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- From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz)
- Newsgroups: sci.math,news.answers,sci.answers
- Subject: sci.math FAQ: Fundamentals
- Followup-To: sci.math
- Date: 17 Feb 2000 22:51:57 GMT
- Organization: University of Waterloo
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- Summary: Part 1 of 31, New version
- Keywords: Algebraic structures/What are numbers
- Originator: alopez-o@neumann.uwaterloo.ca
- Originator: alopez-o@daisy.uwaterloo.ca
- Xref: senator-bedfellow.mit.edu sci.math:347413 news.answers:177533 sci.answers:11241
-
- Archive-name: sci-math-faq/numbers
- Last-modified: February 20, 1998
- Version: 7.5
-
- FUNDAMENTALS
-
-
- _________________________________________________________________
-
- Algebraic structures
-
- We will attempt to give a brief explanation of the following concepts:
- * N is a monoid
- * Z is an integral domain
- * Q is a field
- * in the field R the order is complete
- * the field C is algebraically complete
-
- If you have been asked by a child to give them arithmetic problems, so
- they could show off their newly learned skills in addition and
- subtraction I'm sure that after a few problems such as: 2 + 3, 9 - 5,
- 10 + 2 and 6 - 4, you tried tossing them something a little more
- difficult: 4 - 7 only to be told `` That's not allowed.''
-
- What you may not have realized is that you and the child did not just
- have different objects in mind (negative numbers) but entirely
- different algebraic systems. In other words a set of objects (they
- could be natural numbers, integers or reals) and a set of operations,
- or rules regarding how the numbers can be combined.
-
- We will take a very informal tour of some algebraic systems, but
- before we define some of the terms, let us build a structure which
- will have some necessary properties for examples and counterexamples
- that will help us clarify some of the definitions.
-
- We know that any number that is divided by six will either leave a
- remainder, or will be divided exactly (which is after all the
- remainder 0). Let us write any number by the remainder n it leaves
- after division by six, denoting it as [ n ]. This means that, 7, 55
- and 1 will all be written [1], which we call the class to which they
- all belong: i.e. 7 in [1], 55 in [1], or, a bit more technically, they
- are all equivalent to 1 modulo 6. The complete set of class will
- contain six elements, and this is called partitioning numbers into
- equivalent classes because it separates (or partitions) all of our
- numbers into these classes, and any one number in a class is
- equivalent to any other in the same class.
-
- One interesting thing we can do with these classes is to try to add or
- to multiply them. What can [1] + [3] mean? We can, rather naively try
- out what they mean in ``normal'' arithmetic: [1] + [3] = [1 + 3] =
- [4]. So far so good, let us try a second example 25 in [1] and 45 in
- [3], their sum is 70 which certainly belongs to [4]. Here we see what
- we meant above by equivalence, 25 is equivalent to 1 as far as this
- addition is concerned. Of course this is just one example, but
- fortunately it can be proven that the sum of two classes is always the
- class of the sums.
-
- Now this is the kind of thing we all do when we add hours for example,
- 7 (o' clock) plus 6 hours is 1 (o' clock), and all we are really doing
- is adding hours (modulo 12).
-
- The neat part comes with multiplication, as we will see later on. But
- for now just remember, it can be proven that something like [4] x [5]
- = [2] will work: the product of two classes is the class of the
- product.
-
- Now for some of the necessary terminology.
-
-
- Monoids and Groups
-
- We need to define a group.
-
- Let us take a set of objects and a rule (called a binary operation)
- which allows us to combine any two elements of this set. Addition is
- an example from math, or ANDing in some computer language.
-
- The set must be closed under the operation. That means that when two
- elements are combined the result must also be in the set. For example
- the set containing even numbers will always give us an even number
- when two elements are added together. But if we restrict ourselves to
- odd numbers, their sum is not an odd number and so we know right off
- the bat that the set of odd numbers and addition cannot constitute a
- group. Some books will consider closure in the definition of binary
- operation, and others add it as one of the requirements for a group
- along with the ones that follow below.
