Download Numbers: Real, Imaginary, Complex, and beyond

Mathematical Structures:
Groups, Rings, and Fields
Roger House
Scientific Buzz Café
Coffee Catz
Sebastopol, CA
2008 September 18
Copyright © 2008 Roger House
1
Why haven't I ever heard of this?
“Nothing which is taught at a secondary
school was discovered later than the year
1800. ... Even among physicists, apart from
those who work in quantum theory or
relativity, I believe that those who do
experimental work use hardly any more
mathematics than was known to Maxwell in
1860.”
Jean Dieudonné (1906-1992)
Mathematics – The Music2of Reason
6 – Our first group
• We begin with a set of six numbers
6 = { 0, 1, 2, 3, 4, 5 }
• We define an operation on the set which we
call addition and denote by +
• This is similar to, but not really the same as,
the usual operation of addition
• To see this, let's add 1 to each element of
6
3
Add 1 to each element of 6
0+1=1
1+1=2
2+1=3
3+1=4
4+1=5
5+1=?
• What is 5 + 1?
• Usually it's 6, but there is no 6 in our set
4
• What to do?
Getting nothing from something
• Let's be bold and define
5+1=0
• This might seem unreasonable, but
consider what time it is 3 hours after 11am:
11 + 3 = 2
• Out of context this seems unreasonable
too, but when doing "clock arithmetic" it
makes sense
• Remember we said + is not exactly
addition
5
Add 2 to each element of 6
0+2=2
1+2=3
2+2=4
3+2=5
4+2=0
5+2=1
• Now we seem to have two strange results,
but keep thinking about clock arithmetic
6
Addition table for 6


0 1 2 3 4 5
1 2 3 4 5 0
2 3 4 5 0 1
3 4 5 0 1 2
4 5 0 1 2 3
5 0 1 2 3 4
Pick an element x in the left column and an
element y in the top row
7
Where the row and column meet is x + y
Addition made easy
• Given any two elements x and y of 6 the
addition table shows us how to add them
together to get the sum x + y
• There's another way to do it: Just add x
plus y in the usual way as if they were
ordinary whole numbers to get a sum z
If z < 6, then z is the sum
If z ≥ 6, then z – 6 is the sum
8
Examples of addition
3+2=5
3 + 3 = 6, so 6 - 6 = 0 is the sum
3 + 4 = 7, so 7 - 6 = 1 is the sum
4+1=5
4 + 4 = 8, so 8 - 6 = 2 is the sum
5 + 5 = 10, so 10 - 6 = 4 is the sum
Note how similar this is to clock arithmetic
9
0 is an identity


0+0=0
1+0=1
2+0=2
3+0=3
4+0=4
5+0=5
So x + 0 = x for every x
We say 0 is an identity element for +
10
Every element has an inverse

0+0=0
1+5=0
2+4=0
3+3=0
4+2=0
5+1=0
Every element x has an inverse element x'
such that
11
x + x' = 0
Be associative
• How to compute 5 + 4 + 2 ?
(5 + 4) + 2 = (3) + 2 = 5
5 + (4 + 2) = 5 + (0) = 5
• For any three elements x, y, z of 6 we
have
(x + y) + z = x + (y + z)
• Another way to say this: Addition is
associative
12
Can we prove + is associative?
• There are 6∙6∙6 = 216 different triples of
numbers from 6 and + appears 4 times in
(x + y) + z = x + (y + z)
 So by using the addition table 4∙216 = 864
times, we can prove that + is associative
 This is a proof by “main strength and
awkwardness”
 There are better ways
13
Where are we?



