Mathematics and Its History

Frontmatter

1. The Theorem of Pythagoras

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The Pythagorean theorem is the most appropriate starting point for a book on mathematics and its history. It is not only the oldest mathematical theorem, but also the source of three great streams of mathematical thought numbers, geometry, and infinity. The number stream begins with Pythagorean triples; triples of integers \((a, b, c)\) such that \(a^2 + b^2 = c^2\). The geometry stream begins with the interpretation of \(a^2, b^2,\ \rm{and}\ c^2\) as squares on the sides of a right-angled triangle with sides a, b, and hypotenuse c. The infinity stream begins with the discovery that \(\sqrt{2}\), the hypotenuse of the right-angled triangle whose other sides are of length 1, is an irrational number. These three streams are followed separately through Greek mathematics in Chapters 2, 3, and 4. The geometry stream resurfaces in Chapter 7, where it takes an algebraic turn. The basis of algebraic geometry is the possibility of describing points by numbers—their coordinates—and describing each curve by an equation satisfied by the coordinates of its points. This fusion of numbers with geometry is briefly explored at the end of this chapter, where we use the formula \(a^2 + b^2 = c^2\) to define the concept of distance in terms of coordinates.

John Stillwell

2. Greek Geometry

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Geometry was the first branch of mathematics to become highly developed. The concepts of “theorem” and “proof” originated in geometry, and most mathematicians until recent times were introduced to their subject through the geometry in Euclid’s Elements. In the Elements one finds the first attempt to derive theorems from supposedly self-evident statements called axioms. Euclid’s axioms are incomplete and one of them, the so-called parallel axiom, is not as obvious as the others. Nevertheless, it took over 2000 years to produce a clearer foundation for geometry. The climax of the Elements is the investigation of the regular polyhedra, five symmetric figures in three-dimensional space. The five regular polyhedra make several appearances in mathematical history, most importantly in the theory of symmetry—group theory—discussed in Chapters 19 and 23. The Elements contains not only proofs but also many constructions, by ruler and compass. However, three constructions are conspicuous by their absence: duplication of the cube, trisection of the angle, and squaring the circle. These problems were not properly understood until the 19th century, when they were resolved (in the negative) by algebra and analysis. The only curves in the Elements are circles, but the Greeks studied many other curves, such as the conic sections. Again, many problems that the Greeks could not solve were later clarified by algebra. In particular, curves can be classified by degree, and the conic sections are the curves of degree 2, as we will see in Chapter 7.

John Stillwell

3. Greek Number Theory

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Number theory is the second large field of mathematics that comes to us from the Pythagoreans via Euclid. The Pythagorean theorem led mathematicians to the study of squares and sums of squares; Euclid drew attention to the primes by proving that there are infinitely many of them. Euclid’s investigations were based on the so-called Euclidean algorithm, a method for finding the greatest common divisor of two natural numbers. Common divisors are the key to basic results about prime numbers, in particular unique prime factorization, which says that each natural number factors into primes in exactly one way. Another discovery of the Pythagoreans, the irrationality of \(\sqrt{2}\), has repercussions in the world of natural numbers. Since\(\sqrt{2}\neq m/n\) for any natural numbers m, n, there is no solution of the equation \(x^2 - 2y^2 = 0\) in the natural numbers. But, surprisingly, there are natural number solutions of \(x^2 - \rm{2}y^2 = 1\), and in fact infinitely many of them. The same is true of the equation \(x^2 - Ny^2 = 1\) for any nonsquare natural number N. The latter equation, called Pell’s equation, is perhaps second in fame only to the Pythagorean equation \(x^2 + y^2 = z^2\), among equations for which integer solutions are sought. Methods for solving the Pell equation for general N were first discovered by Indian mathematicians, whose work we study in Chapter 5. Equations for which integer or rational solutions are sought are called Diophantine, after Diophantus. The methods he used to solve quadratic and cubic Diophantine equations are still of interest. We study his method for cubics in this chapter, and take it up again in Chapters 11 and 16.

