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E-320: Teaching Math with a Historical Perspective O. Knill, 2010-
Lecture 1: Mathematical roots
Similarly, as one has distinguished the canons of rhetorics: memory, invention, delivery, style, and arrangement, or combined the trivium: grammar, logic and rhetorics, with the quadrivium: arithmetic, geometry, music, and astronomy, to obtain the seven liberal arts and sciences, one has tried to organize all mathematical activities.
Historically, one has dis- tinguished eight ancient roots of mathematics. Each of these 8 activities in turn suggest a key area in mathematics:
counting and sorting arithmetic spacing and distancing geometry positioning and locating topology surveying and angulating trigonometry balancing and weighing statics moving and hitting dynamics guessing and judging probability collecting and ordering algorithms
To morph these 8 roots to the 12 mathematical areas covered in this class, we complemented the ancient roots with calculus, numerics and computer science, merge trigonometry with geometry, separate arithmetic into number theory, algebra and arithmetic and turn statics into analysis.
Lets call this modern adap- tation the
12 modern roots of Mathematics:
counting and sorting arithmetic spacing and distancing geometry positioning and locating topology dividing and comparing number theory balancing and weighing analysis moving and hitting dynamics guessing and judging probability collecting and ordering algorithms slicing and stacking calculus operating and memorizing computer science optimizing and planning numerics manipulating and solving algebra
While relating math- ematical areas with human activities is useful, it makes sense to select specific top- ics in each of this area. These 12 topics will be the 12 lectures of this course.
Arithmetic numbers and number systems Geometry invariance, symmetries, measurement, maps Number theory Diophantine equations, factorizations Algebra algebraic and discrete structures Calculus limits, derivatives, integrals Set Theory set theory, foundations and formalisms Probability combinatorics, measure theory and statistics Topology polyhedra, topological spaces, manifolds Analysis extrema, estimates, variation, measure Numerics numerical schemes, codes, cryptology Dynamics differential equations, maps Algorithms computer science, artificial intelligence
Like any classification, this chosen division is rather arbitrary and a matter of personal preferences. The 2010 AMS classification distinguishes 63 areas of mathematics. Many of the just defined main areas are broken off into even finer pieces. Additionally, there are fields which relate with other areas of science, like economics, biology or physics:
00 General 01 History and biography 03 Mathematical logic and foundations 05 Combinatorics 06 Lattices, ordered algebraic structures 08 General algebraic systems 11 Number theory12 Field theory and polynomials 13 Commutative rings and algebras 14 Algebraic geometry 15 Linear/multi-linear algebra; matrix theory 16 Associative rings and algebras 17 Non-associative rings and algebras 18 Category theory, homological algebra 19 K-theory 20 Group theory and generalizations
45 Integral equations 46 Functional analysis 47 Operator theory 49 Calculus of variations, optimization 51 Geometry 52 Convex and discrete geometry 53 Differential geometry54 General topology 55 Algebraic topology 57 Manifolds and cell complexes 58 Global analysis, analysis on manifolds 60 Probability theory and stochastic processes 62 Statistics 65 Numerical analysis 68 Computer science 70 Mechanics of particles and systems 22 Topological groups, Lie groups 26 Real functions 28 Measure and integration 30 Functions of a complex variable 31 Potential theory 32 Several complex variables, analytic spaces 33 Special functions 34 Ordinary differential equations 35 Partial differential equations 37 Dynamical systems and ergodic theory 39 Difference and functional equations 40 Sequences, series, summability 41 Approximations and expansions 42 Fourier analysis 43 Abstract harmonic analysis 44 Integral transforms, operational calculus
74 Mechanics of deformable solids 76 Fluid mechanics 78 Optics, electromagnetic theory 80 Classical thermodynamics, heat transfer 81 Quantum theory 82 Statistical mechanics, structure of matter 83 Relativity and gravitational theory 85 Astronomy and astrophysics 86 Geophysics 90 Operations research, math. programming 91 Game theory, Economics Social and Behavioral Sciences 92 Biology and other natural sciences 93 Systems theory and control94 Information and communication, circuits 97 Mathematics education
What are
fancy developments
in mathematics today? Michael Atiyah identified in the year 2000 the following six hot spots:
local and global low and high dimension commutative and non-commutative linear and nonlinear geometry and algebra physics and mathematics
Also this choice is of course highly personal. One can easily add 12 other polarizing quantities which help to distinguish or parametrize different parts of mathematical areas, especially the ambivalent pairs which produce a captivating gradient:
regularity and randomness integrable and non-integrable invariants and perturbations experimental and deductive polynomial and exponential applied and abstract
discrete and continuous existence and construction finite dim and infinite dimensional topological and differential geometric practical and theoretical axiomatic and case based
An other possibility to refine the fields of mathematics is to combine different of the 12 areas. Examples are probabilistic number theory, algebraic geometry, numerical analysis, ge- ometric number theory, numerical algebra, algebraic topology, geometric probability, algebraic number theory, dynamical probability = stochastic processes. Almost every pair is an actual field. Finally, lets give a short answer to the question: What is Mathematics?
