Numbers

The numbers must be constructed with the rigor demanded by the mathematics and not only with our intuition

That is why we need to build them from concepts and primitive properties, because as Leopold Kronecker said:

God made the natural numbers; everything else is the work of man

Starting from a "basic" mathematics, there is a concept called a set and each set is made up of a collection of elements (which are unique and different from each other) which belong to the set. In the event that no element appears in the set, we will have an empty set and it is denoted by $\emptyset$

In case of not seeing clearly the logical necessity to introduce the numbers, it is suggested to try to answer the simple question of what is a number? and try to answer it intuitively

Egyptian unit fractions (Ahmes / Rhind Papyrus)

In this papyrus acquired by Henry Rhind in 1858 whose contents date from 2000 to 1800 BC. C. in addition to the number system, we find fractions. Only unit fractions (inverse of natural $\frac{1}{20}$) that are represented by an oval sign above the number, the fraction $\frac{2}{3}$ that is represented by a special sign, and in some cases fractions of the type $\frac{n}{n+1}$. There are decomposition tables of $\frac{2}{n}$ from n=1 to n=101, such as $\frac{2}{5}=\frac{1}{3}+\frac{1}{15}$ or $\frac{2}{7}=\frac{1}{4}+\frac{1}{28}$, but it is not known why they did not use $\frac{2}{n}=\frac{1}{n}+\frac{n+1}{n}$ but it seems that they were trying to use fractions unit lower than $\frac{1}{n}$

Being a summative system the notation is: $1+\frac{1}{2}+\frac{1}{4}$. The fundamental operation is the addition and our multiplication and division was done by "duplication" and "mediation", for example $69\cdot 19=69\cdot (16+2+1)$, where 16 represents 4 duplications and 2 a duplication

Babylonian sexagesimal fractions (cuneiform documents)

In the cuneiform tablets of the Hammurabi dynasty (1800-1600 BC) the positional system appears, an extension of the fractions, but XXX is valid for $2\cdot 60+2, 2+2\cdot 60-1$ or $2\cdot 60-1+2\cdot 60-2$ with a representation based on the interpretation of the problem

To calculate they resorted to the numerous tables at their disposal: of multiplication, of inverses, of squares and cubes, of square and cubic roots, of successive powers of a given number not fixed, etc. For example to calculate a, they took their best integer approximation $a_1$, and they calculated $b_1=\frac{a}{a_1}$ (one major and one minor) and then $a_2=\frac{(a_1+b_1)}{2}$ is a better approximation, proceeding in the same way we obtain $b_2=\frac{a}{a_2}$ and $a_3=\frac{(a_2+b_2)}{2}$ obtaining in the Yale-7289 tablet 2 = 1; 24,51,10 (in decimal base 1.414222) as value of $a_3$ on the basis of $a_1=1;30$

They carried out the operations in a similar way to today, the division multiplying by the inverse (for which they use their inverse tables). Missing from the table of inverses are those of 7 and 11 that have an infinitely long sexagesimal expression. If they are $\frac{1}{59}=;1,1,1$ (our $\frac{1}{9}=0,\stackrel{\frown}{1}$) and $\frac{1}{61}=;0,59,0,59$ (our $\frac{1}{11}=0,\stackrel{\frown}{09}$) but did not notice the periodic development

Basic concepts of set theory

If we denote a set using X, the expression $\displaystyle x\in X$ means that the element x belongs to set X; so what $\displaystyle x\notin X$ that does not belong

A common notation is to mark the elements of a set in braces, $\displaystyle X=\{a,b,c\}$ or $\displaystyle X=\{x| x\text{ meets a certain property}\}$ where the symbol is read as such that

Given two sets X, Y, their intersection $\displaystyle X\cap Y$ and your union $\displaystyle X\cup Y$ are two new sets defined by:

$\displaystyle X\cap Y=\{a|a\in X\cap a\in Y\}$
$\displaystyle X\cup Y=\{a|a\in X\cup a\in Y\}$

If all the elements of a set X are in another set Y, X is said to be subset of Y and denotes how $\displaystyle X \subset Y\text{ or }X \subseteq Y$; its negation is denoted as $\displaystyle X \not\subset Y\text{ or }X \not\subseteq Y$

If all the elements of a set X are equal to those of another set Y, which happens when $\displaystyle X \subseteq Y\text{ and }Y \subseteq X$, X is said to be equal to Y and denoted as X=Y; its negation is denoted as $\displaystyle X \not\subseteq Y\text{ or }X \neq Y$

When $\displaystyle X \subset Y$, the set of elements of Y that are not in X is denoted as $\displaystyle Y \backslash X =\{a| a\in Y\cap a\not\in X\}$

In some cases, the set Y is a kind of implicitly present total set, in those cases we have a set complementary of X, which has the same meaning as $\displaystyle Y \backslash X$

We could go further into set theory to achieve more rigor and depth, but there are two important concepts that we are going to emphasize: relationships and applications

Relationships

Relations relate two or more sets

Given two sets X and Y, their Cartesian product is denoted as $X x Y=\{(x, y)|x\in X, y\in Y\}$ where $(x, y)$ denotes an ordered pair consisting of x and y (this Generalization of Cartesian product can be applied to more than two sets)

A subset XxY is called relationship

When we have $\mathbb{R}\subset X x X$ a relation in X, where $(a, b)\in \mathbb{R}$ denotes aRb. These relationships can perform the following properties:

• Property reflective when all the $a \in X$ meet aRa
• The property of symmetric if aRb is met bRa must also be met (it can be abbreviated how $a R b\Rightarrow b R a$)
• Property transitive if aRb and bRc are met then it must be met that aRc (you can shortly denote how $a R b, a R c\Rightarrow a R c$)

