The math is related at all times with any human society; arithmetic and geometry arise in them almost immediately in the face of the need to count and measure in the commercial, productive and legal operations that occur within these human groups.
In the process of evolution of this science it is possible to say that arithmetic is first given, which is a branch of mathematics that allows counting the objects and establishing a numerical order through the abstraction of nature that arises from the numbers; Likewise, in arithmetic the elementary operations that can be performed with numbers are defined: addition, subtraction, multiplication and division.
Through the numbers, the arithmetic allows counting, establishes a numerical order and defines the four basic operations that can be performed with them
Arithmetic evolved in various stages to pass through the numerical systems with and without a relative position, decimal base, vigesimal and sexagesimal, the appearance of zero and the mechanization of certain operations that allowed some complex calculations such as area and volume.
In an associated way, and in a higher stage of human development, geometry emerges as a mathematical conception of nature related to the visual stimulation of the human environment. Through this branch of mathematics, it is possible to make an approximation of the real world from the abstraction of nature by means of geometric entities (points, lines, triangles, squares, etc.); likewise, through it, various properties and relationships of these geometric entities are determined.
Representation of geometry in everyday life
Geometry emerges as a mathematical conception of nature, originating from the visual stimulation of the human environment. This branch of mathematics allows us to make an approximation of the real world
From a practical point of view, arithmetic and geometry are sufficient to solve most of the problems that gave rise to mathematics; However, arithmetic itself cannot develop general models of a given problem and it is only possible to pose and solve with it particular cases of these problems. Therefore, algebra emerges as a branch of mathematics that allows modelling and determining the general behaviour of mathematical structures that can be raised by arithmetic means; from the development of the algebraic language, new mathematical operations arise exponentiation, radiation and logarithms.
This evolution can be illustrated in a simple way in the case of exponentiation. For the calculation of areas and volumes, it is required in some cases to perform arithmetic operations where a number is multiplied several times by itself, for example, extracting the square or cube of some number to obtain the result. In ancient arithmetic and geometry, this knowledge was highly valued, since it was used in commerce and was also applied to quickly perform multiplication of whole numbers from the squares of other numbers, as shown below. The rule applied for these cases was the following: "To multiply two numbers, you must add both and take a quarter of your square, subtracting this result a quarter of the difference of both squared". Therefore, there were tables of squares of numbers like the one partially illustrated below: Tables of squares of numbers Table with squares of some numbers.
The square of a quantity was obtained by means of geometrical supports such as those that appear on the right because, by the numerical system used or by lack of an algorithm, it was easier to develop them in this way.
Thanks to the development of algebra (and from this its generality is observed), it is possible to know that the algorithm known in the past for multiplication was not an exotic property of numbers, and has its equal support.
The knowledge of negative exponents, fractions, logarithms and many other properties and algorithms was obtained thanks to the development of algebra. Similar problems were also found for geometry, and the case of the Pythagorean theorem can be a way of visualizing them.
In the case of geometry, there is a problem analogous to that of arithmetic: it lacks generality and resources to describe certain geometric structures that are observed in nature with the advance of societies. In this way, by joining algebra and geometry, analytic geometry emerges. Analytical geometry arises from the algebraic generalization of geometry and the need to describe and operate with complex figures in artisan, industrial and productive processes
With arithmetic, geometry, algebra and analytical geometry, the most advanced societies have managed to solve most of their mathematical enigmas; however, there are some practical problems that can not be completely resolved with these resources. The lack of consistency and generality in the solutions found until then for these problems requires a revision of the mathematical foundation.
An example of the type of problems mentioned is shown below.
Suppose that arithmetic, algebra and geometry are sufficient to solve any problem and, through them, we look for an alternative way to find the area of a circle without using the formula learned in primary education (π r 2). To do this, we propose the alternative of approximating the area of a circle through a regular polygon with the use of the exhaustive method proposed by Eudoxus in the 4th century BC.
Although intuition indicates that it should be possible to apply this type of reasoning in problems, mathematics says something that does not agree with this perception, or one can think that mathematically there is no way to express the proposed idea. Here we begin to observe the inconsistencies of the mathematical tool, particularly if everything is limited to the resources provided by arithmetic, geometry and algebra.
Throughout the topic, elements of human intuition have been mentioned; however, it should be noted that common sense is not the main support of mathematics, which is a precise language that uses well-defined symbols and fixed rules that allow to determine and deductively establish more complex relationships between their abstract entities (arithmetic and geometric) without breaking or twisting those rules. Faced with this type of scenarios, the elements that will originate the emergence of a new branch of mathematics, called differential calculus, will be presented.
The old mathematical problems (500-300 BC) on the calculation of areas, raised by Archimedes of Syracuse, and the paradoxes of Zeno and Eudoxus are associated with the emergence of differential calculus.
Arithmetic and geometric approximation to the solution of various practical problems, related to industrial and productive processes. On the left, the roof of a building is illustrated and, on the right, a skein of thread
As mentioned, the differential calculation arises from problems that could not be modelled mathematically and, for this reason, it is not known how to solve them correctly. Generally, they involve the management of algebraic operations where amounts that increase or decrease indefinitely or an infinity of addendums or subtracts are involved; even those related to fractions where their denominators become successively larger, smaller or null may be the epicentre of the problem.
It is important to note at this time that there is a real connection for these mathematical unknowns and is associated in a practical way to the approach of problems that involve events that occur in extremely short or very long term, but it can also be given in situations where the positions between objects they approach continuously or are indistinguishably close.
The conditions outlined represent the core of these mathematical problems, several of ancient origin and others more recent, and induce naturally (after many years of associated research) the study of new ideas related to the fact that the numerical cardinality associated with the variables of an algebraic problem can grow or decrease indefinitely, something not solved in the previous mathematics.
These new approaches point to the conception of differential calculus, whose birth is established in the concept of limit, which addresses these enigmas. They also show that, due to the need for algebraic generality, this also implies the development of the concept of function and establishing within its context, in both cases, it's arithmetical, algebraic and geometric connotations. The differential calculus arises associated with arithmetic, geometric and algebraic problems that imply the indefinite growth or decrease of the variables of an algebraic problem.
From the previous description, it is possible to observe the convenience of studying the differential calculus from the ideas of function and limit; In this voyage, you will see that the other important concepts such as derivation and integration of functions, which have become powerful and efficient mathematical tools to solve a wide range of practical problems in various areas of the exact sciences, administrative and social.
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