# Formulas and Tables

Number Sets# Real Numbers

Set of positive real numbers: \(\mathbb{R^ + }\)

Set of negative real numbers: \(\mathbb{R^ – }\)

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

Real numbers: \(a\), \(b\), \(c\), \(d\), \(\xi\)

Natural number: \(n\)

- The real numbers consist of positive real numbers, negative real numbers and zero.

\(\mathbb{R} = \mathbb{R^ – } \cup \left\{ 0 \right\} \cup \mathbb{R^ + }\) - Real numbers include the rational numbers and the irrational numbers.

\(\mathbb{R} = \mathbb{Q} \cup \mathbb{I}\) - Examples of irrational numbers

\(\pi = 3.141592653 \ldots ,\) \(e = 2.718281828 \ldots ,\) \(\sqrt 2 = 1.414213562 \ldots, \) \(\ln 3 = 1.098612289 \ldots \) - Ordering property

For any pair of real numbers \(a\) and \(b,\) one and only one of the following relations is true: \(a = b,\;a \gt b,\;a \lt b\) - Transitivity property

If \(a \le b\) and \(b \le c\), then \(a \le c\) - If \(a \le b\), then \(a + c \le b + c\)
- If \(a \gt 0\) and \(b \gt 0\), then \(ab \gt 0\)
- Commutativity of addition \(a + b = b + a\)
- Associativity of addition \(a + (b + c) =\) \( (a + b) + c\)
- Existence of an additive identity element \(a + 0 = a\)
- Existence of an opposite element

For every real number \(a\) there exists an opposite number \(-a\) such that \(a + (-a) = 0\) - Commutativity of multiplication \(a \cdot b = b \cdot a\)
- Associativity of multiplication \(a \cdot \left( {b \cdot c} \right) =\) \( \left( {a \cdot b} \right) \cdot c\)
- Distributivity of multiplication over addition \(a \cdot \left( {b + c} \right) =\) \( a \cdot b + a \cdot c\)
- Existence of a multiplicative identity element \(a \cdot 1 = a\)
- \(a \cdot 0 = 0\)
- 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 \(a \cdot {a^{ – 1}} = 1\) - 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 \({\large\frac{1}{n}\normalsize} \lt a.\) - 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 \(b \in B.\)