Numbers

Understanding of numbers, especially natural numbers, is one of the oldest mathematical skills. Many cultures, even some contemporary ones, attribute some mystical properties to numbers because of their huge significance in describing the nature. Although mathematics and the modern science don't confirm such views, the significance of the theory of numbers is undisputed.

Historically, first occurred the set of natural numbers; rather quickly expanded with fractions, and even with positive irrational numbers; zero and negative numbers were discovered only after these subsets of real numbers. The last in the series, a set of complex numbers, occurs only with the development of modern science.

On the other hand, modern mathematics does not introduce numbers chronologically; even though the order of introduction is quite similar.

Natural Numbers $\mathbb{N}$

A set of natural numbers is often denoted by $\mathbb{N}=\lbrace 1,2,3,4... \rbrace$, and it is often expanded with $0$ in which case is denoted by $\mathbb{N}_0$.

In $\mathbb{N}$ operations of addition (+) and multiplication ($\cdot$) are defined with the following properties for each $a,b,c\in \mathbb{N}$:

1. $a+b\in \mathbb{N}$, $a\cdot b \in \mathbb{N}$ set $\mathbb{N}$ is closed for addition and multiplication
2. $a+b=b+a$, $a\cdot b=b\cdot a$ commutativity
3. $(a+b)+c=a+(b+c)$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ associativity
4. $a\cdot (b+c)=a\cdot b+a\cdot c$ distributivity
5. $a\cdot 1=a$ there is a neutral element for multiplication

Since there is a neutral element for multiplication in a set $\mathbb{N}$, but not for addition as well, this is the exact reason why this set is often expanded with 0, which is a neutral element for addition.

Apart from these two operations in set $\mathbb{N}$, relations strictly-less-than ($<$) and less-than-or-equal ($\leq$) are defined, with the following properties for each $a,b,c\in \mathbb{N}$:

1. $a < b$ or $a=b$ or $a > b$ trichotomy
2. if $a\leq b$ and $b\leq a$ then $a=b$ antisymmetry
3. if $a\leq b$ and $b\leq c$ then $a\leq c$ transitivity
4. if $a\leq b$ then $a+c\leq b+c$
5. if $a\leq b$ then $a\cdot c\leq b\cdot c$

Integer $\mathbb{Z}$

Examples of integer numbers:
$1, -20, -100, 30, -40, 120...$

Solving the equation $a+x=b$, where $a$ and $b$ are given natural numbers, and $x$ an unknown natural number, requires an introduction of a new arithmetic operation: deduction (-). If there exists a natural number $x$ which satisfies the equation, then it is $x=b-a$. However, this particular equation does not necessarily have a solution in a set $\mathbb{N}$, so it is required, out of practical reasons, to expand the set of natural numbers with solutions to this equation; which basically leads to the integer set: $\mathbb{Z}=\lbrace 0,1,-1,2,-2,3,-3...\rbrace$.

Since $\mathbb{N}\subset \mathbb{Z}$, it is natural to introduce operations $+$ and $\cdot$ and relations $<$ and $\leq$ which would 'inherit' the same properties as in the set $\mathbb{N}$. Apart from these properties, there are also two new ones that refer to addition:
1. $0+a=a+0=a$ there is a neutral element for addition
2. $a+(-a)=(-a)+a=0$ there is an opposite number $-a$ to a number $a$

Property 5.
5. if $0\leq a$ and $0\leq b$ then $0\leq a\cdot b$

A set $\mathbb{Z}$ is closed for deduction as well, i.e. $(\forall a,b\in \mathbb{Z})(a-b\in \mathbb{Z} )$.

Rational Numbers $\mathbb{Q}$

Examples of rational numbers:
$\frac{1}{2}, \frac{4}{7}, -\frac{5}{8}, \frac{10}{20}...$

Apart from the mentioned equation, it is required to find the solution to this type of equations $a\cdot x=b$, where $a$ and $b$ are given integers, and $x$ an unknown numbers. For the purpose of solution to this type of equations, an operation of division is introduced ($:$), and the solution to this equations is $x=b:a$, that is $x=\frac{b}{a}$. Again, the problem that $x$ does not always belong to to $\mathbb{Z}$ occurs, so it is necessary to expand the set with solutions of this type of equations. Thus, a set $\mathbb{Q}$ is introduced which elements are $\frac{p}{q}$, where $p\in \mathbb{Z}$ and $q\in \mathbb{N}$. A subset of this set where for each element $q=1$ is a set $\mathbb{Z}$, so $\mathbb{Z}\subset \mathbb{Q}$ and therefore operations of addition and multiplication are being expanded to this set based on the following rules that preserve all aforementioned properties also in a set $\mathbb{Q}$:
$\frac{p_1}{q_1}+\frac{p_2}{q_2}=\frac{p_1\cdot q_2+p_2\cdot q_1}{q_1\cdot q_2}$
$\frac{p-1}{q_1}\cdot \frac{p_2}{q_2}=\frac{p_1\cdot p_2}{q_1\cdot q_2}$