John Stillwell

4. Infinity in Greek Mathematics

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Perhaps the most interesting—and most modern—feature of Greek mathematics is its treatment of infinity. The Greeks feared infinity and tried to avoid it, but in doing so they laid the foundations for a rigorous treatment of infinite processes in 19th century calculus. The most original contributions to the theory of infinity in ancient times were the theory of proportions and the method of exhaustion. Both were devised by Eudoxus and expounded in Book V of Euclid’s Elements. The theory of proportions develops the idea that a “quantity” π (what we would now call a real number) can be known by its position among the rational numbers. That is, π is known if we know the rational numbers less than π and the rational numbers greater than π. The method of exhaustion generalizes this idea from “quantities” to regions of the plane or space. A region becomes “known” (in area or volume) when its position among known areas or volumes is known. For example, we know the area of a circle when we know the areas of the polygons inside it and the areas of polygons outside it; we know the volume of a pyramid when we know the volumes of stacks of prisms inside it and outside it. Using this method, Euclid found that the volume of a tetrahedron equals 1/3 of its base area times its height, and Archimedes found the area of a parabolic segment. Both of them relied on an infinite process that is fundamental to many calculations of area and volume: the summation of an infinite geometric series.

John Stillwell

5. Number Theory in Asia

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In the next three chapters we see algebra, in the form of techniques for manipulating equations, becoming firmly established in mathematics. The present chapter shows equations applied to number theory, Chapter 6 shows equations studied for their own sake, and Chapter 7 shows equations applied to geometry. As we saw in Chapter 3, Diophantus had methods for finding rational solutions of quadratic and cubic equations. But when integer solutions are sought, even linear equations are not trivial. The first general solutions of linear equations in integers were found in China and India, along with independent discoveries of the Euclidean algorithm. The Indians also rediscovered Pell’s equation \(x^2 - Ny^2 = 1\), and found methods of solving it for general natural number values of N. The first advance on Pell’s equation was made by Brahmagupta, who in 628 ce found a way of “composing” solutions of \(x^2 - Ny^2 = k_1\) and\(x^2 - Ny^2 = k_2\) to produce a solution of \(x^2 - Ny^2 = k_1k_2\). (We also touch on a curious formula of Brahmagupta that gives all triangles with rational sides and rational area.) In 1150 ce, Bhâaskara II found an extension of Brahmagupta’s method that finds a solution of \(x^2 - Ny^2 = 1\) for any nonsquare natural number N. He illustrated it with the case N = 61, for which the least nontrivial solution is extraordinarily large.

John Stillwell

6. Polynomial Equations

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The first phase in the history of algebra was the search for solutions of polynomial equations. The “degree of difficulty” of an equation corresponds rather neatly to the degree of the corresponding polynomial. Linear equations are easily solved, and 2000 years ago the Chinese were even able to solve n equations in n unknowns by the method we now call “Gaussian elimination.” Quadratic equations are harder to solve, because they generally require the square root operation. But the solution—essentially the same as that taught in high schools today—was discovered independently in many cultures more than 1000 years ago. The first really hard case is the cubic equation, whose solution requires both square roots and cube roots. Its discovery by Italian mathematicians in the early 16th century was a decisive breakthrough, and equations quickly became the language of virtually all mathematics. (See, for example, analytic geometry in Chapter 7 and calculus in Chapter 9.) Despite this breakthrough, the problem of polynomial equations remained incompletely solved. The obstacle was the quintic equation—the general equation of degree 5. In the 1820s it finally became clear that the quintic equation is not solvable in the sense that equations of lower degree are solvable. But explaining why this is so requires a new, and more abstract, concept of algebra (see Chapter 19).

