and the denominator are combined in a single sign, according to the rule of the sign, described above, and set in front of the fraction, so the above fractions become:
^-7/_-6 = (-)(-)⁷/₆ = (+)⁷/₆ = ⁷/₆
^-3/_-4 = (-)(-)³/₄ = (+)³/₄ = ³/₄
^-13/_-20 = (-)(-)¹³/₂₀ = (+)¹³/₂₀ = ¹³/₂₀
^-7/₆ = (-)(+)⁷/₆ = (-)⁷/₆ = - ⁷/₆
³/_-4 = (+)(-)³/₄ = (-)³/₄ = - ³/₄
^-13/_-20 = (-)(-)¹³/₂₀ = (+)¹³/₂₀ = ¹³/₂₀

Absolute value of a number = the numerical value of a number without considering its sign. For example, the absolute value of -7 (written as │-7│) is 7. More examples: |-17| = 17; |10| = 10; |-123| = 123;
Proper fractions: ²/₃, ¹/₇, ⁵/₉, - ¹¹/₁₃, ¹⁰/₁₁, ^-15/_-16 - the absolute value of the numerator is less than the absolute value of the denominator, so the absolute value of the fraction is less than 1.
Improper or top-heavy fractions: ⁴/₃, ¹⁶/₃, ⁹/₈, ¹²³/_-13 - the absolute value of the numerator is larger than or equal to the absolute value of the denominator, so the absolute value of the fraction is larger than or equal to 1;
- The improper fractions can be written as mixed numbers (also called mixed fractions):
- ⁴/₃ = ³/₃ + ¹/₃ = 1 + ¹/₃, which is written as: 1 ¹/₃
- ¹⁶/₃ = ¹⁵/₃ + ¹/₃ = 5 + ¹/₃, which is written as: 5 ¹/₃
- ⁹/₈ = ⁸/₈ + ¹/₈ = 1 + ¹/₈, written as: 1 ¹/₈
- ¹²³/_-13 = - ¹²³/₁₃ = - ^{(117 + 6)}/₁₃ = - ¹¹⁷/₁₃ - ⁶/₁₃ = - 9 - ⁶/₁₃, written as: - 9 ⁶/₁₃
- Notice that a mixed fraction consists of an integer number and a proper fraction, both having the same sign.

If the numerator of a fraction is equal to the denominator of another fraction and vice versa, then the fractions are called reciprocal. Ex: ³/₅ and ⁵/₃; ¹⁷/₆ and ⁶/₁₇ - the product of a fraction and its reciprocal is 1, the reciprocal is the multiplicative inverse of a fraction.

If two positive fractions have like denominators then the larger the fraction's numerator the larger the fraction: ²/₇ < ⁶/₇. Why? 7 parts of a larger number (6) is always larger than 7 parts of a smaller number (2);
If two positive fractions have like numerators then the larger the fraction's denominator the smaller the fraction: ⁵/₉ < ⁵/₇. Why? When we divide the same quantity (5) into fewer parts (7), the result is larger than when we divide it into several more parts (9);
If two positive fractions have unlike numerators and denominators:
- any positive proper fraction (less than 1) is smaller than any positive improper one (larger than or equal to 1):
- ³/₇ < 1 < ⁵/₂
- if the fractions are both proper or both improper, first the fractions are built up to the same denominator, then follow the rule: the larger the new numerator, the larger the fraction:
- ⁸/₉ ? ⁵/₇
- ^{(8 × 7)} / _{(9 × 7)} ? ^{(5 × 9)} / _{(7 × 9)}
- ⁵⁶/₆₃ > ⁴⁵/₆₃
- ⁸/₉ > ⁵/₇

If two negative fractions have like denominators then the larger the fraction's numerator the smaller the fraction: - ²/₇ > - ⁶/₇
If two negative fractions have like numerators then the larger the fraction's denominator the larger the fraction: - ⁵/₉ > - ⁵/₇
If two negative fractions have unlike numerators and denominators:
- any negative proper fraction (larger than -1) is larger than any negative improper one (less than or equal to -1):
- - ³/₇ > -1 > - ⁵/₂
- if the fractions are both proper or both improper, first the fractions are built up to the same denominator and the fraction with larger numerator is the smaller one:
- - ⁸/₉ vs. - ⁵/₇
- - ^{(8 × 7)} / _{(9 × 7)} vs. - ^{(5 × 9)} / _{(7 × 9)}
- - ⁵⁶/₆₃ < - ⁴⁵/₆₃
- - ⁸/₉ < - ⁵/₇