Mathematics is the science of structure.
The goal is to illustrate some of these structures from a historical point of view.
survived the thousands of years. The Roman system improved the tally stick concept by intro- ducing new symbols for larger numbers like V = 5, X = 10, L = 40, C = 100, D = 500, M = 1000. in order to avoid bundling too many single sticks. The system is unfit for computations as sim- ple calculations V III + V II = XV show. Clay tablets, some as early as 2000 BC and others from 600 - 300 BC are known. They feature Akkadian arithmetic using the base 60. The hexadecimal system with base 60 is convenient becuase of many factors. It survived: we use 60 minutes per hour. The Egyptians used the base 10. The most important source on Egyptian mathematics is the Rhind Papyrus of 1650 BC. Hieratic numerals were used to write on pa- pyrus from 2500 BC on. Egyptian numerals are hieroglyphics. They were found in carvings on tombs and monuments and are 5000 years old. The modern way to write numbers like 2015 is the Hindu-Arab system which diffused to the West only during the late Middle ages. It replaced the more primitive Roman system. Greek arithmetic used a primitive number system with no place values: 9 Greek letters for 1, 2 ,... 9, nine for 10, 20 ,... , 90 and nine for 100, 200 ,... , 900.
Integers. Indian Mathematics morphed the place-value system into a modern method of writing numbers. Hindu astronomers used words to represent digits, but the numbers would be written in the opposite order. Sometimes after 500, the Hindus changed to a digital notation which included the symbol 0. Negative numbers were introduced around 100 BC in the Chinese text ”Nine Chapters on the Mathematica art”. Also the Bakshali manuscript, written around 300 AD subtracts numbers carried out additions with negative numbers, where + was used to indicate a negative sign. In Europe, negative numbers were avoided until the 15’th century.
Fractions: Babylonians could handle fractions. The Egyptians also used fractions, but wrote every fraction a as a sum of fractions with unit numerator and distinct denominators, like 4 /5 = 1/2 + 1/4 + 1/20 or 5/6 = 1/2 + 1/3. Maybe because of such cumbersome computa- tion techniques, Egyptian mathematics failed to progress beyond a primitive stage. The modern decimal fractions used nowadays for numerical calculations were adopted only in 1595 in Europe.
Real numbers: The Greeks who noticed first that the diagonal of the square is not a rational number. It produced a crisis. Only much later, it became clear that ”most” numbers are not rational. Georg Cantor realized first that the cardinality of all real numbers is much larger than the cardinality of the integers: while one can count all rational numbers but not enumerate all real numbers. One consequence is that most real numbers are transcendental: they do not occur as solutions of polynomial equations with integer coefficients. The number π is an example. The concept of real numbers is related to th concept of limit. Sums like 1+1/4+1/9+1/16+1/25+... approach real numbers which are not rational any more.
Complex numbers: Some polynomials have no real root. To solve x^2 = −1 for example, we need new numbers. One idea is to use pairs of numbers (a, b) where (a, 0) = a are the usual numbers and extend addition and multiplication (a, b)+(c, d) = (a+c, b+d) and (a, b)·(c, d) = (ac−bd, ad+bc). With this multiplication, the number (0, 1) has the property that (0, 1) · (0, 1) = (− 1 , 0) = −1. It is more convenient to write a + ib where i = (0, 1) satisfies i^2 = −1. One can now use the common rules of addition and multiplication.
Surreal numbers: Similarly as real numbers fill in the gaps between the integers, the surreal numbers fill in the gaps between Cantors ordinal numbers. They are written as (a, b, c, ...|d, e, f, ...) meaning that the ”simplest” number is larger than a, b, c... and smaller than d, e, f, ... We have (|) = 0, (0|) = 1, (1|) = 2 and (0|1) = 1/2 or (|0) = −1. Surreals contain already transfinite num- bers like (0, 1 , 2 , 3 ...|) or infinitesimal numbers like (0| 1 / 2 , 1 / 3 , 1 / 4 , 1 / 5 , ...). They were introduced in the 1970’ies by John Conway. The late appearance confirms the pedagogical principle: late human discovery manifests in increased difficulty to teach it.
E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010-
Lecture 3: Geometry
Geometry is the science of shape, size and symmetry. While arithmetic dealt with numerical structures, geometry deals with metric structures. Geometry is one of the oldest mathemati- cal disciplines and early geometry has relations with arithmetics: we have seen that that the implementation of a commutative multiplication on the natural numbers is rooted from an inter- pretation of n × m as an area of a shape that is invariant under rotational symmetry. Number systems built upon the natural numbers inherit this. Identities like the Pythagorean triples 32 + 4^2 = 5^2 were interpreted geometrically. The right angle is the most ”symmetric” angle apart from 0. Symmetry manifests itself in quantities which are invariant. Invariants are one the most central aspects of geometry. Felix Klein’s Erlanger program uses symmetry to classify geome- tries depending on how large the symmetries of the shapes are. In this lecture, we look at a few results which can all be stated in terms of invariants. In the presentation as well as the worksheet part of this lecture, we will work us through smaller miracles like special points in triangles as well as a couple of gems: Pythagoras, Thales,Hippocrates, Feuerbach, Pappus, Morley, Butterfly which illustrate the importance of symmetry.
