Category Archives: Numbers

Description of the numbers

Numbers

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

Set

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

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:

  • Property antisymmetric if aRb and bRa are met, it must be fulfilled that a=b (you can shortly denote how a R b, b R a\Rightarrow a = b)
  • 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

    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 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

    Natural numbers

    Natural numbers

    Natural numbers are the ordered set of numbers that humanity has used to count, which are assigned ordinal names to name each particular number:

    Units
    Number Ordinal
    1 First
    2 Second
    3 Third
    4 Fourth
    5 Fifth
    6 Sixth
    7 Seventh
    8 Eighth
    9 Ninth
    11 to 19
    Number Ordinal
    11 Eleventh
    12 Twelfth
    13 Thirteenth
    14 Fourteenth
    15 Fifteenth
    16 Sixteenth
    17 Seventeenth
    18 Eighteenth
    19 Nineteenth
    Tens
    Number Ordinal
    10 Tenth
    20 Twenty
    30 Thirty
    40 Forty
    50 Fifty
    60 Sixty
    70 Seventy
    80 Eightieth
    90 Ninetieth
    Hundreds
    Number Ordinal
    100 Hundredth
    200 Two hundredth
    300 Three hundredth
    400 Four hundredth
    500 Five hundredth
    600 Six hundredth
    700 Seven hundredth
    800 Eight hundredth
    900 Nine hundredth

    And so we could continue listing them until we get bored, since the set of natural numbers is infinite

    However, there is no universal agreement on whether zero should be considered a natural number. Historically, 0 as a number had a much later origin than the rest of numbers. The Babylonians, in the 7th century BC. they had a symbol for zero, but just to leave no gaps when representing quantities in their base 60 positional numbering system in cuneiform script

    The Hindu mathematician Brahmagupta, already considered it a number more, in the 7th century; and possibly, the Olmec and Mayan civilizations already used it as a number, centuries earlier. On the other hand, the Greeks, from which the mathematics of Western culture derive, had no concept of zero; and for that reason, there is no way to represent it in Roman numerals, a culture that drank from Greek, which prevailed in Europe until the base 10 decimal system (of Indian origin and adopted by the Arabs) began to slowly prevail from the 13th century onwards

    For this reason, you can define natural numbers in two ways:

    \begin{cases}\mathbb{N}=\{1,2,3,4,5,\cdots,n\} & \text{(if we do not include 0) } \\ \mathbb{N}_0=\{0,1,2,3,4,5,\cdots,n\} & \text{(if we do include 0) } \end{cases}

    The axioms of Peano

    The axioms of Peano

    The Peano axioms (also known as Peano's postulates) were a proposal by the Italian mathematician Geuseppe Peano in 1889, in order to axiomatically formalize the natural numbers, based on the set theory developed by Georg Cantor

    Axioms, which is still used in the present

    Natural numbers are defined as a set (called \mathbb{N}_0), an element (which assumes the role of zero and that we will denote as 0) and a “next” (or “successor”) element that is an application denoted by S so that it fulfills:

    1. The zero is a natural number
    2. The next natural number is also a natural number
    3. There is No natural number whose next is zero
    4. If the next of two natural numbers are equal, then the two numbers are equal
    5. If S is a set of natural numbers such that zero is of S and whenever a natural number is of S also its next one is in S, then S is the set of natural numbers

    Using a more formal or algebraic language, these five axioms can be stated like this:

    1. 0\in \mathbb{N}_0
    2. \exists s| \mathbb{N}_0\rightarrow \mathbb{N}_0 and in addition satisfies the axioms following
    3. \not\exists n\in \mathbb{N}_0 |s(n)=0
    4. s(n)=s(m)\Rightarrow n=m (s is injective)
    5. S\in \mathbb{N}, 0\in S |\forall n\in S=s(n)\in S\Rightarrow S=\mathbb{N}_0

    The method of induction

    The method of induction

    From the concept of induction, the idea of mathematical induction appears, which allows demonstrations by the induction method

    You can express the natural numbers as a triplet (\mathbb{N}_0,0,s) with \mathbb{N}_0 a set, 0\in \mathbb{N}_0, s| \mathbb{N}_0\rightarrow\mathbb{N}_0\backslash\{0\} an injective application, and in such a way that the fifth of the previous axioms is fulfilled, which is called the axiom of induction or the principle of induction

    To do this, S is taken as the set of natural numbers that satisfy a certain property that wants to be proved, it is verified that zero fulfills the property (that is 0\in S), and that if a number n fulfills it, the next will also fulfill it (that is n\in S\Rightarrow s(n)\in S); and as a consequence of the axiom of induction, all natural numbers fulfill the property (S=\mathbb{N}_0)

