- 1. 1 LONG LIVE INTERNATIONAL MATHEMATICS DAY, THE QUEEN OF SCIENCES Fernando Alcoforado* International Mathematics Day is a worldwide celebration. Yesterday, March 14, International Mathematics Day was celebrated around the world, created by UNESCO (The United Nations Educational, Scientific and Cultural Organization) in 2019 at the suggestion of the International Mathematical Union (IMU). This date aims to encourage educational institutions, museums and other entities to promote activities to demonstrate how Mathematics is essential for the society in which we live. The date of March 14 was chosen because in many countries Pi Day (π) is celebrated, a mathematical constant whose value corresponds to 3.14. In the North American standard, the month is written before the day: thus, 3/14. The idea is to expand the celebration so that everyone remembers the importance of Mathematics in our lives. Mathematics has been studied and applied throughout the history of humanity. Today, it has become such a sophisticated tool that people don't even realize its omnipresence in our lives, for example, in computer programming algorithms and logic, in GPS (Global Positioning System), in research tools on the internet, in medical examinations, in astronomy applications in the search for extraterrestrial life, in the air traffic system, in cryptography, in the analysis of epidemics, in the launch of satellites and rockets into space, among other applications. Mathematics is present in everything. If someone pays for a ticket or purchases an object with a magnetic card, responds to messages via WhatsApp, searches for films currently showing and listens to music with headphones, arrives at the laboratory and does a computed tomography scan and, at the end, calls a car using the app and go watch an animation at the cinema, what all these actions have in common is Mathematics, which is present in everything, including music. What does Mathematics have to do with music? When observing musical rhythms, for example, time and its divisions (which are mathematical concepts) appear. Frequencies, sounds and timbres also have mathematical roots and are present in music, as well as measures, which are times that are repeated. The time figures (duration) of the notes, for example, are fractions of a measure such as 1/2, 1/4, 1/8, etc. The pitch (tuning) of the notes is established by an exponential relationship, like "2 raised to x/12", where x is the distance from one note to another. When a frequency is multiplied by 2, the note remains the same. For example, the note musical A (440 Hz) multiplied by 2 is 880 Hz, which is still an A note, but an octave higher. If the objective is to lower an octave, it would be enough to divide by 2. In other words, a note and its respective octave maintain a relationship of ½. In Ancient Greece, Pythagoras made very important discoveries for Mathematics, such as the Pythagorean Theorem, and also for music). For example, Pythagoras discovered that stretching a string, attaching it to its ends and touching it causes it to vibrate. He also decided to divide this string into two parts and played each end again. The sound produced was exactly the same, only higher pitched (as it was the same note an octave higher). Pythagoras decided to analyze what the sound would be like if the string were divided into 3 parts. A new sound emerged, different from the previous one. Pythagoras realized that it was not the same note an octave higher, but a different note, which needed to be given another name. Despite being different, the sound matched the previous one, creating a pleasant harmony to the ear. Thus, he continued making subdivisions and combining sounds mathematically, creating scales that, later, stimulated the creation of musical
- 2. 2 instruments capable of reproducing these scales. What we can understand is that music works mathematically, being the result of a numerical organization [BLOG COM CIÊNCIA. A relação entre música, física e matemática (The relationship between music, physics and mathematics). Available on the website <https://museuweg.net/blog/a- relacao-entre-musica-fisica-e- matematica/#:~:text=Na%20Gr%C3%A9cia%20Antiga%2C%20Pit%C3%A1goras%20 fez,(e%20para%20a%20m%C3%BAsica>]. As Mathematics is present in everything in people's lives, it deserves the applause of all humanity. It is because Mathematics is present in everything that it is considered the queen of sciences. It was the great mathematician Carl Gauss who stated that mathematics is the queen of sciences. Mathematics is the science of logical reasoning whose development