Much of geometry is based on our ability to measure length, the distance between two points. A modern way to measure distance is to determine how long light needs to get from one point to the other. This geodesic distance generalizes to curved spaces like the sphere and is also a practical way to measure distances, for example with lasers. It bypasses the problem to determine first the underlying nature of the space in which we do geometry. Having a distance d(A, B) between any two points A, B, we can look at the next more complicated object, which is a set A, B, C of 3 points, a triangle. Given an arbitrary triangle ABC, are there relations between the 3 possible distances a = d(B, C), b = d(A, C), c = d(A, B)? If we fix the scale by c = 1, then a + b ≥ 1 , a + 1 ≥ b, b + 1 ≥ a. For any pair of (a, b) in this region, there is a triangle. After an identification, we get an abstract space, which represent all triangles uniquely up to similarity. Mathematicians call this an example of a moduli space.
A sphere Sr(x) is the set of points which have distance r from a given point x. In the plane, the sphere is called a circle. A natural problem is to find the circumference L = 2π of a unit circle, or the area A = π of a unit disc, the area F = 4π of a unit sphere and the volume V = 4 = π/3 of a unit sphere. Measuring the length of segments on the circle leads to new concepts like angle or curvature. Because the circumference of the unit circle in the plane is L = 2π, angle questions are tied to the number π, which Archimedes already approximated by fractions.
Also volumes were among the first quantities, Mathematicians wanted to measure and com- pute. A problem on Moscow papyrus dating back to 1850 BC explains the general formula h(a^2 +ab+b^2 )/3 for a truncated pyramid with base length a, roof length b and height h. Archimedes achieved to compute the volume of the sphere: place a cone inside a cylinder. The complement of the cone inside the cylinder has on each height h the area π − πh^2. The half sphere cut at height h is a disc of radius (1 − h^2 ) which has area π(1 − h^2 ) too. Since the slices at each height have the same area, the volume must be the same. The complement of the cone inside the cylinder has volume π − π/3 = 2π/3, half the volume of the sphere.
The first geometric playground was planimetry, the geometry in the flat two dimensional space. Highlights are Pythagoras theorem, Thales theorem, Hippocrates theorem, and Pappus
E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010-
Lecture 4: Number Theory
Number theory studies the structure of integers like prime numbers and solutions to Diophantine equations. Gauss called it the ”Queen of Mathematics”. Here are a few theorems and open prob- lems. An integer larger than 1 which is divisible by 1 and itself only is called a prime number. The number 2^57885161 − 1 is the largest known prime number. It has 17425170 digits. Euclid proved that there are infinitely many primes: [Proof. Assume there are only finitely many primes p 1 < p 2 <... < pn. Then n = p 1 p 2 · · · pn + 1 is not divisible by any p 1 ,... , pn. Therefore, it is a prime or divisible by a prime larger than pn.] Primes become more sparse as larger as they get. An important result is the prime number theorem which states that the n’th prime number has approximately the size n log(n). For example the n = 10^12 ’th prime is p(n) = 29996224275833 and n log(n) = 27631021115928. 545 ... and p(n)/(n log(n)) = 1. 0856 ... Many questions about prime numbers are unsettled: Here are four problems: the third uses the notation (∆a)n = |an+1 − an| to get the absolute difference. For example: ∆^2 (1, 4 , 9 , 16 , 25 ...) = ∆(3, 5 , 7 , 9 , 11 , ...) = (2, 2 , 2 , 2 , ...). Progress on prime gaps has been done recently: a paper which just appears showed pn+1 − pn is smaller than 100’000’000 eventually (Yitang Zhang April 2013) pn+1 −pn is smaller than 600 even- tually (Maynard). The largest known gap is 1476 which occurs after p = 1425172824437699411.
Twin prime there are infinitely many primes p such that p + 2 is prime. Goldbach every even integer n > 2 is a sum of two primes. Gilbreath If pn enumerates the primes, then (∆kp) 1 = 1 for all k > 0. Andrica The prime gap estimate
pn+1 −
pn < 1 holds for all n.