    In practice, the principle of induction is usually applied in terms of properties rather than in terms of sets. To carry it out, we will carry out the following steps:

    Suppose that for each natural number n\geq n_0 it has a certain property P_n which may or may not be true. We assume that:

    1. P_n it is certain
    2. If for some n\geq n_0 the property P_n is true then P_{n+1} also what is

    Then P_n it is certain for all n\geq n_0

    The demonstration will be over, because we will have been able to test the property for all natives

    The Gaussian sum

    Induction Example: The Gaussian Sum

    The Gaussian sum is a practical example of the induction method

    In 1789. Carl Friedrich Gauss (who would become a great mathematician and physicist), while in school, at the age of nine, his teacher wanted to keep the children busy for a while, he ordered them to add the first hundred numbers. He had barely finished assigning homework when Gauss got up and handed over his whiteboard. On the board there was a single number: 5050. It turned out that 5050 was precisely the sum of the numbers from one to one hundred. How had he found the solution so quickly?

    Gauss realized something curious. The sum to be made was \sum\limits_{k=1}^{100} k, which can take a long time if done in that order, but if you add the first and last numbers, you get 101. The same happens if you add the second with the penultimate and the third with the penultimate, and so on. You can see that all these sums have the same result: 101

    Since there are obviously 50 pairs whose sum is 101, the result of adding from one to one hundred is 50\times 101=5050. This way of dealing with the problem is an excellent example of an elegant solution

    This solution is not only valid for numbers from one to one hundred. In general, the sum of the first n numbers (where n is an even number) is the last number plus one times the number of pairs, that is \sum\limits_{k=1}^{n} k=\frac{n\cdot(n+1)}{2}

    Now we are going to prove that \sum\limits_{k=1}^{n} k=\frac{n\cdot(n+1)}{2} is true for all natural numbers using the induction method:

    We take it as a P_n the left side of the equality, we check that for P_1 is true: P_1=\sum\limits_{k=1}^{1} k=1

    Now compare it with the right side \frac{1\cdot(1+1)}{2}=\frac{1\cdot 2}{2}=\frac{2}{2}=1. Since 1=1, we have that the equality was true for P_1

    We assume that it is true up to n: \sum\limits_{k=1}^{n} k=\frac{n\cdot(n+1)}{2}

    Now we check that it is true for n+1:

    \sum\limits_{k=1}^{n+1} k=\frac{(n+1)\cdot((n+1)+1)}{2}=\frac{(n+1)\cdot(n+2)}{2}=\frac{n^2+2\cdot n+n+2}{2}=\frac{n^2+3\cdot n+2}{2}
    (\sum\limits_{k=1}^{n} k)+(n+1)=\frac{n^2+3\cdot n+2}{2}

    We apply the induction hypothesis: (\frac{n\cdot(n+1)}{2})+(n+1)=\frac{n^2+3\cdot n+2}{2}

    \frac{n\cdot(n+1)+2\cdot(n+1)}{2}=\frac{n^2+3\cdot n+2}{2}
    \frac{n^2+n+2\cdot n+2}{2}=\frac{n^2+3\cdot n+2}{2}
    \frac{n^2+3\cdot n+2}{2}=\frac{n^2+3\cdot n+2}{2}

    It is fulfilled, then P_n it is certain for all n\geq n_0 and therefore is true for all natural numbers

    Principle of good ordination

    Proposition: principle of good ordination

    The principle of good management that if (\mathbb{N}_0, \leq) is a well-ordered set (that is to say, any subset of \mathbb{N}_0 non empty) has minimal

    Demonstration: principle of good ordination

    We are going to prove by reduction to absurdity

    Suppose that there is a subset A\subset\mathbb{N}_0, A\not=\emptyset, which has no minimum

    Then we define the set S=\{n\in\mathbb{N}_0|n\leq a, \forall a\in A\}

    We must realize that if n\in A\cap S, we would have to n=\text{min}(S) and we've assumed that S has no minimum, therefore A\cap S=\emptyset

    Now let's test it for the whole set using the induction axiom:

    We found that 0\in S
    0+a=a then 0\leq a, \forall a\in A

    \text{If }n\in S, as A\cap S=\emptyset then n < a, \forall a\in A, (by defining order in \mathbb{N}_0) in each case there will be a natural number n_a \geq 1 such that n+n_a=a, which is done whenever s(n)=n+1\leq n+n_a=a, and therefore s(n)\in S

    In compliance with these conditions, the axiom of completeness assures us that S=\mathbb{N}_0

    Which is absurd, for it A\cap S=\emptyset then A=\emptyset, which is clearly a contradiction of the original assumption A\not=\emptyset