Real Numbers

• Set of real numbers: $$\mathbb{R}$$, $$A$$, $$B$$
Set of positive real numbers: $$\mathbb{R^ + }$$
Set of negative real numbers: $$\mathbb{R^ – }$$
Set of rational numbers: $$\mathbb{Q}$$
Set of irrational numbers: $$\mathbb{I}$$
Real numbers: $$a$$, $$b$$, $$c$$, $$d$$, $$\xi$$
Natural number: $$n$$
1. The real numbers consist of positive real numbers, negative real numbers and zero.
$$\mathbb{R} = \mathbb{R^ – } \cup \left\{ 0 \right\} \cup \mathbb{R^ + }$$
2. Real numbers include the rational numbers and the irrational numbers.
$$\mathbb{R} = \mathbb{Q} \cup \mathbb{I}$$
3. Examples of irrational numbers
$$\pi = 3.141592653 \ldots ,$$ $$e = 2.718281828 \ldots ,$$ $$\sqrt 2 = 1.414213562 \ldots,$$ $$\ln 3 = 1.098612289 \ldots$$
4. Ordering property
For any pair of real numbers $$a$$ and $$b,$$ one and only one of the following relations is true: $$a = b,\;a \gt b,\;a \lt b$$
5. Transitivity property
If $$a \le b$$ and $$b \le c$$, then $$a \le c$$
6. If $$a \le b$$, then $$a + c \le b + c$$
7. If $$a \gt 0$$ and $$b \gt 0$$, then $$ab \gt 0$$
8. Commutativity of addition $$a + b = b + a$$
9. Associativity of addition $$a + (b + c) =$$ $$(a + b) + c$$
10. Existence of an additive identity element $$a + 0 = a$$
11. Existence of an opposite element
For every real number $$a$$ there exists an opposite number $$-a$$ such that $$a + (-a) = 0$$
12. Commutativity of multiplication $$a \cdot b = b \cdot a$$
13. Associativity of multiplication $$a \cdot \left( {b \cdot c} \right) =$$ $$\left( {a \cdot b} \right) \cdot c$$
14. Distributivity of multiplication over addition $$a \cdot \left( {b + c} \right) =$$ $$a \cdot b + a \cdot c$$
15. Existence of a multiplicative identity element $$a \cdot 1 = a$$
16. $$a \cdot 0 = 0$$
17. Existence of reciprocal
For any real number $$a \ne 0$$ there exists a unique real number called the reciprocal and denoted by $${a^{ – 1}}$$ such that $$a \cdot {a^{ – 1}} = 1$$
18. Archimedean property
For any real number $$a$$ there exists a natural number $$n$$ greater than $$a.$$ Alternatively, for any real number $$a$$ there exists a natural number $$n$$ such that $${\large\frac{1}{n}\normalsize} \lt a.$$
19. Continuity property
Consider two non-empty sets $$A \subset \mathbb{R}$$ and $$B \subset \mathbb{R}$$, and suppose that for any two numbers $$a \in A$$ and $$b \in B$$ the following inequality is satisfied: $$a \le b$$. Then there exists a number $$\xi \in \mathbb{R}$$ such that the relation $$a \le \xi \le b$$ is true for all numbers $$a \in A$$ and $$b \in B.$$