The numbers are classified into sets or number systems and these are: natural numbers, integer numbers, rational numbers, real numbers, and complex numbers. All main categories are given in the following table.
Natural Numbers - the most familiar set of numbers i.e. \(1,2,3,...\). Initially the sequence of the natural numbers started with number 1 since 0 was not considered a number in Ancient Greece. In 19th century mathematicians started including 0 in a set of natural numbers, so today different mathematicians use the term to describe both sets (with and without 0). The symbol for all natural numbers is \(\mathbb{N}\). Sometimes the \(\mathbb{N}_0\) or \(\mathbb{N}_1\) are used to indicate whether the set should start with 0 or 1, respectively.
Integers - is the set of numbers that consist of natural numbers (\(1,2,3,...\)) their additve inverses i.e negative numbers (\(-1,-2,-3,...\)) and zero (0). The set of integer numbers is denoted by letter \(\mathbb{Z}\) which stands for German word "Zahlen" which is translated as "number". The \(\mathbb{Z}\) is a subset of rational numbers \(\mathbb{Q}\), which is a subset of the real numbers \(\mathbb{R}\). The integer numbers like natural numbers are countably infinte.
Rational Numbers - are the numbers that can be expressed as the quotient or fraction of two integers i.e. \(\frac{a}{b}.\) The rational numbers are denoted by a letter \(\mathbb{Q}.\) This type of numbers were denoted in 1895 by Giuseppe Peano after Italian word quoziente - "quotient".
Real Numbers - are the values of a continuos quantity that can represent a disance along a line. The adjective real was introduced by Rene Descartes and includes all the rational numbers, suas as the integer -123, fraction \(-42/34\) and all the irrational numbers \(\sqrt{3} = 1.732...\)
Complex Numbers - is a number that can be expressed in the form \(a+bi\) where \(a\) and \(b\) are real numbers, and \(i\) is a symbol called the imaginary unit $$ i^2 = -1.$$ In a complex number \(a+bi,\) \(a\) is the real part and \(b\) is the imaginary part. As allready noted in Table 1 the complex numbers is denoted by the symbol \(\mathbb{C}.\)
| Name | Symbol | Example |
|---|---|---|
| Natural Numbers | $$\mathbb{N}$$ | |
| Integer Numbers | $$\mathbb{Z}$$ | |
| Rational Numbers | $$\mathbb{Q}$$ | |
| Real Numbers | $$\mathbb{R}$$ | |
| Complex Numbers | $$\mathbb{C}$$ | \(x+yi\) - the \(x\) and \(y\) are real numbers while \(i\) is the square root of \(-1.\) |
Natural Numbers - the most familiar set of numbers i.e. \(1,2,3,...\). Initially the sequence of the natural numbers started with number 1 since 0 was not considered a number in Ancient Greece. In 19th century mathematicians started including 0 in a set of natural numbers, so today different mathematicians use the term to describe both sets (with and without 0). The symbol for all natural numbers is \(\mathbb{N}\). Sometimes the \(\mathbb{N}_0\) or \(\mathbb{N}_1\) are used to indicate whether the set should start with 0 or 1, respectively.
Integers - is the set of numbers that consist of natural numbers (\(1,2,3,...\)) their additve inverses i.e negative numbers (\(-1,-2,-3,...\)) and zero (0). The set of integer numbers is denoted by letter \(\mathbb{Z}\) which stands for German word "Zahlen" which is translated as "number". The \(\mathbb{Z}\) is a subset of rational numbers \(\mathbb{Q}\), which is a subset of the real numbers \(\mathbb{R}\). The integer numbers like natural numbers are countably infinte.
Rational Numbers - are the numbers that can be expressed as the quotient or fraction of two integers i.e. \(\frac{a}{b}.\) The rational numbers are denoted by a letter \(\mathbb{Q}.\) This type of numbers were denoted in 1895 by Giuseppe Peano after Italian word quoziente - "quotient".
Real Numbers - are the values of a continuos quantity that can represent a disance along a line. The adjective real was introduced by Rene Descartes and includes all the rational numbers, suas as the integer -123, fraction \(-42/34\) and all the irrational numbers \(\sqrt{3} = 1.732...\)
Complex Numbers - is a number that can be expressed in the form \(a+bi\) where \(a\) and \(b\) are real numbers, and \(i\) is a symbol called the imaginary unit $$ i^2 = -1.$$ In a complex number \(a+bi,\) \(a\) is the real part and \(b\) is the imaginary part. As allready noted in Table 1 the complex numbers is denoted by the symbol \(\mathbb{C}.\)
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