If the sum of the proper divisors of a n is equal to n, then n is called a perfect number. For example, 6 is perfect as its proper divisors 1, 2 , 3 sum up to 6. All currently known per- fect numbers are even. The question whether odd perfect numbers exist is probably the oldest open problem in mathematics and not settled. Perfect numbers were familiar to Pythagoras and his followers already. Calendar coincidences like that we have 6 work days and the moon needs ”perfect” 28 days to circle the earth could have helped to promote the ”mystery” of per- fect number. Euclid of Alexandria (300-275 BC) was the first to realize that if 2p^ − 1 is prime then k = 2p−^1 (2p^ − 1) is a perfect number: [Proof: let σ(n) be the sum of all factors of n, including n. Now σ(2n^ − 1)2n−^1 ) = σ(2n^ − 1)σ(2n−^1 ) = 2n(2n^ − 1) = 2 · 2 n(2n^ − 1) shows σ(k) = 2k and verifies that k is perfect.] Around 100 AD, Nicomachus of Gerasa (60-120) classified in his work ”Introduction to Arithmetic” numbers on the concept of perfect numbers and lists four perfect numbers. Only much later it became clear that Euclid got all the even perfect numbers: Euler showed that all even perfect numbers are of the form (2n^ − 1)2n−^1 , where 2 n^ − 1 is prime. The factor 2n^ − 1 is called a Mersenne prime. [Proof: Assume N = 2km is perfect where m is odd and k > 0. Then 2k+1m = 2N = σ(N) = (2k+1^ − 1)σ(m). This gives σ(m) = 2k+1m/(2k+1^ − 1) = m(1 + 1/(2k+1^ − 1)) = m + m/(2k+1^ − 1). Because σ(m) and m are integers, also m/(2k+1^ − 1) is an integer. It must also be a factor of m. The only way that σ(m) can be the sum of only two of its factors is that m is prime and so 2k+1^ − 1 = m.] The first 39 known Mersenne primes are of the form 2n^ − 1 with n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917. There are 8 more known from which one does not know the rank of the corresponding Mersenne prime: n = 20996011, 24036583, 25964951, 30402457, 32582657,
37156667, 42643801,43112609,57885161. The last was found in January 2013 only. It is unknown whether there are infinitely many.
A polynomial equations for which all coefficients and variables are integers is called a Diophantine equation. The first Diophantine equation studied already by Babilonians is x^2 + y^2 = z^2. A solution (x, y, z) of this equation in positive integers is called a Pythagorean triple. For example, (3, 4 , 5) is a Pythagorean triple. Since 1600 BC, it is known that all solutions to this equation are of the form (x, y, z) = (2st, s^2 − t^2 , s^2 + t^2 ) or (x, y, z) = (s^2 − t^2 , 2 st, s^2 + t^2 ), where s, t are different integers. [Proof. Either x or y has to be even because if both are odd, then the sum x^2 + y^2 is even but not divisible by 4 but the right hand side is either odd or divisible by 4. Move the even one, say x^2 to the left and write x^2 = z^2 − y^2 = (z − y)(z + y), then the right hand side contains a factor 4 and is of the form 4s^2 t^2. Therefore 2s^2 = z − y, 2 t^2 = z + y. Solving for z, y gives z = s^2 + t^2 , y = s^2 − t^2 , x = 2st.] Analyzing Diophantine equations can be difficult. Only 10 years ago, one has established that the Fermat equation xn^ + yn^ = zn^ has no solutions with xyz 6 = 0 if n > 2. Here are some open problems for Diophantine equations. Are there nontrivial solutions to the following Diophantine equations?
x^6 + y^6 + z^6 + u^6 + v^6 = w^6 x, y, z, u, v, w > 0 x^5 + y^5 + z^5 = w^5 x, y, z, w > 0 xk^ + yk^ = n!zk^ k ≥ 2 , n > 1 xa^ + yb^ = zc, a, b, c > 2 gcd(a, b, c) = 1
The last equation is called Super Fermat. A Texan banker Andrew Beals once sponsored a prize of 100′000 dollars for a proof or counter example to the statement: ”If xp^ + yq^ = zr^ with p, q, r > 2, then gcd(x, y, z) > 1.” Given a prime like 7 and a number n we can add or subtract multiples of 7 from n to get a number in { 0 , 1 , 2 , 3 , 4 , 5 , 6 }. We write for example 19 = 12 mod 7 because 12 and 19 both leave the rest 5 when dividing by 7. Or 5 ∗ 6 = 2 mod 7 because 30 leaves the rest 2 when dividing by 7. The most important theorem in elementary number theory is Fermat’s little theorem which tells that if a is an integer and p is prime then ap^ − a is divisible by p. For example 2^7 − 2 = 126 is divisible by 7. [Proof: use induction. For a = 0 it is clear. The binomial expansion shows that (a + 1)p^ − ap^ − 1 is divisible by p. This means (a + 1)p^ − (a + 1) = (ap^ − a) + mp for some m. By induction, ap^ − a is divisible by p and so (a + 1)p^ − (a + 1).] An other beautiful theorem is Wilson’s theorem which allows to characterize primes: It tells that (n − 1)! + 1 is divisible by n if and only if n is a prime number. For example, for n = 5, we verify that 4! + 1 = 25 is divisible by 5. [Proof: assume n is prime. There are then exactly two numbers 1, −1 for which x^2 − 1 is divisible by n. The other numbers in 1,... , n − 1 can be paired as (a, b) with ab = 1. Rearranging the product shows (n − 1)! = −1 modulo n. Conversely, if n is not prime, then n = km with k, m < n and (n − 1)! = ...km is divisible by n = km. ] The solution to systems of linear equations like x = 3 (mod 5), x = 2 (mod 7) is given by the Chinese remainder theorem. To solve it, continue adding 5 to 3 until we reach a number which leaves rest 2 to 7: on the list 3, 8 , 13 , 18 , 23 , 28 , 33 , 38, the number 23 is the solution. Since 5 and 7 have no common divisor, the system of linear equations has a solution. For a given n, how do we solve x^2 − yn = 1 for the unknowns y, x? A solution produces a square root x of 1 modulo n. For prime n, only x = 1, x = −1 are the solutions. For composite n = pq, more solutions x = r · s where r^2 = −1 mod p and s^2 = −1 mod q appear. Finding x is equivalent to factor n, because the greatest common divisor of x^2 − 1 and n is a factor of n. Factoring is difficult if the numbers are large. It assures that encryption algorithms work and that bank accounts and communications stay safe. Number theory, once the least applied discipline of mathematics has become one of the most applied one in mathematics.