is linked to research, the interest in discovering new things and investigating highly complex situations [ALCOFORADO, Fernando. As Grandes Revoluções Científicas, Econômicas e Sociais que Mudaram o Mundo (The Great Scientific, Economic and Social Revolutions that Changed the World). Curitiba: Editora CRV, 2016]. Since ancient times, man's need to relate natural events to his daily life has sparked an interest in calculations and numbers. Around the 9th and 8th centuries BC, Mathematics was in its infancy in Babylon. The Babylonians and Egyptians already had algebra and geometry, but only what was sufficient for their practical needs, and not an organized science. Mathematics only came to be considered a science, in the modern sense of the word, from the 6th and 5th centuries BC in Greece. Greek Mathematics differs from Babylonian and Egyptian Mathematics because the Greeks made it a science in the strict sense without worrying about its practical applications. From a structural point of view, Greek Mathematics differs from the previous one, as it took into account problems related to infinite processes, movement and continuity. The various attempts by the Greeks to solve such problems gave rise to the axiomatic- deductive method. This method consists of admitting certain propositions (more or less evident) as true and from them, through a logical chain, arriving at more general propositions. The difficulties that the Greeks encountered when studying problems related to infinite processes (especially problems about irrational numbers) were perhaps the causes that diverted them from Algebra, directing them towards Geometry. Indeed, it is in Geometry that the Greeks stand out, culminating in Euclid's Geometry. Archimedes develops Geometry by introducing a new method that would be a true germ from which an important branch of Mathematics (theory of limits) would later sprout. Apollonius of Perga, a contemporary of Archimedes, began studying the so-called conic curves: the ellipse, the parabola, and the hyperbola, which play a very important role in current mathematics. After Apollonius and Archimedes, Greek Mathematics entered its twilight. In India, another type of mathematical culture was developed: Algebra and Arithmetic. The Hindus introduced a completely new symbol into the hitherto known number system: ZERO. This caused a true revolution in the "art of calculation". The culture of the Hindus was propagated by the Arabs. These brought to Europe the so-called "Arabic numerals" invented by the Hindus. In the year 1202, the Italian mathematician Leonardo of Pisa, known as "Fibonacci", diffuse Mathematics in his work entitled "Leber abaci" in which he described the "art of calculating" (Arithmetic and Algebra). In this book Leonardo presents solutions to 1st, 2nd and 3rd degree equations. At this time, Algebra begins to take on its formal aspect. A German monk Jordanus Nemorarius already begins to use letters to signify any number, and also introduces the signs of + (plus) and - (minus) in the form of the letters p (plus = more) and m (minus = less). German mathematician Michael Stifel starts using the plus (+) and minus (-) signs, as we
- 3. 3 use them today. It is Algebra that is born and develops. This development is finally consolidated in the work of the French mathematician, François Viète. In the 17th century, Mathematics took on a new form, with René Descartes and Pierre Fermat standing out initially. René Descartes' great discovery was undoubtedly "Analytical Geometry", which, in short, consists of the applications of algebraic methods to geometry. Pierre Fermat developed the theory of prime numbers and solved the important problem of tracing a tangent to any plane curve, thus laying the seeds for what would later be called, in Mathematics, the theory of maxima and minima. Thus, in the 17th century, we see one of the most important branches of Mathematics begin to germinate, known as Mathematical Analysis. At this time, problems in Physics still arise: the study of the movement of a body, previously studied by Galileo Galilei. Such problems give rise to one of the first descendants of Mathematical Analysis: Differential Calculus. Differential Calculus appears for the first time in the hands of Isaac Newton and also by the German mathematician Gottfried Wihelm Leibniz. Analytical Geometry and Calculus give a great boost to Mathematics. In the 18th century, there was a critical attitude towards reviewing the fundamental facts of Mathematics. It can be said that such a review was the "cornerstone" of Mathematics. Cauchy carried out notable