Formulas and Tables

Number Sets

Rational Numbers

Set of rational numbers: \(\mathbb{Q}\)
Set of integers: \(\mathbb{Z}\)
Set of positive rational numbers: \(\mathbb{Q^ + }\)
Set of negative rational numbers: \(\mathbb{Q^ – }\)

Rational numbers: \(x\)
Integers: \(a\), \(b\), \(c\), \(d\)
Natural numbers: \(n\)

  1. The rational numbers are those that are represented as a common fraction \(\large\frac{a}{b},\) where \(a\) and \(b\) are integers and \(b \ne 0.\)
    \(\mathbb{Q} = \Big\{ {x \mid x = \large\frac{a}{b},\;\normalsize a \in \mathbb{Z},\,}\) \({b \in \mathbb{Z},\;b \ne 0} \Big\}\)
  2. The rational numbers include positive rational numbers, negative rational numbers, and zero.
    \(\mathbb{Q} = \mathbb{Q^ – } \cup \left\{ 0 \right\} \cup \mathbb{Q^ + }\)
  3. Proper fraction
    A common fraction a/b is said to be a proper fraction if the absolute value of its numerator is less than the absolute value of the denominator: \(\left| a \right| \lt \left| b \right|\).
  4. Improper fraction
    A common fraction a/b is said to be an improper fraction if the absolute value of its numerator is greater than or equal to the absolute value of the denominator: \(\left| a \right| \ge \left| b \right|\).
  5. Height of a fraction
    The height of a fraction \(\large\frac{a}{b}\) is the sum of the absolute value of the numerator and the absolute value of the denominator: \(\left| a \right| + \left| b \right|\).
  6. Reciprocal of a fraction \({\large\frac{1}{{a/b}} = \frac{b}{a}\;\normalsize}\) \( \left( {a \ne 0,\;b \ne 0} \right)\)
  7. Any integer can be written as a rational number: \(a = \large\frac{a}{1}\)
  8. \(\large\frac{0}{a}=\normalsize 0\)
  9. Equality of rational numbers
    \(\large\frac{a}{b} = \frac{c}{d}\) if and only if \(ad = bc\) (a property of proportion).
  10. Equivalent fractions \({\large\frac{a}{b} = \frac{{na}}{{nb}}\;\;\normalsize} \left( {n \ne 0} \right)\)
  11. Reducing fractions \(\large\frac{{na}}{{nb}} = \frac{a}{b}\)
  12. Ordering rational numbers
    \(\large\frac{a}{b} \gt \frac{c}{d}\) if and only if \(ad \gt bc\).
  13. \(\large\frac{a}{b} \gt \frac{c}{b}\), if \(a \gt c\) (\(a \gt 0\), \(b \gt 0\), \(c \gt 0\text{).}\)
  14. \(\large\frac{a}{b} \lt \frac{a}{c}\), if \(b \gt c\)  (\(a \gt 0\), \(b \gt 0\), \(c \gt 0\)).
  15. Addition of rational numbers \(\large\frac{a}{b} + \frac{c}{d} = \frac{{ad + bc}}{{bd}}\)
  16. Subtraction of rational numbers \(\large\frac{a}{b} – \frac{c}{d} = \frac{{ad – bc}}{{bd}}\)
  17. Multiplication of rational numbers \(\large\frac{a}{b} \cdot \frac{c}{d} = \frac{{ac}}{{bd}}\)
  18. Multiplication of an integer by a rational number
    \(a \cdot \large\frac{b}{c} = \frac{a}{1} \cdot \frac{b}{c} =\) \( \frac{{ab}}{c}\)
  19. Division of rational numbers
    \({\large\frac{a}{b}:\frac{c}{d}\normalsize} = {\large\frac{a}{b} \cdot \frac{d}{c}\normalsize} =\) \( {\large\frac{{ad}}{{bc}}\;\;\normalsize} \left( {c \ne 0} \right)\)
  20. Division of an integer by a rational number
    \(a:{\large\frac{b}{c}\normalsize} = {\large\frac{a}{1}:\frac{b}{c}\normalsize} =\) \( a \cdot {\large\frac{c}{b}\normalsize} =\) \( {\large\frac{{ac}}{b}\;\;\normalsize} \left( {b \ne 0} \right)\)
  21. Division of a rational number by an integer
    \({\large\frac{a}{b}\normalsize} : c = {\large\frac{a}{b}:\frac{c}{1}\normalsize} =\) \({\large\frac{a}{b} \cdot \frac{1}{c}\normalsize} =\) \( {\large\frac{a}{{bc}}\;\;\normalsize} \left( {c \ne 0} \right)\)
  22. Exponentiation of a rational number to a natural power
    \(\large{\left( {\frac{a}{b}} \right)^n} = \frac{{{a^n}}}{{{b^n}}}\)