We have a set of six elements
6 = { 0, 1, 2, 3, 4, 5 }
We have an operation + defined on the set
which takes any two elements of the set
and combines them to produce another
element of the set
The operation is associative, so for any
three elements x, y, and z we have
(x + y) + z = x + (y + z)
14
Still in Kansas?
• An identity element 0 exists so that for any
element x we have
x+0=x
• Every element x has an inverse element x'
so that
x + x' = 0
• In short, we have a group
15
Groups
• Note that the word “group” is being used
here in a highly technical sense
• A group is a non-empty set with a binary
operation defined on it such that the
operation is associative, an identity element
exists, and every element has an inverse
• The group is one of the most fundamental
structures in mathematics
• Groups sprout like weeds in modern math
16
Examples of groups
• Let n = { 0, 1, 2, ..., n-1 } for any integer n
> 0, and define + in analogy with + for 6
• n is a group
• n is called the cyclic group of n elements
• Note that for every positive integer n there
exists at least one group with n elements
• So we don't have to worry about running
out of groups
17
Some groups go on forever ...
• Now consider  = { ..., -2, -1, 0, 1, 2, ... },
the set of all integers, and let + be the usual
addition of integers
• Is <,+> a group?
•
•
•
•
Is + a binary operation on ?
Is + associative?
Which element is the identity?
For x an integer, what is its inverse?
18
 is indeed a group
•
•
•
•
+ on integers is a binary operation
It is associative
0 is the identity element
For any integer x, its inverse is -x
•  is an infinite group, the first such we
have seen
19
Is <,*> a group?
• Now consider  and the operation *, the
usual multiplication of integers
• Is <,*> a group?
•
•
•
•
Is * a binary operation on ?
Is * associative?
Which element is the identity?
For x an integer, what is its inverse?
20
<,*> is NOT a group
•
•
•
•
•
* on integers is a binary operation
It is associative
1 is the identity element
For any integer x, its inverse is 1/x
Whoops! 1/x is not an integer (except when
x = -1 or x = +1), so the vast majority of
integers have no multiplicative inverses
21
Is <+,*> a group?
• Let + denote the set of all positive real
numbers, and let * be the usual
multiplication of real numbers
• Is <+,*> a group?
•
•
•
•
Is * a binary operation on +?
Is * associative?
Which element is the identity?
For x a positive real, what is its inverse?
22
<+,*> is a group
•
•
•
•
* on real numbers is a binary operation
It is associative
1 is the identity element
For any real x, its inverse is 1/x
• <+,*> is an infinite group, the second
such we have seen
• [It's way more infinite than <,+>, but that's
the subject of another talk.]
23
Time for some geometry
• All the groups we have seen so far are
arithmetic in nature
• It's time to consider a geometrical example
• We start with a regular hexagon
• hexa- means six (Greek)
• -gon means angled (Greek)
• regular means all sides are equal and all
angles are equal
24
A regular hexagon
2
1
3
0
4
5
25
Transformations of the hexagon
• We are going to consider transformations of
the hexagon which map vertices to vertices
• We will restrict our transformations to
counterclockwise rotations by multiples of
60º
• Any two rotations which map vertex 0 to the
same vertex v are considered identical
26
CCW rotations of the hexagon
• There are six possible rotations:
R0 – rotate the hexagon CCW by 0º
R1 – rotate the hexagon CCW by 60º
R2 – rotate the hexagon CCW by 120º
R3 – rotate the hexagon CCW by 180º
R4 – rotate the hexagon CCW by 240º
R5 – rotate the hexagon CCW by 300º
27
H - A set of rotations
• Let's define a set whose members are the
six rotations of the hexagon:
H = { R0, R1, R2, R3, R4, R5 }
• We define an operation on the set called
composition and denote it by *
• The composition Ri * Rj of two rotations is
simply the rotation obtained by first doing
rotation Ri and then doing rotation Rj
28
Examples of composition
R1 * R2 = R3
R 2 * R2 = R 4
R 3 * R4 = R 1
R5 * R4 = R3
R 3 * R3 = R 0
R 2 * R5 = R 1
29
R0 is an identity rotation


R0 * R0 = R 0
R1 * R0 = R 1
R2 * R0 = R 2
R3 * R 0 = R 3
R4 * R0 = R 4
R5 * R0 = R 5
So x * R0 = x for every x
R0 is an identity element for *
30
Every rotation has an inverse

R0 * R0 = R 0
R1 * R5 = R 0
R2 * R4 = R 0
R3 * R 3 = R 0
R4 * R2 = R 0
R5 * R1 = R 0
Every element x has an inverse element x'
x * x' = R0
31
Composition is associative
• How to determine R5 * R4 * R2 ?
(R5 * R4) * R2 = (R3) * R2 = R5
R5 * (R4 * R2) = R5 * (R0) = R5
• For any three elements x, y, z of H we have
(x * y) * z = x * (y * z)
• So composition is associative (but we
haven't proved this)
32
So, what have we got?