Odd perfect numbers
Probably the oldest open problem in mathematics is the question
There is an odd perfect number.
A perfect number is equal to the sum of all its proper positive divisors. Like 6 = 1 + 2 + 3. The search for perfect numbers is related to the search of large prime numbers. The largest prime number known today is p = 2^43112609 − 1. It is called a Mersenne prime. Every even perfect number is of the form 2n−^1 (2n^ − 1) where 2 n^ − 1 is prime.
Diophantine equations
Many problems about Diophantine equations, equations with integer solutions are unsettled. Here is an example:
Solve x^5 + y^5 + z^5 = w^5 for x, y, z, w ∈ N.
Also x^5 + y^5 = u^5 + v^5 has no nontrivial solutions yet. Probabilistic considerations suggest that there are no solutions. The analogue equation x^4 + y^4 + z^4 = w^4 had been settled by Noam Elkies in 1988 who found the identity 2682440^4 +15365639^4 +18796760^4 = 20615673^4.
ABC Conjecture
The abc conjecture is:
If a + b = c, then c ≤ (
∏ p|abc p)
For example, for 10+22 = 32, the prime factors of abc = 7040 are 2, 5 , 11 and indeed 32 ≤ (2 ∗ 5 ∗ 11)^2 = 12100. The abc-conjecture is open but implies Fermat’s theo- rem for n ≥ 6: assume xn^ +yn^ = zn^ with coprime x, y, z. Take a = xn, b = yn, c = zn. The abc-conjecture gives zn^ ≤ (
∏ p|abc p)^ ≤^ (abc) (^2) < z (^6) establishing Fermat for n ≥ 6. The cases n = 3, 4 , 5 to Fermat have been known for a long time. In August 2012, there were rumors of an attack by Shinichi Mochizuki. During 2013 various mathematicians have tried to understand and verify the theory.
E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010-
Lecture 5: Algebra
Algebra studies algebraic structures like ”groups” and ”rings”. The theory allows to solve polynomial equations, characterize objects by its symmetries and is the heart and soul of many puzzles. Lagrange claims Diophantus to be the inventor of Algebra, others argue that the subject started with solutions of quadratic equation by Mohammed ben Musa Al-Khwarizmi in the book Al-jabr w’al muqabala of 830 AD. Solutions to equation like x^2 + 10x = 39 are solved there by completing the squares: add 25 on both sides go get x^2 + 10x + 25 = 64 and so (x + 5) = 8 so that x = 3. The use of variables introduced in school in elementary algebra were introduced later. Ancient texts only dealt with particular examples and calculations were done with concrete numbers in the realm of arithmetic. Francois Viete (1540-1603) used first letters like A, B, C, X for variables.
The search for formulas for polynomial equations of degree 3 and 4 lasted 700 years. In the 16’th century, the cubic equation and quartic equations were solved. Niccolo Tartaglia and Gerolamo Cardano reduced the cubic to the quadratic: [first remove the quadratic part with X = x−a/3 so that X^3 + aX^2 + bX + c becomes the depressed cubic x^3 + px+ q. Now substitute x = u − p/(3u) to get a quadratic equation (u^6 + qu^3 − p^3 /27)/u^3 = 0 for u^3 .] Lodovico Ferrari shows that the quartic equation can be reduced to the cubic. For the quintic however no formulas could be found. It was Paolo Ruffini, Niels Abel and Evariste Galois´ who independently realized that there are no formulas in terms of roots which allow to ”solve” equations p(x) = 0 for polynomials p of degree larger than 4. This was an amazing achievement and the birth of ”group theory”.
Two important algebraic structures are groups and rings.
In a group G one has an operation ∗, an inverse a−^1 and a one-element 1 such that a ∗ (b ∗ c) = (a ∗ b) ∗ c, a ∗ 1 = 1 ∗ a = a, a ∗ a−^1 = a−^1 ∗ a = 1. For example, the set Q∗^ of nonzero fractions p/q with multiplication operation ∗ and inverse 1/a form a group. The integers with addition and inverse a−^1 = −a and ”1”-element 0 form a group too. A ring R has two compositions + and ∗, where the plus operation is a group satisfying a + b = b + a in which the one element is called 0. The multiplication operation ∗ has all group properties on R∗^ except the existence of an inverse. The two operations + and ∗ are glued together by the distributive law a ∗ (b + c) = a ∗ b + a ∗ c. An example of a ring are the integers or the rational numbers or the real numbers. The later two are actually fields, rings for which the multiplication on nonzero elements is a group too. The ring of integers are no field because an integer like 5 has no multiplicative inverse. The ring of rational numbers however form a field.
