See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/355444457 A History of Real Analysis Preprint · October 2021 DOI: 10.13140/RG.2.2.15699.09764 CITATIONS READS 0 3,451 1 author: Harris R. Dela Cruz Bulacan State University 4 PUBLICATIONS 0 CITATIONS SEE PROFILE All content following this page was uploaded by Harris R. Dela Cruz on 21 October 2021. The user has requested enhancement of the downloaded file. A History of Real Analysis Harris R. Dela Cruz College of Science, Bulacan State University Abstract Real analysis is a branch of mathematical analysis dealing with the real numbers, sequence and series of real numbers, and real-valued functions of a real variable. In particular, it deals with theories on limits, convergence, continuity, differentiation, and integration. Although real analysis is distinguished from complex analysis which deals with theories concerning the properties of complex numbers and functions of complex variables, the two were not totally separated and were developed almost simultaneously especially toward the recent century. We present here a short history of real analysis. Analysis in the Ancient Times The history of analysis is intertwined with the quest of early mathematicians to understand real numbers. In Greek mathematics, it was believed initially that all things [with non-zero magnitude] can be measured using the natural numbers 1, 2, 3, and so on, and their ratios, the rational numbers. This was first challenged by the discovery of irrational numbers about 500 BCE when the Pythagorean Hippasus of Metapontum pointed out to Pythagoras that the square root of 2 (the length of the diagonal of a square with sides of length 1 unit) cannot be represented as a rational number (Tesleff, 2020). They have shown that the set of rational numbers is not sufficient in representing measurements of even simple geometric objects. While 2 challenged the concept of measurement, Zeno’s paradox of the dichotomy challenged the concept of motion. According to Zeno, as recounted by Aristotle (Physics, VI:9, 239b11), “That which is in locomotion must arrive at the half-way stage before it arrives at the goal.” Suppose one wants to walk to the end of a path. Before the end is reached, one must get halfway there. Before one can get halfway there, one must get a quarter of the way there. Before traveling a quarter, one must travel one-eight, before that a one-sixteenth, and so on. The total distance one needs to traverse is therefore 1= 1 2 + 1 4 + 1 8 + 1 16 ···. This would mean that one is required to perform [and finish] unlimited tasks in a limited amount of time, which to Zeno at the time, was impossible. (Huggett, 2018) Due to Zeno’s paradoxes, Greek mathematicians started to realize that the concept of infinity was indispensable in the mathematics of continuous magnitude (one which admits of being cut into an indefinite number of parts). In 350 BCE, the theory of proportions was created by Eudoxus of Cnidus (as claimed in Euclid’s Element). He developed the method of exhaustion to compute [and proved the formula] for the area of a circle, and the volume of a sphere, a pyramid, and a cone (O'Connor & Robertson, 1999). He found the area of a circle by inscribing inside it a sequence of regular polygons with increasing number of sides, As the number of sides becomes arbitrarily large, the difference between the area of the polygon and that of the square becomes arbitrarily small, and the possible values for the area of the circle are systematically exhausted by the inscribed polygon. This was the first explicit, but informal, use of the concept of limit and convergence. Using the same method, in 225 BCE, Archimedes gave the first known example of an infinite series with a finite sum. He showed that the area of a segment of a parabola is 4/3 the area of a triangle with the same base and vertex by constructing an infinite sequence of triangles between the existing ones and the parabola. He used this same method of exhaustion to find an approximation of the area of a circle and an ellipse, as well as the volume and the surface area of a sphere, a cone, and any segment of a paraboloid and of a hyperboloid of revolution. (O'Connors & Robertson, 1996) After Archimedes, the attention of Greek mathematics was shifted either to number theory, mechanics, or astronomy, mainly inspired by Euclid. However, the emergence of analysis during these times is not limited to Ancient Greece only. According to Basant & Panda (2013), Acarya Bhadrabahu gives the sum of a geometric series around 433-355 BCE in his Kalpasutra, but Singh (1936) suggested that the Jian literature may contain accounts of Hindus having possession of the formula for the sum of arithmetic and geometric series as early as 4th century BCE or even earlier. Analysis during the Medieval Period At the turn of the millennia, the method of exhaustion used by Eudoxus and Archimedes continued to play a big role in furthering analysis as a different approach to solving geometry problems. In 263 AD, the Chinese mathematician Liu Hui edited and published the book Jiuzhang suanshu (Nine Chapters on the Mathematical Art) which contains his use of the method of exhaustion in finding the area of a circle (Straffin, Jr., 1998). The 5th-century mathematician Zu Chonzhi, together with his son Zu Genzhi, wrote Zhui Shu (Methods for Interpolation), a mathematical text which is said to contain formulas for the volume of a sphere, cubic equations, and an approximation of the value of π (Ho, 1985). The method was later became known as Cavalieri’s principle, a modern implementation of the method of indivisibles. Meanwhile, during the 12th century, Indian mathematician Bhaskara II gave some examples of derivatives using what is now known as Rolle’s Theorem. Although most of