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Book: What Is ADE? Drew Armstrong Section 1: What Is a Number? Version: July 12, 2013

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Book: What Is ADE? Drew Armstrong Section 1: What Is a Number? Version: July 12, 2013 Book: What is ADE? Drew Armstrong Section 1: What is a number? Version: July 12, 2013 Question: What is a number? Answer: R, C, H, and sometimes O. (1.A) In order of logical dependence, we have \numbers that can be added and multiplied" N = f0 < 1 < 2 < · · · g; followed by \numbers that can be subtracted" Z = {· · · < −2 < −1 < 0 < 1 < 2 < · · · g; followed by \numbers that can be divided" na o = : a; b 2 ; b 6= 0 ; Q b Z a c where we declare that the abstract symbols b and d are equal if and only if ad = bc. In this book we also want to know when two numbers are \close together", so we need some kind of topology. The idea of topology arises in the transition from N to Z. Recall that Z is defined as the set of ordered pairs 2 Z := N = f(a; b): a; b 2 Ng; where we declare (a; b) and (c; d) equal if and only if a + d = b + c. We are supposed to think that (a; b) = \a − b". Then the absolute value function Z ! Z≥0 is defined by ( (a; b) if a ≤ b j(a; b)j := ; (b; a) if b ≤ a as you know. This function extends to Q ! Q≥0 in the natural way by defining a jaj := 2 : b jbj Q≥0 Now things get a bitp tricky. If Q is going to be our definition of number, then many interesting things like 2 are unfortunately not numbers. People puzzled over this for a long time until Cantor came up with the following (admittedly quite scary) solution. We say that (xi)i = x1; x2; x3;::: 2 Q is a Cauchy sequence if for each (presumably small) rational number " 2 Q>0 there exists a natural number N" 2 N such that for all natural numbers m; n > N" we have jxm − xnj < ". The thing about a Cauchy sequence is that it seems to be going somewhere. Cantor's idea was to arbitrarily declare that it is going somewhere. We define the set R := f(xi)i in Q :(xi)i is a Cauchy sequence g; and we declare that (xi)i and (yi)i are equal in R if their componentwise difference tends to zero|that is, if for each " 2 Q>0 there exists N" 2 N such that for all n > N" we have jxn − ynj < ". We are supposed to think that (xi)i = \x1". Now this is a real number system. (1.B) We say that R is the topological completion of Q with respect to the absolute value function j · j : Q ! Q≥0. More generally, a function k · k : Q ! Q is called a norm if it satisfies the following three properties: • 8 x 2 Q; kxk = 0 , x = 0, • 8 x; y 2 Q; kxyk = kxkkyk, and • 8 x; y 2 Q; kx + yk ≤ kxk + kyk. Exercise: Show that these three properties imply that kxk ≥ 0 for all x 2 Q. One can also fiddle around with the definitions to show that the absolute value is a norm. It turns out that the norm axioms are the only properties needed in Cantor's construction of R from Q. Thus, if one is worried about the ontological status of R, one might wonder if there are other (inequivalent) norm functions on Q. Indeed there are. We define the trivial norm on Q by ( 0 x = 0 kxk0 := : 1 x 6= 0 I wouldn't worry about that, but there are worse things. Let p 2 N be a prime number. Then for each nonzero x 2 Q we can factor p from the n a numerator and denominator to write x = p b for some n 2 Z with a; b; p coprime. We define the p-adic norm on Q by ( 0 x = 0 kxkp := : p−n x 6= 0 Exercise: Show that the p-adic norm is a norm. We use the notation k · k1 for the usual absolute value, because it acts like that. Ostrowski's Theorem (1916). The inequivalent norms on Q are precisely k · k0; k · k1; and k · kp for p prime: Let Qs denote the topological completion of Q with respect to the norm k · ks. Then Q0 = Q, Q1 = R, and Qp is called the field of p-adic numbers. I will have no more to say about p-adic numbers, because I am not qualified. The absolute value j · j = k · k1 is sometimes called the Archimedean norm because it satisfies the additional (Archimedean) property • 8 0 6= x 2 Q; 9 n 2 N; k x + x + ··· + x k > 1, | {z } n times and the other norms don't. (Roughly, this says that there are no infinitesimal numbers.) So if you are worried about the ontological status of R, you can say something like \it is the unique complete Archimedean field”. But don't worry. (To