Why is the theory of groups and rings not part of arithmetic? First of all, a crucial ingredient of algebra is the appearance of variables and computations with these algebras without using concrete numbers. Second, the algebraic structures are not restricted to ”numbers”. Groups and rings are general structures and extend for example to objects like the set of all possible sym- metries of a geometric object. The set of all similarity operations on the plane for example form a group. An important example of a ring is the polynomial ring of all polynomials. Given any ring R and a variable x, the set R[x] consists of all polynomials with coefficients in R. The addition and multiplication is done like in (x^2 + 3x + 1) + (x − 7) = x^2 + 4x − 7. The problem to
E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010-
Lecture 6: Calculus
Calculus formalizes the process of taking differences and taking sums. Differences measure change, sums explore how things accumulate. The process of taking differences has a limit called derivative. The process of taking sums will lead to the integral. These two processes are related in an intimate way. In this lecture, we look at these two processes in the simplest possible setup, where functions are evaluated on integers and where we do not take any limits. Several dozen thousand years ago, numbers were represented by units like
1 , 1 , 1 , 1 , 1 , 1 ,...
for example carved in the Ishango bone. It took thousands of years until numbers were represented with symbols like 0 , 1 , 2 , 3 , 4 ,....
Using the modern concept of function, we can say f (0) = 0, f (1) = 1, f (2) = 2, f (3) = 3 and mean that the function f assigns to an input like 1001 an output like f (1001) = 1001. Lets call Df (n) = f (n + 1) − f (n) the difference between two function values. We see that the function f satisfies Df (n) = 1 for all n. We can also formalize the summation process. If g(n) = 1 is the function which is constant 1, then Sg(n) = g(0) + g(1) +... + g(n − 1) = 1 + 1 + · · · + 1 = n. We see that Df = g and Sg = f. Lets start with f (n) = n and apply summation on that function:
Sf (n) = f (0) + f (1) + f (2) + · · · + f (n − 1).
In our example, we get the values:
0 , 1 , 3 , 6 , 10 , 15 , 21 ,....
The new function g = Sf satisfies g(1) = 1, g(2) = 3, g(2) = 6, etc. These numbers are called triangular numbers. From g we can get back f by taking difference:
Dg(n) = g(n + 1) − g(n) = f (n).
For example Dg(5) = g(6) − g(5) = 15 − 10 = 5 which indeed is f (5). Finding a formula for the sum Sf (n) is not so easy. Can you do it? When Karl-Friedrich Gauss was a 9 year old school kid, his teacher, a Mr. B¨uttner gave him the task to sum up the first 100 numbers 1 + 2 + · · ·+ 100. Gauss found the answer immediately by pairing things up: to add up 1 + 2 + 3 +... + 100 he would write this as (1 + 100) + (2 + 99) + · · · + (50 + 51) leading to 50 terms of 101 to get for n = 101 the value g(n) = n(n − 1)/2 = 5050. Taking differences again is easier Dg(n) = n(n + 1)/ 2 − n(n − 1)/2 = n = f (n). Lets add now the triangular numbers up compute h = Sg. We get the sequence
0 , 1 , 4 , 10 , 20 , 35 , ...
called the tetrahedral numbers. One can h(n) balls to build a tetrahedron of side length n. For example, h(4) = 20 golf balls are needed to build a tetrahedron of side length 4. The formula
which holds for h is h(n) = n(n − 1)(n − 2)/ 6. Here is the fundamental theorem of calculus,
which is the core of calculus: SDf (n) = f (n) − f (0), DSf (n) = f (n).
Proof.
SDf (n) =
n∑− 1
k=
[f (k + 1) − f (k)] = f (n) − f (0) ,
DSf (n) = [
n∑− 1
k=
f (k + 1) −
n∑− 1
k=
f (k)] = f (n).
The process of adding up numbers will lead to the integral
∫ (^) x 0 f^ (x)^ dx^. The process of taking
differences will lead to the derivative (^) dxd f (x).
∫ (^) x 0
d dt f^ (t)^ dt^ =^ f^ (x)^ −^ f^ (0),^
d dx
∫ (^) x 0 f^ (t)^ dt^ =^ f^ (x)
k
x
y
0
f HkL-f H 0 L
k
x
y
0
f HkL
Theorem: Sum the differences and get
SDf (kh) = f (kh) − f (0)
Theorem: Difference the sum and get
DSf (kh) = f (kh)
If we define [n]^0 = 1, [n]^1 = n, [n]^2 = n(n − 1)/ 2 , [n]^3 = n(n − 1)(n − 2)/6 then D[n] = [1], D[n]^2 = 2[n], D[n]^3 = 3[n]^2 and in general
d dx [x]
n (^) = n[x]n− 1
The calculus you have just seen, contains the essence of single variable calculus. This core idea will become more powerful and natural if we use it together with the concept of limit.