John Stillwell

7. Analytic Geometry

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The first field of mathematics to benefit from the new language of equations was geometry. Around 1630, both Fermat and Descartes realized that geometric problems could be translated into algebra by means of coordinates, and that many problems could then be routinely solved by algebraic manipulation. The language of equations also provides a simple but natural classification of curves by degree. The curves of degree 1 are the straight lines; the curves of degree 2 are the conic sections; so the first “new” curves are those of degree 3, the cubic curves. Cubic curves exhibit new geometric features—cusps, inflections, and self-intersections—so they are considerably more complicated than the conic sections. Nevertheless, Newton attempted to classify them, and in doing so he discovered that cubic curves, when properly viewed, are not as complicated as they seem. We will find our way to the “right” viewpoint in Chapters 8 and 15. In the meantime we discuss another theorem that depends on the “right” viewpoint: B´ezout’s theorem, according to which a curve of degree m always meets a curve of degree n in mn points.

John Stillwell

8. Projective Geometry

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At about the same time as the algebraic revolution in classical geometry, a new kind of geometry also came to light: projective geometry. Based on the idea of projecting a figure from one plane to another, projective geometry was initially the concern of artists. In the 17th century, only a handful of mathematicians were interested in it, and their discoveries were not seen to be important until the 19th century. The fundamental quantities of classical geometry, such as length and angle, are not preserved by projection, so they have no meaning in projective geometry. Projective geometry can discuss only things that are preserved by projection, such a points and lines. Surprisingly, there are nontrivial theorems about points and lines. One of them was discovered by the Greek geometer Pappus around 300 ce, and another by the French mathematician Desargues around 1640. Even more surprisingly, there is a numerical quantity preserved by projection. It is a “ratio of ratios” of lengths called the cross-ratio. In projective geometry, the cross-ratio plays a role similar to that played by length in classical geometry. One of the virtues of projective geometry is that it simplifies the classification of curves. All conic sections, for example, are “projectively the same,” and there are only five types of cubic curve. The projective viewpoint also removes some apparent exceptions to the theorem of B´ezout. For example, a line (curve of degree 1) always meets another line in exactly one point, because in projective geometry even parallel lines meet.

John Stillwell

9. Calculus

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The shift towards algebraic thinking was not only a revolution in geometry. It was decisive in the second and greatest mathematical revolution of the 17th century: the invention of calculus. It is true that some results we now obtain by calculus were known to the ancients; for example, the area of the parabolic segment was found by Archimedes. But the systematic computation of areas, volumes, and tangents became possible only when symbolic computation—that is, algebra—became available. The dependence of calculus on algebra is particularly clear in the work of Newton, whose calculus is essentially the algebra of infinite polynomials (power series). Moreover, Newton’s starting point was a basic theorem about the polynomial (1 + x)n, the binomial theorem, which he extended to fractional values of n. The calculus of Leibniz was likewise based on algebra—in his case the algebra of infinitesimals. Despite doubts about the meaning and existence of infinitesimals, Leibniz and his followers obtained correct results by computing with them. Results that we now obtain through a combination of algebra and limit processes were obtained by Leibniz through the algebra of infinitesimals. Our derivative dy/dx was, for Leibniz, literally the quotient of the infinitesimal dx by the infinitesimal dy. And our integral ∫ f (x) dx was, for Leibniz, literally the sum of the infinitesimals f (x) dx (hence the symbol ∫, which is an elongated S for “sum”).

John Stillwell

10. Infinite Series

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As we saw in the previous chapter, many calculus problems have a solution that can be expressed as an infinite series. It is therefore useful to be able to recognize important individual series and to understand their general properties and capabilities. This is the aim of the present chapter. Starting with the infinite geometric series, already known to Euclid, we discuss the handful of examples known before the invention of calculus. These include the harmonic series \(1 + 1/2 + 1/3 + 1/4 + \cdot\cdot\cdot\), studied by Oresme around 1350, and the stunning series for the inverse tangent, sine, and cosine, discovered by Indian mathematicians in the 15th century. The invention of calculus in the 17th century released a flood of new series, mostly of the form \(a_0 + a_1x + a_2x^2 + \cdot\cdot\cdot\) (called power series), but also some variations, such as fractional power series. The 18th century brought new applications. De Moivre (1730) used power series to find a formula for the nth term of the Fibonacci sequence 0, 1, 1, 2, 3, 5, 8,.... Euler (1748a) introduced a generalization of the harmonic series, \(1 + 1/2^s + 1/3^s + 1/4^s + \cdot\cdot\cdot\), and showed that, for s > 1, it equals the infinite product \((1 - 1/{2^s})^{-1}(1 - 1/3^s)^{-1}(1 - 1/5^s)^{-1} \cdot\cdot\cdot (1 - 1/p^s)^{-1} \cdot\cdot\cdot\) over all the prime numbers p. This discovery of Euler’s opened a new path to the secrets of the primes, exploration of which continues to this day.