differential and integral calculus was credited to Newton and Leibniz, Goonatilaka (1998) believed that it was Bhaskara who first conceived the differential coefficient and differential calculus, predating the two by over half a millennium. On the other hand, according to Rajagopal & Rangachari (1978), Madhava of Sangamagrama during the 14th century was the first to use infinite series approximation and developed infinite series expansion, now called Taylor series. Although, most of the mathematical works in India during the medieval period were lost, Modern Real Analysis The modern foundations of mathematical analysis were established in the 17th century. Bonaventura Cavalieri was led to his method of indivisibles by Kepler's attempts at integration. It appears that Cavalieri thought of an area as being made up of components which were lines and then summed his infinite number of indivisibles. Roberval considered problems of the same type but was much more rigorous than Cavalieri. Gilles Roberval looked at the area between a curve and a line as being made up of an infinite number of infinitely narrow rectangular strips. On the other hand, Pierre de Fermat generalized the parabola and the hyperbola. He also investigated maxima and minima by considering when the tangent to the curve was parallel to the 𝑥-axis. (O'Connors & Robertson, 1996) It was in 1637 that mathematical analysis was established when Rene Descartes introduced the Cartesian coordinate system and proposed to unify algebra and geometry into one subject, now known as analytic geometry (Descartes, 2006). A few decades later, Isaac Newton and Gottfried Wilhelm Leibniz independently developed infinitesimal calculus. During this period, calculus techniques were applied to approximate discrete problems by continuous ones. It was during these times that differential and integral calculus flourished. Afterward, the development of calculus was continued by Jacob Bernoulli and Johann Bernoulli. However when Berkeley published his Analyst in 1734 attacking the lack of rigor in the calculus and disputing the logic on which it was based much effort was made to tighten the reasoning. Maclaurin attempted to put the calculus on a rigorous geometrical basis but the really satisfactory basis for the calculus had to wait for the work of Cauchy in the 19th Century. (O'Connors & Robertson, 1996) Real analysis began to develop as an independent subject when Bernard Bolzano introduced in 1816 the modern definition of continuity, which was not known until the 1870s. By that time, Cauchy’s work had already put calculus on a firm logical foundation in terms of geometric ideas and infinitesimals. His definition of continuity required an infinitesimal change in 𝑥 to correspond to an infinitesimal change in 𝑦. By the mid-19th century, Bernhard Reimann created what was now known as the Riemann Integral. It was the first rigorous definition of the integral of a function on an interval. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration. In 1854, Karl Theodore William Weierstrass submitted his paper on Abelian functions to the famous Crelle Mathematical Journal. Although most of his work was not published because of his insistence of rigor in all of his proof, Karl Weierstrass is known as the father of modern analysis. He formalized the definition of continuity of a function, proved the intermediate value theorem, the first person to create a continuous function that is nowhere differentiable, developed theory on Abelian functions, devised tests for convergence of series, proved the sequential compactness theorem, now more popularly known as the Bolzano-Weierstrass theorem, and contributed a great deal in the theory of periodic functions, functions of a real variable, elliptic functions, infinite products, and the calculus of variation. (O’Connor & Robertson, 1998) Most of Weierstrass’ work was delivered through his lectures, and which he continued to give until 1890, covering the theory of analytic functions, elliptic functions, Abelian functions, and calculus of variation. Through the years, the courses developed and a number of versions taken from the notes made by students who attended his lectures have been published. Weierstrass' approach and style still dominate how analysis is taught today. References Aristotle. (2000). Physics (R. P. Hardie & R. K. Gaye, Trans.). The Internet Classics Archive. http://classics.mit.edu/Aristotle/physics.html View publication stats Basant, K. B., & Panda, S. (2013). Summation of Convergent Geometric Series and the Concept of Approachable Sunya. Indian Journal of History of Science, 48.2(2013), 291-313. Descartes, R. (2006). A Discourse on the Method (I. Maclean, Trans.). Oxford University Press. (Original work published 1637) Goonatilaka, S. (1998). Toward a Global Science: Mining Civilizational Knowledge. Bloomington: Indiana University Press. Ho, P. Y. (1985). Li, Qi and Shu: An Introduction to Science and Civilization in China. Hong Kong University Press. Huggett, N. (2018). Zeno’s Paradoxes. Stanford Encyclopedia of Philosophy. Retrieved September 21, 2021, from https://plato.stanford.edu/entries/paradox-zeno/#Dic O;Connor, J. J., & Robertson, E. F. (1998). Karl Theodor Wilhelm Weierstrass. MacTutor. Retrieved September 21, 2021, from https://mathshistory.st-andrews.ac.uk/Biographies/Weierstrass/ O.Connor, J. J., & Robertson, E. F. (1999). Eudoxus of Cnidus. MacTutor. Retrieved September 21, 2021, from https://mathshistory.st-andrews.ac.uk/Biographies/Eudoxus/ O'Connors, J. J., & Robertson, E. F. (1996). A History of the Calculus. MacTutor. Retrieved September 21, 2021, from https://mathshistory.st-andrews.ac.uk/HistTopics/The_rise_of_calculus/ Rajagopal, C. T., & Rangachari, M. S. (1978). 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