At the same time division is introduced as:
$\frac{p_1}{q_1}:\frac{p_2}{q_2}=\frac{p_1}{q_1}\cdot \frac{q_2}{p_2}$

In the set $\mathbb{Q}$ an equation $a\cdot x=b$ has a unique solution for each $a\neq 0$, while division with zero is not defined. This means that there is an inverse element, which we call a reciprocal number, denoted by $\frac{1}{a}$ or $a^{-1}$:
$(\forall a\in \mathbb{Q}\setminus\lbrace 0\rbrace)(\exists \frac{1}{a})(a\cdot \frac{1}{a}=\frac{1}{a}\cdot a=a)$

Order of a set $\mathbb{Q}$ can be expanded in the following way:
$\frac{p_1}{q_1} < \frac{p_2}{q_2}\Leftrightarrow p_1\cdot q_2 < p_2\cdot q_1$

The set $\mathbb{Q}$ has one other important property - between any two rational numbers there is an infinite number of rational numbers, which means that there are no two adjacent rational numbers, as was the case with natural numbers and integers.

Irrational Numbers $\mathbb{I}$

Examples of irrational numbers:
$\sqrt{2} \approx 1.41422135...$
$\pi \approx 3.1415926535...$

Due to the fact that between any two rational numbers there is an infinite number of other rational numbers, it can easily lead to the wrong conclusion, that the set of rational numbers is so dense, that there is no need for further expanding of the rational numbers set. Even Pythagoras himself was drawn to this conclusion. However, even Pythagoras' contemporaries disproved this conclusion while trying to solve the equation $x\cdot x=2$, that is $x^2=2$ in the set of rational numbers. In order to solve this equation, it is necessary to introduce a square root function, so the solution to this equation is $x=\sqrt{2}$. An equation of this type $x^2=a$, where $a$ is a given rational number, and x an unknown number, does not always have a solution within the rational number set, and a need for expanding of the number set occurs again. These numbers are called irrational numbers, and $\sqrt{2}$, $\sqrt{3}$, $\pi$... belong to this set.

Real Numbers $\mathbb{R}$

A union of rational and irrational numbers sets is a set of real numbers. Since $\mathbb{Q}\subset \mathbb{R}$ it is again logical that the introduced arithmetical operations and relations should expand onto the new set. It is extremely difficult to formally perform such expansion and to arrange the real number set, so already mentioned properties of arithmetic operations and relations are introduced in the real number set as axioms. In algebra, such structure is called a field, so we say that a real numbers set is an ordered field.

In order to complete the definition of real numbers set, we need an additional axiom which makes the difference between sets $\mathbb{Q}$ and $\mathbb{R}$. Let us assume that S is a non-empty subset of a real numbers set. Element $b\in \mathbb{R}$ is the upper limit of the set $S$ if $\forall x\in S$ is true $x\leq b$, and then we say that the set $S$ above bounded. The least upper bound of set $S$ is called supremum and is denoted by $\sup S$. In analogy to this, such concepts as lower bound, an above bounded set and infimum $\inf S$ are introduced. The remaining axiom is as follows:

Every non-empty and above bounded subset of a real numbers set has a supremum.
It can also be proven that the field of real numbers defined like this is unique.

Complex Numbers $\mathbb{C}$

Examples of complex numbers:
$(1, 2), (4, 5), (-9, 7), (-3, -20), (5, 19),...$
$1 + 5i, 2 - 4i, -7 + 6i...$ where $i = \sqrt{-1}$ or $i^2 = -1$

A set of complex numbers is a set of all ordered pairs of real numbers, ie. $\mathbb{C}=\mathbb{R}^2=\mathbb{R}\times \mathbb{R}$, in which the operations of addition and multiplication are defined in the following way:
$(a,b)+(c,d)=(a+b,c+d)$
$(a,b)\cdot (c,d)=(ac-bd,ad+bc)$

There are multiple ways of writing complex numbers, and the most common way is $z=a+ib$, which is a number $(a,b)$, and the number $i=(0,1)$ is called an imaginary unit.

It is easy to show that $i^2=-1$. Expanding of the set $\mathbb{R}$ to set $\mathbb{C}$ allows the defining of the square root of negative numbers, so this exactly is a reason to introduce a set of complex numbers. It is also easy to show that a subset of the set $\mathbb{C}$, defined as $\mathbb{C}_0=\lbrace (a,0)|a\in \mathbb{R}\rbrace$, satisfies all real numbers axioms, which means that $\mathbb{C}_0=\mathbb{R}$, or that $R\subset\mathbb{C}$.

The algebraic structure of the set $\mathbb{C}$ in relation to addition and multiplication has the following properties:
1. commutativity of addition and multiplication
2. associativity of addition and multiplication
3. $0+i0$ is a neutral element for addition
4. $1+i0$ is a neutral element for multiplication
5. multiplication is distributive over addition
6. there is a unique inverse element for addition and for multiplication

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