John Stillwell

11. The Number Theory Revival

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After the work of Diophantus, number theory in Europe languished for about 1000 years. In Asia there was significant progress, as we saw in Chapter 5, on topics such as Pell’s equation. The first signs of reawakening in Europe came in the 14th century, when Levi ben Gershon found formulas for the numbers of permutations and combinations, using rudimentary induction proofs. Interest in number theory gathered pace with the rediscovery of Diophantus by Bombelli, and the publication of a new edition by Bachet de M´eziriac (1621). It was this book that inspired Fermat and launched number theory as a modern mathematical discipline. Fermat mastered and extended the techniques of Diophantus, such as the chord and tangent method for finding rational points on cubic curves. He also shifted the emphasis from rational solutions to integer solutions. He proved “Fermat’s little theorem” that \(n^p - n\) is divisible by p for any prime p, and claimed “Fermat’s last theorem” that \(x^n + y^n = z^n\) has no positive integer solutions when n> 2. We know that Fermat had a proof of his “last theorem” for n = 4, but he seems to have been mistaken in thinking that he could prove it for arbitrary n. The proof now known uses highly sophisticated ideas, not conceivable in the 17th century. Nevertheless, it is strangely appropriate that the modern proof reduces Fermat’s last theorem to a problem about cubic curves.

John Stillwell

12. Elliptic Functions

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Elliptic functions, like many innovations in mathematics, arose as a way around an impasse. As we saw in Section 9.6, the search for closed-form solutions in integral calculus foundered on integrands such as \(1/\sqrt{1-x^4}\), because no “known” function f(x) has derivative \(1/\sqrt{1-x^4}\). Eventually, mathematicians accepted the fact that \(\int_0^x \frac{dt}{\sqrt{1-t^4}}\) is a new function. It is one of a family called the elliptic integrals, because one of them is the integral that defines the arc length of the ellipse. \(\int_0^x \frac{dt}{\sqrt{1-t^4}}\) is the simplest elliptic integral to investigate, and many of its properties were found by analogy with those of the arcsine integral \(\int_0^x \frac{dt}{\sqrt{1-t^2}}\). However, these were feats of virtuosity, like finding properties of the arcsine integral without using the sine function.The real innovation came around 1800, when Gauss realized that one should not study the elliptic integral \(u = \int_0^x \frac{dt}{\sqrt{1-t^4}}\) but rather its inverse function x as a function of u (just as one should study the sine function rather than the arcsine integral). He wrote x = sl(u) and found that sl, like the sine function, is periodic:

$$sl(u + 2 \varpi) = sl(u), \quad {\rm where} \ \varpi \ \hbox{is a certain real number}$$

. More surprisingly, sl has second period 2iϖ, so sl is better viewed as a function of complex numbers. These results first became widely known when they were rediscovered and published by Abel and Jacobi in the 1820s. Further insights into double periodicity were obtained in the 1850s, as we will see in Chapter 16.