-
- The set and the operation is called a group if the binary operation
- satisfies the following criteria:
- * the operation is associative, which means it doesn't matter how
- you group the elements you are operating on, for example in our
- set of remainders: [1] + ([3] + [4]) = ([1] + [3]) + [4]
- * there is an identity element, meaning: one of the elements
- combined with the others in the set doesn't change them in the
- least. For example the zero in addition, or the one in
- multiplication.
- * every element has an inverse with respect to that operation. If
- you combine an element and its inverse you get the identity (of
- that operation) back.
-
- (Be careful with this last one, -3 is the inverse of 3 in addition,
- since they give us 0 when added, but 1/3 is the inverse of 3 with
- respect to multiplication, since 3 x 1/3 = 1 the identity under
- multiplication.)
-
- So we can see that the set of natural numbers N (with the operation of
- addition) is not even a group, since there is no inverse for 5, for
- example. (In other words there is no natural number which added to 5
- will give us zero.) And so the third rule for our operation is
- violated. But it still has some structure, even if it is not as rich
- as the ones we'll see later on.
-
- Sets with an associative operation (the first condition above) are
- called semigroups, and if they also have an identity element (the
- second condition) then they are called monoids.
-
- Our set of natural numbers under addition is then an example of a
- monoid, a structure that is not quite a group because it is missing
- the requirement that every element have an inverse under the operation
- (Which is why in elementary school 4 - 7 is not allowed.)
-
- What about the set of integers, is it a group?
-
- By itself this question is nonsensical. Why? Well, we have not
- mentioned under what operation. OK, let us say: the set of integers
- with addition.
-
- Now, addition is associative, the zero does not change any number when
- added to it, and for every number n we can add -n and get zero. So
- it's a group all right.
-
- In fact it is a special kind of group. When we can perform the
- operation on the two elements in any order (e.g a + b = b + a) then
- the group is called commutative, or Abelian in honor of Abel. Not
- every operation is commutative, for example three minus two is
- certainly not the same as two minus three. Our set of integers under
- addition is then an Abelian group.
-
- Rings
-
- If we take an Abelian group (remember: a set with a binary operation)
- and we define a second operation on it we get a bit more of a
- structure than we had with just a group.
-
- If the second operation is associative, and it is distributive over
- the first then we have a ring. Note that the second operation may not
- have an identity element, nor do we need to find an inverse for every
- element with respect to this second operation. As for what
- distributive means, intuitively it is what we do in math when perform
- the following change: a x (b + c) = (a x b) + (a xc).
-
- If the second operation is also commutative then we have what is
- called a commutative ring. The set of integers (with addition and
- multiplication) is a commutative ring (with even an identity - called
- unit element - for multiplication).
-
- Now let us go back to our set of remainders. What happens if we
- multiply [5] x [1]? We see that we get [5], in fact we can see a
- number of things according to our definitions above, [5] is its own
- inverse, and [1] is the multiplicative element. We can also show
- easily enough (by creating a complete multiplication table) that it is
- commutative. But notice that if we take [3] and [2], neither of which
- are equal to the class that the zero belongs to [0], and we multiply
- them, we get [3]x[2] = [0]. This bring us to the next definition. In a
- commutative ring, let us take an element which is not equal to zero
- and call it a. If we can find a non-zero element, say b that combined
- with a equals zero ( a x b = 0) then a is called a zero divisor.
-
- A commutative ring is called an integral domain if it has no zero
- divisors. Well the set Z with addition and multiplication fullfills
- all the necessary requirements, and so it is an integral domain.
- Notice that our set of remainders is not an integral domain, but we
- can build a similar set with remainders of division by five, for
- example, and voil`, we have an integral domain.
-
- Let us take, for example, the set Q of rational numbers with addition
- and multiplication - I'll leave out the proof that it is a ring, but I
- think you should be able to verify it easily enough with the above
- definitions. But to give you a head start, notice the addition of
- rationals follow all the requirements for an abelian group. If we
- remove the zero we will have another abelian group, and that implies
- that we have something more than a ring, in fact, as we will see in
- the next section.