works, leaving more than 500 written works, of which we highlight two in Mathematical Analysis: "Notes on the development of functions in series" and "Lessons on the application of calculus to geometry". Around 1900, we highlight David Hilbert, with his work "Fundamentals of Geometry" published in 1901. Algebra and Arithmetic take on a new impulse. A problem that worried mathematicians was whether or not it was possible to solve algebraic equations using formulas that appeared with radicals. It was already known that in 2nd and 3rd degree equations this was possible; hence the following question arose: do equations from the 4th degree onwards admit solutions through radicals? In the first third of the 19th century, Niels Abel and Evariste de Galois solved the problem, demonstrating that equations from the fourth and fifth degrees onwards could not be solved by radicals. Galois's work, only published in 1846, gave rise to the so-called "Group Theory" and the so-called "Modern Algebra”. Georg Cantor began the so-called Set Theory, and in a bold way approached the notion of infinity, revolutionizing it. From the 19th century onwards, Mathematics began to branch out into different disciplines, which became increasingly abstract. This move towards the "abstract", even though it may not seem practical at all, was intended to move "Science" forward. History has shown that what seems to us to be pure abstraction, pure mathematical fantasy, later turns out to be a true storehouse of practical applications. Currently, Mathematics is the most important science in the modern world because it is present in all scientific areas. Mathematics had great contributions from the great mathematicians of Babylon, Egypt, Greece, China, India, Islam and, modernly, Europe and the United States. The Scientific Revolution, which began in the 15th century, made knowledge more structured and more practical, absorbing empiricism as a mechanism to consolidate findings. Amid all the effervescence favorable to the Scientific Revolution, Mathematics gained space and developed with great relevance for the development of a more rigorous and critical scientific method. Mathematics began to describe scientific truths applied to all branches of science. The development of Mathematics was fundamental to the development of Physics, Chemistry and Engineering, which culminated in all the industrial and technological progress of recent centuries.
- 4. 4 The most important mathematicians in history were: 1) PYTHAGORAS, Greek, who developed work in the areas of mathematics, geography, music, medicine and philosophy. Observing the pyramids, he developed the important “Pythagorean Theorem”, which says that the sum of the squares of the legs (smaller sides) is equal to the square of the hypotenuse (the larger side); 2) EUCLIDES, Greek, who presented the foundations of Geometry in the 3rd century BC; 3) ARCHIMEDES, Greek, who applied Geometry, uniting the abstract world of numbers with the real world. He was the first to notice the constant relationship between the diameter and radius of any circle: the number π (pi) = 3.14; 4) AL-KHWARIZMI, Persian, who created the theoretical foundations of modern Algebra in the 8th century. The Italian Fibonacci took Khwarizmi's teachings to Europe, propagating the use of Arabic numerals and the numerals from 0 to 9 to represent them; 5) RENÉ DESCARTES, Frenchman, who created Analytical Geometry in the 17th century and was responsible for representing the numbers in that graph with x and y axes, named Cartesian in his honor; 6) ISAAC NEWTON, Englishman, created Calculus in the 17th century and was responsible for scientific advances, such as the law of universal gravitation; 7) GOTTFRIED LEIBNIZ, German, also created Calculus in the 17th century; 8) LEONHARD EULER, Swiss, revolutionized almost all of Mathematics in the 18th century. He founded Graph Theory, which enabled the emergence of Topology; 9) HENRI POINCARÉ, French, invented Algebraic Topology in the 19th century, considered an extension of Geometry; 10) ÉVARISTE GALOIS, French, created algebraic structures in the 19th century. His main work was related to polynomials and algebraic structures, which led him to solve open mathematical problems since Antiquity; 11) CARL GAUSS, German, who was the most complete mathematician of the first half of the 19th century, published, at the age of 21, his masterpiece on Number Theory, contributing to areas such as Statistics, Analysis, Differential Geometry and Geodesy. One of his "inventions" was the Gauss curve, which always appears in statistical graphs; 