John Stillwell

13. Mechanics

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In Chapter 9 we introduced the concepts of derivative and integral geometrically, as tangents and areas respectively. Geometry was certainly an important source of calculus problems and concepts, but not the only one. From the beginning, mechanics was just as important. Mechanics is conceptually important because the derivative and the integral are inherent in the concept of motion: velocity is the derivative of displacement (with respect to time), and displacement is the integral of velocity. Also, mechanics was initially the only source of nonalgebraic curves; for example, the cycloid, which is generated by rolling a circle along a line. The “mechanical” curves spurred the development of calculus for the simple reason that they were not accessible to pure algebra. An even greater spur was the development of continuum mechanics, which studies the behavior of such things as flexible and elastic strings, fluid motion, and heat flow. Continuum mechanics involves functions of several variables, and their various derivatives, hence partial differential equations. Some of the most important partial differential equations, such as the wave equation and the heat equation, are clearly inseparable from their origins in continuum mechanics. Yet these very equations confronted mathematicians with basic questions in pure mathematics: for example, what is a function?

John Stillwell

14. Complex Numbers in Algebra

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The next three chapters revisit the topics of algebra, curves, and functions, observing how they are simplified by the introduction of complex numbers. That’s right: the so-called “complex” numbers actually make things simpler. In the present chapter we see where complex numbers came from (not from quadratic equations, as you might expect, but from cubic equations) and observe how they simplify the study of polynomial equations. Equations become simpler because they always have solutions in the complex numbers, and it follows that they have the “right” number of solutions. One of the reasons for the simplifying power of complex numbers is their two-dimensional nature. The extra dimension gives more room for solutions of equations to exist. For example, the equation x ⁿ = 1, which has only one or two solutions in the real numbers, has n different solutions in the complex numbers, equally spaced around the unit circle. More generally, complex numbers give a way to divide any angle into n equal parts. This comes about because multiplication of complex numbers involves addition of angles, and is related to the famous de Moivre formula in trigonometry. The equation x ⁿ = 1 is not the only one with the “right” number of solutions in the complex numbers. In fact, any equation of degree n has n complex solutions, when solutions are properly counted. This is the fundamental theorem of algebra, and it follows from intuitively simple properties of the plane and continuous functions.

John Stillwell

15. Complex Numbers and Curves

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The fundamental theorem of algebra—that a polynomial of degree k has exactly k complex roots—enables us to get the “right” number of intersections between a curve of degree m and a curve of degree n. However, it is not enough to introduce complex coordinates: getting the right count of intersections also requires us to adjust our viewpoint in two other ways.

1.

We must count intersections according to their multiplicity, which amounts to counting a root x = r of a polynomial equation p(x) = 0 as many times as the factor (x - r) occurs in p(x).

 

2.

We must view curves projectively, so that intersections at infinity are included.

 

For these reasons, and others, algebraic geometry moved to the setting of complex projective space in the 19th century. In this chapter we see how this viewpoint affects our picture of algebraic curves. The simplest such curve is the complex projective line, which turns out to look like a sphere. Other algebraic curves also look like surfaces, but they can be more complicated than the sphere. It was discovered by Riemann that rational curves (curves that can be parameterized by rational functions) are essentially the same as the sphere, but nonrational curves have “holes” and hence are essentially different. This discovery reveals the role of topology in the study of algebraic curves.

John Stillwell

16. Complex Numbers and Functions

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The insight into algebraic curves afforded by complex coordinates—that a complex curve is topologically a surface—has important repercussions for functions defined as integrals of algebraic functions, such as the logarithm, exponential, and elliptic functions. The complex logarithm turns out to be “many-valued,” due to the different paths of integration in the complex plane between the same endpoints. It follows that its inverse function, the exponential function, is periodic. In fact, the complex exponential function is a fusion of the real exponential function with the sine and cosine: e ^x+iy = e ^x (cos y + i sin y). The double periodicity of elliptic functions also becomes clear from the complex viewpoint. The integrals that define them are taken over paths on a torus surface, on which there are two independent closed paths. The two-dimensional nature of complex numbers imposes interesting and useful constraints on the nature of differentiable complex functions. Such functions define conformal (angle-preserving) maps between surfaces. Also, their real and imaginary parts satisfy equations, called the Cauchy–Riemann equations, that govern fluid flow. So complex functions can be used to study the motion of fluids. Finally, the Cauchy–Riemann equations imply Cauchy’s theorem. This fundamental theorem guarantees that differentiable complex functions have many good features, such as power series expansions.

