Set of rational numbers: \(\mathbb{Q}\)

Set of integers: \(\mathbb{Z}\)

Set of positive rational numbers: \(\mathbb{Q^ + }\)

Set of negative rational numbers: \(\mathbb{Q^ – }\)

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: \(n\)

Integers: \(a\), \(b\), \(c\), \(d\)

Natural numbers: \(n\)

- The rational numbers are those that are represented as a common fraction \(\large\frac{a}{b},\) where \(a\) and \(b\) are integers and \(b \ne 0.\)

\(\mathbb{Q} = \Big\{ {x \mid x = \large\frac{a}{b},\;\normalsize a \in \mathbb{Z},\,}\) \({b \in \mathbb{Z},\;b \ne 0} \Big\}\) - The rational numbers include positive rational numbers, negative rational numbers, and zero.

\(\mathbb{Q} = \mathbb{Q^ – } \cup \left\{ 0 \right\} \cup \mathbb{Q^ + }\) - 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: \(\left| a \right| \lt \left| b \right|\). - 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: \(\left| a \right| \ge \left| b \right|\). - 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: \(\left| a \right| + \left| b \right|\). - Reciprocal of a fraction

\({\large\frac{1}{{a/b}} = \frac{b}{a}\;\normalsize}\) \( \left( {a \ne 0,\;b \ne 0} \right)\) - Any integer can be written as a rational number: \(a = \large\frac{a}{1}\)
- \(\large\frac{0}{a}=\normalsize 0\)
- Equality of rational numbers

\(\large\frac{a}{b} = \frac{c}{d}\) if and only if \(ad = bc\) (a property of proportion). - Equivalent fractions \({\large\frac{a}{b} = \frac{{na}}{{nb}}\;\;\normalsize} \left( {n \ne 0} \right)\)
- Reducing fractions \(\large\frac{{na}}{{nb}} = \frac{a}{b}\)
- Ordering rational numbers

\(\large\frac{a}{b} \gt \frac{c}{d}\) if and only if \(ad \gt bc\). - \(\large\frac{a}{b} \gt \frac{c}{b}\), if \(a \gt c\) (\(a \gt 0\), \(b \gt 0\), \(c \gt 0\text{).}\)
- \(\large\frac{a}{b} \lt \frac{a}{c}\), if \(b \gt c\) (\(a \gt 0\), \(b \gt 0\), \(c \gt 0\)).
- Addition of rational numbers \(\large\frac{a}{b} + \frac{c}{d} = \frac{{ad + bc}}{{bd}}\)
- Subtraction of rational numbers \(\large\frac{a}{b} – \frac{c}{d} = \frac{{ad – bc}}{{bd}}\)
- Multiplication of rational numbers \(\large\frac{a}{b} \cdot \frac{c}{d} = \frac{{ac}}{{bd}}\)
- Multiplication of an integer by a rational number

\(a \cdot \large\frac{b}{c} = \frac{a}{1} \cdot \frac{b}{c} =\) \( \frac{{ab}}{c}\) - Division of rational numbers

\({\large\frac{a}{b}:\frac{c}{d}\normalsize} = {\large\frac{a}{b} \cdot \frac{d}{c}\normalsize} =\) \( {\large\frac{{ad}}{{bc}}\;\;\normalsize} \left( {c \ne 0} \right)\) - Division of an integer by a rational number

\(a:{\large\frac{b}{c}\normalsize} = {\large\frac{a}{1}:\frac{b}{c}\normalsize} =\) \( a \cdot {\large\frac{c}{b}\normalsize} =\) \( {\large\frac{{ac}}{b}\;\;\normalsize} \left( {b \ne 0} \right)\) - Division of a rational number by an integer

\({\large\frac{a}{b}\normalsize} : c = {\large\frac{a}{b}:\frac{c}{1}\normalsize} =\) \({\large\frac{a}{b} \cdot \frac{1}{c}\normalsize} =\) \( {\large\frac{a}{{bc}}\;\;\normalsize} \left( {c \ne 0} \right)\) - Exponentiation of a rational number to a natural power

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