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