John Stillwell

17. Differential Geometry

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As mentioned in Chapter 13, calculus made it possible to study nonalgebraic curves: the “mechanical” curves, or transcendental curves as we now call them. Calculus computes not only their basic features, such as tangents and area, but also more sophisticated properties such as curvature. Curvature turns out to be a fundamental concept of geometry, not only for curves, but also for higher-dimensional objects. The concept of curvature is particularly interesting for surfaces, because it can be defined intrinsically. The intrinsic curvature, or Gaussian curvature as it is known, is unaltered by bending the surface, so it can be defined without reference to the surrounding space. This opens the possibility of studying the intrinsic surface geometry. On any smooth surface one can define the distance between any two points (sufficiently close together), and hence “lines” (curves of shortest length), angles, areas, and so on. The question then arises, to what extent does the intrinsic geometry of a curved surface resemble the classical geometry of the plane? For surfaces of constant curvature, the difference is reflected in two of Euclid’s axioms: the axiom that straight lines are infinite, and the parallel axiom. On surfaces of constant positive curvature, such as the sphere, all lines are finite and there are no parallels. On surfaces of zero curvature there may also be finite straight lines; but if all straight lines are infinite the parallel axiom holds. The most interesting case is constant negative curvature, because it leads to a realization of non-Euclidean geometry, as we will see in Chapter 18.

John Stillwell

18. Non-Euclidean Geometry

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Surprisingly, the geometry of curved surfaces throws light on the geometry of the plane. More than 2000 years after Euclid formulated axioms for plane geometry, differential geometry showed that the parallel axiom does not follow from the other axioms of Euclid. It had long been hoped that the parallel axiom followed from the others, but no proof had ever been found. In particular, no contradiction had been derived from the contrary hypothesis, P ₂, that there is more than one parallel to a given line through a given point. In the 1820s, Bolyai and Lobachevsky proposed that the consequences of P ₂ be accepted as a new kind of geometry—non-Euclidean geometry. To prove that no contradiction follows from P ₂, however, one needs to find a model for P ₂ and the other axioms of Euclid. One seeks a mathematical structure, containing objects called “points” and “lines,” that satisfies Euclid’s axioms with P ₂ in place of the parallel axiom. Such a structure was first found by Beltrami (1868a), in the form of a surface of constant negative curvature with geodesics as its “lines.” By various mappings of this surface, Beltrami found other models, including a projective model in which “lines” are line segments in the unit disk, and conformal models in which “angles” are ordinary angles. Finally, Poincaré (1882) showed that Beltrami’s conformal models arise naturally in complex analysis. Papers had already been published with pictures of patterns of non-Euclidean “lines,” most notably Schwarz (1872). Thus, non-Euclidean geometry was actually a part of existing mathematics, but a part whose geometric nature had not previously been understood.

John Stillwell

19. Group Theory

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The next three chapters are concerned with the emergence of “modern,” or abstract, algebra from the old algebra of equations. In the present chapter we look at group theory. Group theory today is often described as the theory of symmetry, and indeed groups have been inherent in symmetric objects since ancient times. However, extracting algebra from a symmetric object is a highly abstract exercise, and groups first appeared in situations where some algebra was already present. One of the first nontrivial examples was the group of integers mod p, for prime p, used by Euler (1758) to prove Fermat’s little theorem. Of course, Euler had no idea that he was using a group. But he did use one of the characteristic group properties, namely, the existence of inverses. Likewise, Lagrange (1771) was not aware of the group concept when he studied permutations of the roots of equations. But he was using the group S _n of permutations of n things, and some of its subgroups. It was Galois (1831a) who first truly grasped the group concept, and he used it brilliantly to explain what makes an equation solvable by radicals. In particular, he was able to explain why the general quintic equation is not solvable by radicals. These discoveries changed the face of algebra, though few mathematicians realized it at first. In the second half of the 19th century the group concept spread from algebra to geometry, following the observation of Klein (1872) that each geometry is characterized by a group of transformations. This very fruitful idea is explored further in Chapter 23.

