Number Sets

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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 \(b \in B.\)