-
- Fields
-
- Now we can make one step further. If the elements of a ring, excluding
- the zero, form an abelian group (with the second operation) then it is
- a field. For example, write the multiplication table of the remainders
- of division by 5, and you will see that it satisfies all the
- requirements for a group: (You will probably have noticed that the
- group does not contain the number five itself since [5] = [0].)
- (tabular)(c | c c c c) 1 2 3 4 ; 1 2 3 4 ; 2 2 4 1 3 ; 3 3 1 4 2 ; 4 4
- 3 2 1(tabular)
-
- (Why isn't the set of divisors of six - excluding the zero and under
- multiplication - a group? That's easy enough, since we have excluded
- the zero we do not have the result of [2] x [3] in our set, so it
- isn't closed.)
-
- Ordering
-
- Given a ring, we can say that it is ordered when you have a special
- subset of that ring behaves in a very special way. If any two elements
- of that special subset are added or multiplied their sum and their
- product are again in the special subset. Take the negative numbers in
- R , can they be that special subset? Well the sum seems to be
- allright, it is also a negative number. But things don't work with the
- product: it is positive. What about the positive numbers? Yep, and in
- fact we call that special subset, the set of positive elements. Now,
- we gave the definition for an ordered ring, we can also define an
- ordered field the same way.
-
- But what does a complete ordered field mean? Well the definition looks
- rather nasty: it is complete if every non-empty subset which posesses
- an upper bound has a least upper bound.
-
- Let's translate some of that, trying to lose as little information on
- the way. A bound is something that guarantees that all of the elements
- of your set are on one side of it (reasonably enough). For example,
- certainly all negative reals are less than 100, so 100 is a bound (it
- is in fact an upper bound 'cause all negatives are ``below'' it). But
- there are lots of other bounds, 1, 5, 26 will all do nicely. The
- question now is, of all of these (upper bounds) which is the smallest,
- that is which one is ``the border'' so to speak? Does it always
- exists?
-
- Let's take the rationals, and look at the following numbers:
-
- 1.4, \; 1.41, \; 1.414, \; 1.4142, \; 1.41421, ...
-
- Now each of these is a rational number (it can be written as a
- fraction), and they are getting closer and closer to a number we've
- probably seen before (just take out your calculator and find the
- square root of two). So we can write the shorthand for this series as
- sqrt(2). Certainly we can find an upper bound for this series, 3 will
- do nicely, but so can 1.5, or 1.42. But what is the smallest. Well
- there isn't any. Not among the rational at least, because no matter
- what fraction you give I can give you one closer to the square root of
- two. What about the square root of two itself? Well it's not a
- rational number (I'll skip the proof, but it is really rather easy) so
- you can't use it. If you want another series which is really neat look
- at the section on ``Euler's formula'' in the FAQ.
-
- And that is where the reals come in. Any set or reals that is bounded
- you can certainly find the smallest of these bounds. (By the way this
- ``least upper bound'' is abbreviated ``l.u.b.'', or ``sup'' for
- supremum.) We can also turn things around and talk of lower bounds,
- and of the largest of these etc. but most of that will be just a
- mirror image of what we have dealt with so far.
-
- So that should be it. And for years that did seem to be it, we seemed
- to have all the numbers we'd ever care to have.
-
- There was just one small stick in the works, but most people just sort
- of pretended not to notice, and that was that not all polynomials had
- solutions. One simple polynomial of this kind is x^2 + 1 = 0. It's so
- simple, yet there's no self respecting number that would solve this
- polynomial. There were these funny answers which seemed like they
- should be solutions but no one could make any sense out of them, so
- they were considered imaginary solutions. Which was really too bad
- because they were given the name of imaginary numbers and now that the
- name stuck we realize that they are numbers just as good as any of the
- ones we have been using for centuries. And in fact that takes us to
- the last great pinnacle in this short excursion. The field of complex
- numbers.