John Stillwell

20. Hypercomplex Numbers

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This chapter is the story of a generalization with an unexpected outcome. In trying to generalize the concept of real number to n dimensions, we find only four dimensions where the idea works: n = 1, 2, 4, 8. “Numberlike” behavior in ℝn, far from being common, is a rare and interesting exception. Our idea of “numberlike” behavior is motivated by the cases n = 1, 2 that we already know: the real numbers ℝ and the complex numbers ℂ. The number systems ℝ and ℂ have both algebraic and geometric properties in common. The common algebraic property is that of being a field, and it is captured by nine laws governing addition and multiplication, such as ab = ba and a(bc) = (ab)c (commutative and associative laws for multiplication). The common geometric property is the existence of an absolute value, |u|, which measures the distance of u from O and is multiplicative: |uv| = |u||v|. In the 1830s and 1840s, Hamilton and Graves searched long and hard for “numberlike” behavior in ℝ _n , but they came up short. Beyond ℝ and ℂ, only two hypercomplex number systems even come close: for n = 4 the quaternion algebra ℍ, which has all the required properties except commutative multiplication, and for n = 8 the octonion algebra \(\mathbb{O}\), which has all the required properties except commutative and associative multiplication. Despite lacking some of the field properties, ℍ and \(\mathbb{O}\) can serve as coordinates for projective planes. In this setting, the missing field properties have a remarkable geometric meaning. Failure of the commutative law corresponds to failure of the Pappus theorem, and failure of the associative law corresponds to failure of the Desargues theorem.

John Stillwell

21. Algebraic Number Theory

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Another concept of abstract algebra that emerged from the old algebra of equations was that of ring, which arose from attempts to find integer solutions of equations. The first steps towards the ring concept were taken by Euler (1770b), who discovered equations whose integer solutions are most easily found with the help of irrational or imaginary numbers. Gauss realized that these auxiliary numbers work because they behave like integers. In particular, they admit a concept of “prime” for which unique prime factorization holds. In the 1840s and 1850s the idea of “algebraic integers” was pushed further by various mathematicians, and it reached maturity when Dedekind (1871) defined the concept of algebraic integer in a number field of finite degree. By this time, considerable experience with number fields had been acquired, and Kummer had noticed that such fields do not always admit unique prime factorization. Kummer found a way around this difficulty by introducing new objects that he called ideal numbers (in analogy with “ideal” objects in geometry, such as points at infinity). Dedekind replaced Kummer’s undefined “ideal numbers” by concrete sets of numbers that he called ideals. He was then able to restore unique prime factorization by proving that it holds for ideals. Ring theory as we know it today is largely the result of building a general setting for Dedekind’s theory of ideals. It owes its existence to Emmy Noether, who used to say that “it’s already in Dedekind.”

John Stillwell

22. Topology

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In Chapter 15 we saw how Riemann found the topological concept of genus to be important in the study of algebraic curves. In the present chapter we will see how topology became a major field of mathematics, with its own methods and problems. Naturally, topology interacts with geometry, and it is common for topological ideas to be noticed first in geometry. An important example is the Euler characteristic, which was originally observed as a characteristic of polyhedra, then later seen to be meaningful for arbitrary closed surfaces. Today, we tend to think that topology comes first, and that it controls what can happen in geometry. For example, the Gauss–Bonnet theorem seems to show that the Euler characteristic controls the value of the total curvature of a surface. Topology also interacts with algebra. In this chapter we focus on the fundamental group, a group that describes the ways in which flexible loops can lie in a geometric object. On a sphere, all loops can be shrunk to a point, so the fundamental group is trivial. On the torus, however, there are many closed loops. But they are all combinations of two particular loops, a and b, such that ab = ba. In 1904, Poincaré famously conjectured that a closed three-dimensional space with trivial fundamental group is topologically the same as the threedimensional sphere. This Poincaré conjecture was proved only in 2003, with the help of methods from differential geometry. Thus the interaction between geometry and topology continues.

