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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
      \(\large{\left( {\frac{a}{b}} \right)^n} = \frac{{{a^n}}}{{{b^n}}}\)