-
- We can define an algebraically closed field as a field where every
- nonconstant polynomial (i.e. one with an x in it from high school
- days) has a zero in the field. Whew! This in short means that as long
- as the polynomial is not a constant number (which is no fun anyways)
- but something which looks like it wants a solution, like 5 x^3 - 2 x^2
- + 6 = 0 it will always have one, if you are working with complex
- numbers and not just reals.
-
- There is another definition which is probably just as good, but may or
- may not be easier: A field is algebraically closed if every polynomial
- splits into linear factors. Linear factors are briefly factors not
- containing x to any power of two or higher, in other words in the
- form: ax + b. For example x^2 + x - 6 can be factored as (x + 3)(x -
- 2), but if we are in the field of reals we cannot factor x^2 + 1, but
- we can in the field of complex numbers: x^2 + 1 = (x - i)(x + i),
- where, you may recall, i^2 = -1.
-
-
- _________________________________________________________________
-
- What are numbers?
-
- Introduction
-
- Informally:
- * N = { 0,1,... } or N = { 1,2,... }
- Wether 0 is in N depends on where you live and what is your field
- of interest. At the informal level it is a religious topic.
- * Z = { ..., - 1,0,1,... }
- * Q = { p/q | p, q in Z and q != 0 }
- * R = { d_0.d_1d_2... | d_0 in Z and 0 <= d_i <= 9 for i > 0 }
- * C = { a + b o i | a, b in R and i^2 = -1 }
-
- Construction of the Number System
-
- Formally (following the mainstream in math) the numbers are
- constructed from scratch out of the axioms of Zermelo Fraenkel set
- theory (a.k.a. ZF set theory) [Enderton77, Henle86, Hrbacek84]. The
- only things that can be derived from the axioms are sets with the
- empty set at the bottom of the hierarchy. This will mean that any
- number is a set (it is the only thing you can derive from the axioms).
- It doesn't mean that you always have to use set notation when you use
- numbers: just introduce the numerals as an abbreviation of the formal
- counterparts.
-
- The construction starts with N and algebraically speaking, N with its
- operations and order is quite a weak structure. In the following
- constructions the structures will be strengthen one step at the time:
- Z will be an integral domain, Q will be a field, for the field R the
- order will be made complete, and field C will be made algebraically
- complete.
-
- Before we start, first some notational stuff:
- * a pair (a,b) = { { a } , { a,b } } ,
- * an equivalence class [a] = { b | a == b } ,
- * the successor of a is s(a) = a U { a } .
-
- Although the previous notations and the constructions that follow are
- the de facto standard ones, there are different definitions possible.
-
- Construction of N
-
- * { } in N
- * if a in N then s(a) in N
- * N is the smallest possible set such that the preceding rules hold.
-
- Informally n = { 0,...,n - 1 } (thus 0 = { } , 1 = { 0 } , 2 = { 0,1 }
- , 3 = { 0,1,2 } ). We will refer to the elements of N by giving them a
- subscript _n. The relation <_n on N is defined as: a_n <_n b_n iff a_n
- in b_n. We can define +_n as follows:
- * a_n +_n 0_n = a_n
- * a_n +_n s(b_n) = s(a_n +_n b_n)
-
- Define *_n as:
- * a_n *_n 0_n = 0_n
- * a_n *_n s(b_n) = (a_n *_n b_n) +_n a_n
-
- Construction of Z
-
- We define an equivalence relation on N x N: (a_n,b_n) ==_z(c_n,d_n)
- iff a_n +_n d_n = c_n +_n b_n. Note that ==_z ``simulates'' a
- subtraction in N . Z = { [(a_n,b_n)]_z | a_n, b_n in N } . We will
- refer to the elements of Z by giving them a subscript _z. The elements
- of N can be embedded as follows: embed_n : N --> Z such that
- embed_n(a_n) = [(a_n,0_n)]_z. Furthermore we can define:
- * [(a_n,b_n)]_z <_z [(c_n,d_n)]_z iff a_n +_n d_n <_n c_n +_n b_n
- * [(a_n,b_n)]_z +_z [(c_n,d_n)]_z = [(a_n +_n c_n, b_n +_n d_n)]_z
- * [(a_n,b_n)]_z *_z [(c_n,d_n)]_z =
- [((a_n *_n c_n) +_n (b_n *_n d_n), (a_n *_n d_n) +_n (c_n *_n
- b_n))]_z
-
- Construction of Q
-
- We define an equivalence relation on Z x (Z { 0_z }): (a_z,b_z) ==_q
- (c_z,d_z) iff a_z *_z d_z = c_z *_z b_z. Note that ==_q ``simulates''
- a division in Z . Q = { [(a_z,b_z)]_q | a_z in Z and b_z in Z { 0_z }
- } . We will refer to the elements of Q by giving them a subscript _q.