John Stillwell

23. Simple Groups

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We saw in Chapter 19 that the group concept came to light when Galois used it to explain why some equations are solvable and some are not. Solving an equation corresponds to “simplifying” a group by forming quotients, so knowing which equations are not solvable depends on knowing which groups cannot be “simplified.” These are the so-called simple groups. The groups associated with polynomial equations are finite, so one would like to classify the finite simple groups. Galois found one infinite family of such groups—the alternating groups An for n≥ 5—and three other provocative examples that we now view as the symmetry groups of finite projective lines. However, classification of the finite simple groups was much harder than could have been foreseen in the 19th century. It turned out to be easier (though still very hard) to classify continuous simple groups. This was done by Lie, Killing, and Cartan in the 1880s and 1890s. Each continuous simple group is the symmetry group of a space with hypercomplex coordinates, either from ℝ, ℂ,ℍ, or \(\mathbb{O}\). While this classification was in progress, it was noticed that a single continuous simple group can yield infinitely many finite simple groups, obtained by replacing the hypercomplex number system by a finite field. These “finite groups of Lie type” were completely worked out by 1960. Together with the alternating groups and the cyclic groups of prime order, they account for all but finitely many of the finite simple groups. But identifying all the exceptions—the 26 sporadic simple groups— turned out to be the hardest problem of all…

John Stillwell

24. Sets, Logic, and Computation

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In the 19th century, perennial concerns about the role of infinity in mathematics were finally addressed by the development of set theory and formal logic. Set theory was proposed as a mathematical theory of infinity and formal logic was proposed as a mathematical theory of proof (partly to avoid the paradoxes that seem to arise when reasoning about infinity). In this chapter we discuss these two developments, whose interaction led to mind-bending consequences in the 20th century. Both set theory and logic throw completely new light on the question, “What is mathematics?” But they turn out to be double-edged swords.

• Set theory brings remarkable clarity to the concept of infinity, but it shows infinity to be unexpectedly complicated–in fact, more complicated than set theory itself can describe.
• Formal logic encompasses all known methods of proof, but at the same time it shows these methods to be incomplete. In particular, any reasonably strong system of logic cannot prove its own consistency.
• Formal logic is the origin of the concept of computability, which gives a rigorous definition of an algorithmically solvable problem. However, some important problems turn out to be unsolvable.

It might be thought that the limits of formal proof are too remote to be of interest to ordinary mathematicians. But in the next chapter we will show how these limits are now being reached in one of the most down-to-earth fields of mathematics: combinatorics.

John Stillwell

25. Combinatorics

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In this final chapter we look at another field that came to maturity in the 20th century: combinatorics. Like number theory before the 19th century, combinatorics before the 20th century was thought to be an elementary topic without much unity or depth. We now realize that, like number theory, combinatorics is infinitely deep and linked to all parts of mathematics. Here we emphasize the parts that link nicely to topics from earlier chapters, but without completely sacrificing the distinctive features of the subject. Combinatorics is often called “finite mathematics” because it studies finite objects. But there are infinitely many finite objects, and it is sometimes convenient to reason about all members of an infinite collection at once. In fact, combinatorics pioneered this idea with the use of generating functions (already seen in Section 10.6). Other important infinite principles in combinatorics are the infinite pigeonhole principle and the Kőnig infinity lemma. We illustrate these first by some classical proofs in number theory and analysis, then in the 20thcentury fields of graph theory and Ramsey theory. Ramsey theory leads us to a proof of the Paris–Harrington theorem, mentioned in Section 24.8 as a theorem that cannot be proved in the strictly finite reasoning of PA. Infinite reasoning is likewise essential for graph theory. The field had its origins in topology, and it is still relevant there, but it has expanded extraordinarily far in other directions. Graph theory today is exploring the boundaries of finite provability first exposed by Gödel’s incompleteness theorem.

John Stillwell

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