paraphrase Hadamard:) All of this is merely to sanction and legitimize our intuition about the continuity of a piece of string, and there never was any other reason for it. (1.C) Using R as a foundation, we can build three more amazing number systems. Here I will primarily follow Chapter 20 of Stillwell (2001), a book which I strongly recommend. For more references, see the Notes. The complex numbers C are the first exceptional structure in mathematics. (My use of the term \exceptional" will hopefully become clear later.) Apparently, they were first observed by Diophantus, who knew that (a2 + b2)(α2 + β2) = (aα ∓ βb)2 + (bα ± βa)2: We will call this the two-square identity. In geometric terms, let (a; b) denote the right angled triangle with side lengths a and b. If (a; b) and (α; β) are two such triangles, then we have a rule for producing a third triangle (aα − βb; bα + βa); whose hypotenuse is the product of the hypotenuses of (a; b) and (α; β). It is literally unbelievable how much of modern mathematics comes from this simple rule. Following Hamilton (1835), we will define the complex numbers as pairs of real numbers 2 C := R = f(a; b): a; b; 2 Rg with componentwise addition and with the strange product (a; b) × (α; β) := (aα − βb; bα + βa): Exercise: Explain why the distributive law holds. We can extend the absolute value from 2 2 2 R to C by defining j(a; b)j := a + b , and the two-square identity tells us that this is a norm. Furthermore, we can define the conjugation map (a; b)∗ := (a; −b) and observe that (a; b) × (a; b)∗ = j(a; b)j2: It follows that we can divide by nonzero complex numbers 1 (a; b)∗ a −b = = ; ; (a; b) j(a; b)j2 a2 + b2 a2 + b2 hence C is a field. We can also think of C as a 2-dimensional vector space over R with basis 1 = (1; 0) and i = (0; 1), so that (a; b) = a1 + bi. Then for each (a; b) 2 C, the multiplication map (α; β) 7! (a; b) × (α; β) is linear and we may identify (a; b) with the 2 × 2 matrix a −b 1 0 0 −1 (a; b) = = a + b = a1 + bi: b a 0 1 1 0 Note that 0 −12 −1 0 i2 = = = −1: 1 0 0 −1 2 We will call this a Hermitian structure on R . If you already believe in linear algebra then this is the correct definition of C because it explains the properties of the conjugate (a; b)∗ = (a; b)> (transpose matrix) and norm j(a; b)j = det(a; b) (determinant). Taking all of this structure into account, we can say that C is a real normed division algebra. (1.D) Hamilton was so fascinated by the complex numbers that he spent at least 13 years (from 1830 to 1843) trying to multiply triples of real numbers, in order to define hypercomplex numbers. Had he been a better student of number theory, he wouldn't have bothered. For example, we have 3 = 12 + 12 + 12 and 5 = 02 + 12 + 22; but it is easy to show that 15 = 3 · 5 is not the sum of three integer squares. This implies that there is no such thing as a three-square identity, and hence it is impossible to 3 give R the structure of a normed division algebra. But we are happy that Hamilton didn't know this. In frustration, he was eventually willing to throw away the commutative property of multiplication. On the afternoon of Monday, October 16, 1843, he was walking along the Royal Canal in Dublin with his wife, on the way to a meeting of the Royal Irish Academy, when he suddenly realized that there is a reasonable way to multiply quadruples of real numbers. After recording this insight in his notebook he got out his knife and carved the following formulas into the stone of the Brougham Bridge: i2 = j2 = k2 = ijk = −1: (?) Implicit in these formulas are the equations ij = −ji = k; jk = −kj = i; ki = −ik = j; which we can visualize in the following way: Hamilton had just invented vector calculus|not to mention vectors. To be specific, Hamilton defined the set of quaternions 4 H := R = fa1 + bi + cj + dk : a; b; c; d 2 Rg: The relations (?) induce an R-bilinear associative product operation, which means that the quaternions can be represented as matrices. If you believe in the complex number i 2 C then, following Cayley (1858), we can identify each quaternion with a complex 2 × 2 matrix: a1 + bi c1 + di a1 + bi + cj + dk = −c1 + di a1 − bi 1 0 i 0 0 1 0 i = a + b + c + d : 0 1 0 −i −1 0 i 0 Warning: i 2 H is not the same as i 2 C.
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