A relation that satisfies these three properties is called relationship of equivalence. Instead of using R, to denote them it is used $\sim$

The simplest example of an equivalence relationship is the equality relationship (each element is related only to itself). And if in an equivalence relationship we identify the related elements, we will get a kind of equality

We assume that in X we have a relation of equivalence $\sim$, group each element $a\in R$ with everyone who is related to him. In this way we get for each to the following set: $\hat{a}=\{x \in X| a \sim x\}$. This set is called class of equivalence of a

If we have two elements $a, b\in X$, their respective equivalence classes $\hat{a}$ and $\hat{b}$ are the same $(\hat{a} =\hat{b})$ or disjoint $(\hat{a}\cap\hat{b} = \emptyset)$, the different types of equivalence form what is called partition of X (by definition, a partition in a set is a series of subsets that are two-to-two disjoints and whose join results in the set)

The set of equivalence classes is a new set that is named set quotient and is denoted: $\displaystyle X\backslash\sim=\{\hat{a}|a\in X\}$

If $a, b \in X$ meet $a \sim b$, your equivalence classes will be $\hat{a} = \hat{b}$ and therefore are the same element in $X\backslash\sim$

If the symmetric property is not met, the following can be true:

A reflective, antisymetric and transitive R relationship is called order relationship, and it is common to denote it by $\leq$; it is said that $(X, \leq)$ is an orderly set. With the same meaning as $a\leq b$ it is also used $b\geq a$; if in addition to $a\leq b$ we want to make sure that a=b, you can use $a < b \text{ o }b > a$

In an orderly set, if it is always true that $a\leq b\text{ o }b\leq a$, is called total order; otherwise we are faced with a partial order

When S is an ordered subset of X, we say that $x \in X$ it is a upper bound of S if $x \geq t$ for any $t \in S$. If there is a, the smallest of the upper dimensions and is called supreme S; if the supreme is in S, it is said to be the maximum S

When S is an ordered subset of X, we say that $x \in X$ it is a lower elevation of S if $x\geq t$ for any $\displaystyle t \in S$. If there is a, the largest of the lower dimensions and is called tiny S; if the still is in S, it is said to be the minimum S

A set in a well-ordered (or that complies with the principle of good sorting), is such an orderly set that any non empty subset has minimal

Applications

An application is a rule that given two sets X and Y, each element of X is associated with an element of Y (and only one)

If we call that rule f, this is denoted as: $f|X\rightarrow Y$

If f is associated $x \in X$ with $y \in Y$ it is denoted $y=f(x)$ (and it is said that y is the image of x), and you can also use the notation $x\rightarrow y$

X is called domain, and the set $f(X)=\{f(x)|x\in X\}$ is the image or the path of f. (With more rigor we can define it from previous concepts, as a relation $f\subset X x Y$ such that, for everything $x\in X$ there is a single $y\in Y$ that meets $(x, y)\in f$)

Applications can be of the type:

• $f|X\rightarrow Y$ is called injective $\text{Si }f(a)\not=f(b)$ always that $a\not= b$ (can be denoted in abbreviated form as $f(a)=f(b) \Rightarrow a = b$)
• $f|X\rightarrow Y$ is called surjective if for each $y \in Y$ there is $x\in X$ such that $y=f(x)$
• $f|X\rightarrow Y$ is called bijective if is injective and suprayectiva at the same time

Instead of applications, and with the same meaning, there is also talk of functions; commonly, the term function is used when dealing with a mapping between sets of numbers. In some Spanish-speaking countries, the term mapping is also used from the English term map. Sometimes an application is also said to be 1−1 with the meaning of being an injective application

The body of the numbers

The set of numbers is what is called a body (sometimes the field designation is also used)

A body is a set with two operations $\displaystyle +\text{ and }\cdot$ (called sum and product), defined on F so that they fulfill the following properties:

• Associative property for addition
$\displaystyle (a+b)+c=a+(b+c)\text{; }\forall a, b, c\in F$
• Commutative property for addition
$\displaystyle a+b=b+a\text{; }\forall a, b\in F$
• Neutral element of the sum
  There is some element $\displaystyle a+b=b+a\text{; }0\in F$ such that $\displaystyle a+0=a \text{; }\forall a\in F$
  For all $\displaystyle a\in F$, exist some element $\displaystyle b\in F$ such that $\displaystyle a+b=0$ (that b is the reverse with respect to the sum from a, which is denoted by −a and is often referred to as the opposite element of a)

• Associative property for the product
  $\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)\text{; }\forall a, b, c\in F$
• Commutative property for the product
  $\displaystyle a\cdot b=b\cdot a\text{; }\forall a, b\in F$
• Neutral element of the product
  There is some element $\displaystyle 1\in F$ such that $\displaystyle a\cdot 1=a\text{; }\forall a\in F$
  For all $\displaystyle a\in F$ with $\displaystyle a\not=0$, exist some element $\displaystyle b\in F$ such that $\displaystyle a\cdot b=1$ (that b is the reverse with respect to the product from a, which is denoted by $\displaystyle a^{-1}$)

When the property of the existence of inverse with respect to the product fails, instead of body, we have what is called ring (which has its own properties, which we are not going to mention at this time, but due to the lack of an inverse, they differ a lot from the multiplication that is usually taught in primary school)

For example, natural, real, and complex numbers are not bodies. Another set that does not form a body are polynomials (with rational, real or complex coefficients), but it is formed by the so called rational functions, that is, the quotients of polynomials