Number Theory is the branch of mathematics that investigates the properties of natural numbers or positive integers: 1, 2, 3, 4, 5,… Natural numbers arise from the counting process and it is impossible to imagine humanity without the ability to count. . The concept of natural number was *axiomatized* (axioms are statements accepted as *early truths* without demonstration) in 1889 by the Italian mathematician Giuseppe Peano (1858-1932), in one of the first manifestations of Modern Axiomatics and Mathematical Abstraction. Mathematicians have extended natural numbers to integers, rationals, irrationals, complexes, quaternions, octonions, Cayley numbers,…

It is impossible to imagine Number Theory devoid of the rich and powerful Function Theory of a Complex Variable. One of the most important examples is the function of a complex variable called *Riemann's Zeta function* which gives information about the distribution of prime numbers. It is defined by:

Where *s* = *ç* + *i* *d* is a complex number and *ç* >1.

This function is the key to the demonstration of the prime number theorem that states that the number cousins *P* such that *P* is less than or equal to *x*, is approximately

When *x* It's very big. This theorem was conjectured by Gauss and Legendre, and demonstrated by Hadamard and La Vallée Poussin in 1898.

The history of complex numbers is fascinating. Historical records show that by 2500 BC the Sumerians already needed subtraction. The numbers we know as *negative integers* they are the result of certain subtractions. For example, in modern notation, the result of subtraction 5 - 10 is -5. Mathematicians have not resisted, throughout history, the pressure of the curiosity of *multiply negative numbers* giving rise to the numeric set we currently call the Whole Numbers set: {0, ± 1, ± 2, ± 3…}. The Pythagoreans (550 BC) believed that the world could be understood for reasons of the form *m*/*no* (rational) with *m* and *no* natural and *no* distinct from zero. However, this model of the world collapsed when it was discovered that the measurement of the diagonal of the square, with sides measuring 1, is . Now, Not natural reason! In addition, the Pythagoreans discovered many others of this type: , , , ,…

Therefore, due to the intrinsic needs of mathematical inquiry, the universe of natural numbers has been expanded widely. During the development of algebra in the Middle Ages, Italian mathematicians explored various types of equations and classified their solutions. This investigation showed that some equations had no solution in terms of known numbers. One of the problems faced consisted of solving the equation x² + 1 = 0. This equation did not seem to have a solution because it contradicted the fact that every real number other than zero, when squared, is positive. The Indian and Arab mathematicians, when faced with these equations, refused to define any symbol to express the square root of a negative number because they considered the problem completely meaningless. In the 16th century, square roots of negative numbers began to appear in algebraic texts, but the authors stressed that the expressions were meaningless and used terms such as "fictitious", "impossible", "sophisticated" to mention them. The German mathematician Leibniz (1646-1716), one of the inventors of Differential Calculus, attributed to the square root of -1 a certain metaphysical character by interpreting it as a manifestation of the "Divine Spirit"; The same sense of amazement happened to the Swiss mathematician Lenhard Euler.

Some European mathematicians, in particular Italians Gerolamo Cardano and Rafaello Bombelli, introduced complex numbers in Algebra during the 16th century when they assumed the existence of square roots of negative numbers, although they considered such roots “impossible numbers” and thus , call them “imaginary numbers”. For this reason, to this day the name of imaginary numbers remains when we refer to square roots of negative numbers. Postulating the existence of square roots of negative integers, and assuming that *i* is solution of the equation *x*² + 1 = 0, that is, axiomatizing that *i* satisfy the relationship *i*² = -1, you can perform operations involving *i* and the real numbers. So for any positive real number *The*, the square root of the negative number -*The* é *i* , that is, = *i* . Given the actual numbers *ç* and *d*, we can multiply *d* per *i* and obtain *i d*, and add to *ç* to get *ç* + *i d*. In general, any complex number is written as *ç* + *d* *i*, Where *ç* is called the “real part” and *d* "Imaginary part". So we get numbers of the form *ç* + *i* *d* forming the *set of complex numbers*. In the set of complex numbers, we can add and multiply by forming an algebraic structure called *body of complex numbers*.

Mathematicians often represent real numbers as points on a line called the real line, where each point corresponds to a single real number and each real number associates a single point on that line. Since the square root of a negative number cannot be represented on this line, a stalemate persisted until the 19th century. The first to propose *a visualization* of the complexes identifying them as points on the two-dimensional plane was Norwegian self-taught Caspar Wessel in 1797. This idea was rediscovered by Jean-Robert Argand, a Swiss accountant, who published a book in 1860 on the subject, and also by the German mathematician Karl Friedrich Gauss. Since it was impossible to associate a point of the real line with the square root of a negative number, the issue was resolved by associating the imaginary numbers with points on a line perpendicular to the real line, passing through zero, and thus creating a Cartesian coordinate system. . In this system, the real numbers are placed on the horizontal axis, called *real axis*, and all imaginary numbers on the line perpendicular to the real line, passing through the zero of the horizontal real line, called *imaginary axis*. How = = *i* , all imaginary numbers can be placed on the imaginary axis as multiples of *i* = . Therefore, not only do the imaginary have a graphic representation, but the possible combinations of real and imaginary, that is, the complex numbers, are represented by points in the plane defined by the real and imaginary axes, called *complex plan*.

Gauss's talent and genius led to one of the most profound results of mathematics, the Fundamental Algebra Theorem, which states that *every polynomial equation has a solution in the body of complex numbers*. In addition to this very important result, complex number algebra has given rise to a new area of research - Complex Analysis - which plays a key role in the development of Algebra and Number Theory. Complex numbers represent one of the most important structures of science. Today, it is impossible to imagine electrical engineering, aerodynamics, or fluid dynamics without complex numbers. Quantum Mechanics makes use of complex numbers and, in Einstein's Theory of Relativity, three-dimensional space is seen as real and the time dimension as imaginary.

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