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.

**Arithmetic**

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.

**Geometry**

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

**Algebra**

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.

**Analytic geometry**

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

**Differential calculus**

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

**Calculation environment**

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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