12) J. WILLARD GIBBS, North American, OLIVER HEAVISIDE, British, and EDWIN BIDWELL WILSON, North American, began at the end of the 19th century and beginning of the 20th century the development of Vector Integral Differential Calculus, widely used in Physics and Engineering; 13) BERNHARD RIEMANN, German mathematician, made several contributions to Differential Geometry and was the father of Elliptical Geometry (one of the non-Euclidean geometries or geometry of curved surfaces and the other is Hyperbolic Geometry) at the end of the 19th century. Differential Geometry and Elliptical Geometry are used in the Theory of Relativity, since space-time is curved; 14) DAVID HILBERT, German, was one of the most influential mathematicians of the 19th and 20th centuries. He created theories in various fields of Mathematics. He created theories used in Quantum Mechanics (Hilbert Space) and Theory of Relativity; 15) JOHN VON NEUMANN, Hungarian, was one of the most brilliant mathematicians of the 20th century and in history. He was the chief mathematician on the atomic bomb project when he performed fundamental calculations for the implosion mechanism and made several contributions to Quantum Mechanics, Statistics, Game Theory and Computer Science. He was also a professor at Princeton University and one of the builders of the ENIAC (the first electronic computer); 16) ANDREW WILES, British mathematician, made history when he announced on June 23, 1993, after 7 years of much study and hard work, the solution to the greatest mathematical puzzle and challenge of all time that lasted 358 years: Last Fermat's theorem formulated in 1637. * Fernando Alcoforado, awarded the medal of Engineering Merit of the CONFEA / CREA System, member of the Bahia Academy of Education, of the SBPC- Brazilian Society for the Progress of Science and of IPB- Polytechnic Institute of Bahia, engineer from the UFBA Polytechnic School and doctor in Territorial
- 5. 5 Planning and Regional Development from the University of Barcelona, college professor (Engineering, Economy and Administration) and consultant in the areas of strategic planning, business planning, regional planning, urban planning and energy systems, was Advisor to the Vice President of Engineering and Technology at LIGHT S.A. Electric power distribution company from Rio de Janeiro, Strategic Planning Coordinator of CEPED- Bahia Research and Development Center, Undersecretary of Energy of the State of Bahia, Secretary of Planning of Salvador, is the author of the books Globalização (Editora Nobel, São Paulo, 1997), De Collor a FHC- O Brasil e a Nova (Des)ordem Mundial (Editora Nobel, São Paulo, 1998), Um Projeto para o Brasil (Editora Nobel, São Paulo, 2000), Os condicionantes do desenvolvimento do Estado da Bahia (Tese de doutorado. Universidade de Barcelona,http://www.tesisenred.net/handle/10803/1944, 2003), Globalização e Desenvolvimento (Editora Nobel, São Paulo, 2006), Bahia- Desenvolvimento do Século XVI ao Século XX e Objetivos Estratégicos na Era Contemporânea (EGBA, Salvador, 2008), The Necessary Conditions of the Economic and Social Development- The Case of the State of Bahia (VDM Verlag Dr. Müller Aktiengesellschaft & Co. KG, Saarbrücken, Germany, 2010), Aquecimento Global e Catástrofe Planetária (Viena- Editora e Gráfica, Santa Cruz do Rio Pardo, São Paulo, 2010), Amazônia Sustentável- Para o progresso do Brasil e combate ao aquecimento global (Viena- Editora e Gráfica, Santa Cruz do Rio Pardo, São Paulo, 2011), Os Fatores Condicionantes do Desenvolvimento Econômico e Social (Editora CRV, Curitiba, 2012), Energia no Mundo e no Brasil- Energia e Mudança Climática Catastrófica no Século XXI (Editora CRV, Curitiba, 2015), As Grandes Revoluções Científicas, Econômicas e Sociais que Mudaram o Mundo (Editora CRV, Curitiba, 2016), A Invenção de um novo Brasil (Editora CRV, Curitiba, 2017), Esquerda x Direita e a sua convergência (Associação Baiana de Imprensa, Salvador, 2018), Como inventar o futuro para mudar o mundo (Editora CRV, Curitiba, 2019), A humanidade ameaçada e as estratégias para sua sobrevivência (Editora Dialética, São Paulo, 2021), A escalada da ciência e da tecnologia e sua contribuição ao progresso e à sobrevivência da humanidade (Editora CRV, Curitiba, 2022), a chapter in the book Flood Handbook (CRC Press, Boca Raton, Florida United States, 2022), How to protect human beings from threats to their existence and avoid the extinction of humanity (Generis Publishing, Europe, Republic of Moldova, Chișinău, 2023) and A revolução da educação necessária ao Brasil na era contemporânea (Editora CRV, Curitiba, 2023).