- The elements of Z can be embedded as follows: embed_z : Z --> Q such
- that embed_z(a_z) = [(a_z,1_z)]_q. Furthermore we can define:
- * [(a_z,b_z)]_q <_q [(c_z,d_z)]_q iff a_z *_z d_z <_z c_z *_z b_z
- when 0_z <_z b_z and 0_z <_z d_z
- * [(a_z,b_z)]_q +_q [(c_z,d_z)]_q = [((a_z *_z d_z) +_z (c_z *_z
- b_z), b_z *_z d_z)]_q
- * [(a_z,b_z)]_q *_q [(c_z,d_z)]_q = [(a_z *_z c_z, b_z *_z d_z)]_q
-
- Construction of R
-
- The construction of R is different (and more awkward to understand)
- because we must ensure that the cardinality of R is greater than that
- of Q .
- Set c is a Dedekind cut iff
- * { } subset c subset Q (strict inclusions!)
- * c is closed downward:
- if a_q in c and b_q <_q a_q then b_q in c
- * c has no largest element:
- there isn't an element a_q in c such that b_q <_q a_q for all b_q
- != a_q in c
-
- You can think of a cut as taking a pair of scissors and cutting Q in
- two parts such that one part contains all the small numbers and the
- other part contains all large numbers. If the part with the small
- numbers was cut in such a way that it doesn't have a largest element,
- it is called a Dedekind cut. R = { c | c is a Dedekind cut } . We will
- refer to the elements of R by giving them a subscript _r. The elements
- of Q can be embedded as follows: embed_q : Q --> R such that
- embed_q(a_q) = { b_q | b_q <_q a_q } . Furthermore we can define:
- * a_r <_r b_r iff a_r subset b_r (strict inclusion!)
- * a_r +_r b_r = { c_q +_q d_q | c_q in a_r and d_q in b_r }
- * -_r a_r = ; { b_q | there exists an c_q in Q such that b_q <_q c_q
- and (-1)_q *_q c_q in a_r }
- * |a_r|_r = a_r U -_r a_r
- * *_r is defined as:
- + if not a_r <_r 0_r and not b_r <_r 0_r
- then a_r *_r b_r = 0_r U { c_q *_q d_q | c_q in a_r and d_q
- in b_r }
- + if a_r <_r 0_r and b_r <_r 0_r then a_r *_r b_r = |a_r|_r *_r
- |b_r|_r
- + otherwise a_r *_r b_r = -_r (|a_r|_r *_r |b_r|_r)
-
- There exists an alternative definition of R using Cauchy sequences: a
- Cauchy sequence is a s : N --> Q such that s(i_n) +_q((-1)_q *_q
- s(j_n)) can be made arbitrary near to 0_q for all sufficiently large
- i_n and j_n. We will define an equivalence relation ==_r on the set of
- Cauchy sequences as: r ==_r s iff r(m_n) +_q((-1)_q *_q s(m_n)) can be
- made arbitrary close to 0_q for all sufficiently large m_n. R = {
- [s]_r | s is a Cauchy sequence } . Note that this definition is close
- to ``decimal'' expansions.