1 Problem: The Fibonnacci sequence 1, 1 , 2 , 3 , 5 , 8 , 13 , 21 ,... satisfies the rule f (x) = f (x −
- f (x − 2). It defines a function on the positive integers. For example, f (6) = 8. What is the function g = Df , if we assume f (0) = 0? We take the difference between successive numbers and get the sequence of numbers
0 , 1 , 1 , 2 , 3 , 5 , 8 , ...
which is the same sequence again. We can deduce from this recursion that f has the property that Df (x) = f (x − 1).
Calculus has many applications: computing areas, volumes, solving differential equations. It even has applications in arithmetic. Here is an example for illustration. It is a proof that π is irrational. This is especially appropriete since next Friday is π day!
We show here the proof by Ivan Niven is given in a book of Niven-Zuckerman-Montgomery. It originally appeared in 1947 (Ivan Niven, Bull.Amer.Math.Soc. 53 (1947),509). The proof illustrates how calculus can help to get results in arithmetic. Proof. Assume π = a/b with positive integers a and b. For any positive integer n define
f (x) = xn(a − bx)n/n!.
We have f (x) = f (π − x) and 0 ≤ f (x) ≤ πnan/n!(∗)
for 0 ≤ x ≤ π. For all 0 ≤ j ≤ n, the j-th derivative of f is zero at 0 and π and for n <= j, the j-th derivative of f is an integer at 0 and π.
The function F (x) = f (x) − f (2)(x) + f (4)(x) − ... + (−1)nf (2n)(x)
has the property that F (0) and F (π) are integers and F + F ′′^ = f. Therefore, (F ′(x) sin(x) − F (x) cos(x))′^ = f sin(x). By the fundamental theorem of calculus,
∫ (^) π 0 f^ (x) sin(x)^ dx^ is an integer. Inequality (*) implies however that this integral is between 0 and 1 for large enough n. For such an n we get a contradiction.
E-320: Teaching Math with a Historical Perspective Oliver Knill, 2015
Lecture 7: Set Theory and Logic
We will mostly focus on the work of two mathematicians: Georg Cantor and Kurt G¨odel. Their mathematics changed our way we think about mathematics. In both cases, the mathematics community needed time to absorb the implications of the revolutions. Hilbert said about Cantor ”Nobody will drive us from the paradise that Cantor has created for us”. Cantor clarified the term ”cardinality” is, showed that certain infinities like that the cardinality of points in the plane or points in space are the same and most importantly showed that different infinities exist. G¨odels theorems show that mathematics and knowledge in general can not be exhausted by listing a sequence of basic truths from which everything follows. Whenever we make such a list, there are statements which are independent of the system. It would be a mistake to take this as a limitation of mathematics, in contrary it shows that mathematics is inexhaustible: there is always something more to explore.
Counting: Set theory
We first demonstrate that one can compute with sets like with numbers. There is an addition, the symmetric difference and a multiplication, the intersection. With these two operations, we prove the familiar rules of arithmetic
A + B = B + A, A · B = B · A, A · (B + C) = A · B + A · C
hold. This is a Boolean algebra. There is a set which plays the role of 0. Which one is it? There is also a set which plays the role of 1. Which one is it?
Counting: Hilbert’s Hotel
Hilbert’s hotel is located on route 8. It has countably many rooms numbered 1, 2 , 3 ,.. .. The hotel is fully booked. As a newcomer arrives. David, the hotel manager is mortified. David has an idea and moves guest in room i to room i + 1 and gives the newcomer the first room 1.
An other day, the hotel is empty but a large group arrives. They are the ”fractions” on their way to a cardinal match with the ”squares”. Can David accommodate them? He thinks hard and finally manages.
In the summer, the ”reals” appear. David is not there but has George, the apprentice is in the office. The group consists of all real numbers between 0 and 1. Can George accomodate them? As much as he tries to shift and renumber, he can not do it.
Counting: the interval
E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010-
Lecture 8: Probability theory
Probability theory is the science of chance. It starts with combinatorics and leads to a theory of stochastic processes. Historically, probability theory initiated from gambling problems as in Girolamo Cardano’s gamblers manual in the 16th century. A great moment of mathematics occurred, when Blaise Pascal and Pierre Fermat jointly laid a foundation of mathematical probability theory.
It took mathematicians longer to formalize ”randomness” precisely. Here is the setup as which it had been put forward by Andrey Kolmogorov: all possible experiments of a situation are modeled by a set Ω, the ”laboratory”. A measurable subset of experiments is called an ”event”. Measurements are done by real-valued functions X. These functions are called random variables and are used to observe the laboratory.