-
- Construction of C
-
- C = R x R. We will refer to the elements of C by giving them a
- subscript _c. The elements of R can be embedded as follows: embed_r :
- R --> C such that embed_r(a_r) = (a_r,0_r). Furthermore we can define:
- * (a_r,b_r) +_c (c_r,d_r) = (a_r +_r c_r, b_r +_r d_r)
- * (a_r,b_r) *_c (c_r,d_r) = ((a_r *_r c_r) +_r -_r (b_r * d_r), (a_r
- *_r d_r) +_r (b_r *_r c_r))
-
- There exists an elegant alternative definition using ideals. To be a
- bit sloppy: C = R [x]/< (x *_r x) +_r 1_r > , i.e. C is the resulting
- quotient ring of factoring ideal < (x *_r x) +_r 1_r > out of the ring
- R [x] of polynomials over R . The sloppy part is that we need to
- define concepts like quotient ring, ideal, and ring of polynomials.
- Note that this definition is close to working with i^2 = -1: (x *_r x)
- +_r 1_r = 0_r can be rewritten as (x *_r x) = (-1)_r.
-
- Rounding things up
-
- At this moment we don't have that N is a subset of Z , Z of Q , etc.
- But we can get the inclusions if we look at the embedded copies of N ,
- Z , etc. Let
- * N' = ran embed_r o embed_q o embed_z o embed_n
- * Z' = ran embed_r o embed_q o embed_z
- * Q' = ran embed_r o embed_q
- * R' = ran embed_r
-
- For these sets we have N' subseteq Z' subseteq Q' subseteq R' subseteq
- C. Furthermore these sets have all the properties that the
- ``informal'' numbers have.
-
- What's next?
-
- Well, for some of the more alien parts of math we can extend this
- standard number system with some exotic types of numbers. To name a
- few:
- * Cardinals and ordinals
- Both are numbers in ZF set theory [Enderton77, Henle86, Hrbacek84]
- and so they are sets as well. Cardinals are numbers that represent
- the sizes of sets, and ordinals are numbers that represent well
- ordered sets. Finite cardinals and ordinals are the same as the
- natural numbers. Cardinals, ordinals, and their arithmetic get
- interesting and ``tricky'' in the case of infinite sets.
- * Hyperreals
- These numbers are constructed by means of ultrafilters [Henle86]
- and they are used in non-standard analysis. With hyperreals you
- can treat numbers like Leibnitz and Newton did by using
- infinitesimals.
- * Quaternions and octonions
- Normally these are constructed by algebraic means (like the
- alternative C definition that uses ideals) [Shapiro75, Dixon94].
- Quaternions are used to model rotations in 3 dimensions.
- Octonions, a.k.a. Cayley numbers, are just esoteric artifacts :-).
- Well, if you know where they are used for, feel free to contribute
- to the FAQ.
- * Miscellaneous
- Just to name some others: algebraic numbers [Shapiro75], p-adic
- numbers [Shapiro75], and surreal numbers (a.k.a. Conway numbers)
- [Conway76].
-
- Cardinals and ordinals are commonly used in math. Most mortals won't
- encounter (let alone use) hyperreals, quaternions, and octonions.
-
- References
-
- J.H. Conway. On Numbers and Games, L.M.S. Monographs, vol. 6. Academic
- Press, 1976.
-
- H.B. Enderton. Elements of Set Theory. Academic Press, 1977.
-
- G.M. Dixon. Division Algebras; Octonions, Quaternions, Complex Numbers
- and the Algebraic Design of Physics. Kluwer Academic, 1994.
-
- J.M. Henle. An Outline of Set Theory. Springer Verlag, 1986.
-
- K. Hrbacek and T. Jech. Introduction to Set Theory. M. Dekker Inc.,
- 1984.
-
- L. Shapiro. Introduction to Abstract Algebra. McGraw-Hill, 1975.
-
- This subsection of the FAQ is Copyright (c) 1994, 1995 Hans de
- Vreught. Send comments and or corrections relating to this part to
- J.P.M.deVreught@cs.tudelft.nl
- _________________________________________________________________
-
- --
- Alex Lopez-Ortiz alopez-o@unb.ca
- http://www.cs.unb.ca/~alopez-o Assistant Professor
- Faculty of Computer Science University of New Brunswick
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