As an example, lets model the process of throwing a coin 5 times. An experiment is a word like httht, where h stands for ”head” and t represents ”tail”. The laboratory consists of all such 32 words. We could look for example at the event A that the first two coin tosses are tail. It is the set A = {ttttt, tttth, tttht, ttthh, tthtt, tthth, tthht, tthhh}. We could look at the random variable which assigns to a word the number of heads. For every experiment, we get a value, like for example, X[tthht] = 2.
In order to make statements about randomness, the concept of a probability measure is needed. This is a function P from the set of all events to the interval [0, 1]. It should have the property that P [Ω] = 1 and P [A 1 ∪ A 2 ∪ ...] = P [A 1 ] + P [A 2 ] + ..., if Ai are disjoint events.
The most natural probability measure on a finite set Ω is P [A] = ‖A‖/‖Ω‖, where ‖A‖ stands for the number of elements in A. It is the ”number of good cases” divided by the ”number of all cases”. For example, to count the probability of the event A that we throw 3 heads during the 5 coin tosses, we have |A| = 10 possibilities. Since the entire laboratory has |Ω| = 32 possibilities, the probability of the event is 10/32. In order to study these probabilities, one needs combina- torics:
How many ways are there to: The answer is: rearrange or permute n elements n! = n(n − 1)... 2 · 1 choose k from n with repetitions nk pick k from n if order matters (^) (n−n!k)!
pick k from n with order irrelevant
( n k
) = (^) k!(nn−!k)!
The expectation of a random variable E[X] is defined as the sum m =
∑ ω∈Ω X(ω)P^ [{ω}]. In our coin toss experiment, this is 5/2. The variance of X is the expectation of (X − m)^2. In our coin experiments, it is 5/4. Its square root is called the standard deviation. This is the expected deviation from the mean. An event happens almost surely if the event has probability 1.
An important case of a random variable is X(ω) = ω on Ω = R equipped with probability P [A] =
∫ A √^1 π e
−x^2 dx, the standard normal distribution. Analyzed first by Abraham de
Moivre in 1733, it was studied by Carl Friedrich Gauss in 1807 and therefore also called Gaussian distribution.
Two random variables X, Y are called decorrelated, if E[XY ] = E[X] · E[Y ]. If for any func- tions f, g also f (X) and g(Y ) are decorrelated, then X, Y are called independent. Two random variables are said to have the same distribution, if for any a < b, the events {a ≤ X ≤ b } and {a ≤ Y ≤ b } are independent. If X, Y are decorrelated, then the relation Var[X] + Var[Y ] = Var[X + Y ] holds which is just Pythagoras theorem, because decorrelated can be understood geometrically: X − E[X] and Y − E[Y ] are orthogonal. A common problem is to study the sum of independent random variables Xn with identical distribution. One abbreviates this IID. Here are the three most important theorems which we formulate in the case, whee all random variables are assumed to have expectatation 0 and standard deviation 1. Let Sn = X 1 + ... + Xn be the n’th sum of the IID random variables. It is also called a random walk.
LLN Law of Large Numbers assures that Sn/n converges to 0. CLT Central Limit Theorem:Sn/
n approaches the Gaussian distribution.
LIL Law of Iterated Logarithm: Sn/
√ 2 n log log(n) accumulates in [− 1 , 1].
The LLN shows that one can find out about the expectation by averaging experiments. The CLT explains why one sees the standard normal distribution so often. The LIL finally gives us a precise estimate how fast Sn grows. Things become interesting if the random variables are no more independent. Generalizing LLN,CLT,LIL to such situations is part of ongoing research.
Here are two open questions in probability theory:
Are π, e,
2 ... normal: do all digits appear with the same frequency? What growth rates Λn can occur in Sn/Λn having limsup 1 and liminf −1?
For the second question, there are examples for Λ√ n = 1 , λn = log(n) and of course λn =
n log log(n) from LIL if the random variables are independent. Examples of random variables
which are not independent are Xn = cos(n
Statistics is the science of modeling random events in a probabilistic setup. Given data points, we want to find a model which fits the data best. This allows to understand the past, pre- dict the future or discover laws of nature. The most common task is to find the mean and the standard deviation of some data. The mean is also called the average and given by m = (^) n^1
∑n k=1 xk. The variance is^ σ
n
∑n k=1(xk^ −^ m) (^2) with standard deviation σ.
A sequence of random variables Xn define a so called stochastic process. Continuous versions of such processes are where Xt is a curve of random random variables. An important example is Brownian motion, which is a model of a random particles.
Besides gambling and analyzing data, also physics was an important motor to develop proba- bility theory. An example is statistical mechanics where laws of nature are studied with prob- abilistic methods. A famous physical law is Ludwig Boltzmann’s relation S = k log(W ) for entropy, a formula which decorates Boltzmann’s tombstone. The entropy of a probability mea- sure P [{k}] = pk on a finite set { 1 , ..., n} is defined as S = −
∑n i=1 pi^ log(pi).^ Today, we would reformulate Boltzmann’s law and say that it is the expectation S = E[log(W )] of the logarithm of the ”Wahrscheinlichkeit” random variable W (i) = 1/pi on Ω = { 1 , ..., n }. Entropy is important because nature tries to maximize it