We have a set of six elements
H = { R0, R1, R2, R3, R4, R5 }
We have an operation * defined on the set
which takes any two elements of the set
and combines them to produce another
element of the set
The operation is associative, so for any
three elements x, y, and z we have
(x * y) * z = x * (y * z)
33
H is a group!
• An identity element R0 exists so that for
any element x we have
x * R0 = x
• Every element x has an inverse element x'
so that
x * x' = R0
• In short, H is a group!
34
Composition table for H
R0
R1
R2
R3
R4
R5
R1
R2
R3
R4
R5
R0
R2
R3
R4
R5
R0
R1
R3
R4
R5
R0
R1
R2
R4
R5
R0
R1
R2
R3
R5
R0
R1
R2
R3
R4
35
Is H starting to look familiar?
• Consider this mapping:
0 ↔ R0
1 ↔ R1
2 ↔ R2
3 ↔ R3
4 ↔ R4
5 ↔ R5
36
Some correspondences
1+2=3
2+2=4
3+4=1
5+4=3
3+3=0
2+5=1
↔ R 1 * R 2 = R3
↔ R2 * R 2 = R 4
↔ R3 * R 4 = R 1
↔ R 5 * R 4 = R3
↔ R3 * R 3 = R 0
↔ R2 * R 5 = R 1
37
It's déjà vu all over again
• Given a true statement about 6 replace
each digit i by Ri and replace + by * and we
have a true statement about H
• And vice versa
• We say that 6 and H are isomorphic
<6,+>  <H,*>
 iso- means same (Greek)
 -morph means shape or form (Greek)
38
<6,+> is isomorphic to <H,*>
• When two mathematical objects are
isomorphic, they have the same structure
• When a statement is proved true about one
structure, the equivalent statement is true
about the other structure
• Sometimes one structure is easier to work
with than another
• Addition in 6 may be easier than rotating
cardboard hexagons
39
More geometry
• Let's consider another geometrical example
• This time we'll work with a regular triangle,
also called an equilateral triangle
• tri- means three (Latin)
• regular means all sides are equal and all
angles are equal, so each angle is 60º
40
An equilateral triangle
B
A
C
41
Transformations of the triangle
• We're going to consider transformations of
the triangle which map vertices to vertices
• One kind of transformation is a
counterclockwise rotation by a multiple of
120º
• Another kind of transformation is a flip
around an axis through a vertex,
perpendicular to the side opposite
42
Rotations and flips
There are six such transformations:
R0 – rotate CCW by 0º
R1 – rotate CCW by 120º
R2 – rotate CCW by 240º
F0 – flip around the axis defined by A
F1 – flip around the axis defined by B
F2 – flip around the axis defined by C
43
Permutations, anyone?
Another view of the transformations:
R0 ≡ ABC
R1 ≡ CAB
R2 ≡ BCA
F0 ≡ ACB
F1 ≡ CBA
F2 ≡ BAC
44
S3 – Triangle transformations
• Let's define a set whose members are the
six transformations of the triangle:
S3 = { R0, R1, R2, F0, F1, F2 }
• We define an operation on the set called
composition and denote it by *
• The composition t*s of two transformations
is simply the transformation obtained by
first doing transformation t followed by
transformation s
45
Examples of composition
R1 * R2 = R0
R 2 * R2 = R 1
F0 * R1 = F2
F2 * F1 = R1
R1 * F2 = F0
F2 * R1 = F1
46
R0 is an identity


R0 * R0 = R 0
R1 * R0 = R 1
R2 * R0 = R 2
F0 * R 0 = F 0
F1 * R 0 = F 1
F2 * R 0 = F 2
So x * R0 = x for every x
R0 is an identity element for *
47
Every element has an inverse

R0 * R0 = R 0
R1 * R2 = R 0
R2 * R1 = R 0
F0 * F 0 = R 0
F1 * F1 = R 0
F2 * F2 = R 0
Every element x has an inverse element x'
x * x' = R0
48
Composition is associative
• How to determine F2 * R2 * F0 ?
(F2 * R2) * F0 = (F0) * F0 = R0
F2 * (R2 * F0) = F2 * (F2) = R0
• For any three elements x, y, z of H we have
(x * y) * z = x * (y * z)
• So composition is associative (but we
haven't proved this)
49
We've been here before



We have a set of six elements
S3 = { R0, R1, R2, F0, F1, F2 }
We have an operation * defined on the set
which takes any two elements of the set
and combines them to produce another
element of the set
The operation is associative, so for any
three elements x, y, and z we have
(x * y) * z = x * (y * z)
50
S3 is a group
• An identity element R0 exists so that for
any element x we have
x * R0 = x
• Every element x has an inverse element x'
so that
x * x' = R0
• In short, S3 is a group!
51
Composition table for S3
R0
R1
R2
F0
F1
F2
R1
R2
R0
F2
F0
F1
R2 F0
R 0 F1
R 1 F2
F1 R0
F2 R1
F0 R2
F1
F2
F0
R2
R0
R1
F2
F0
F1
R1
R2
R0
52
Something a bit odd
You may have noticed something of interest
in earlier examples of composition:
R1 * F2 = F0
F2 * R1 = F1
What is a bit strange about this?
What is going on here?
53
S3 is noncommutative
• Many operations in mathematics are
commutative, i.e., the order of the
operands does not matter:
15 + 6 = 6 + 15 = 21
15 * 6 = 6 * 15 = 90
• But composition on S3 is noncommutative:
R1 * F2 ≠ F2 * R1
 Be careful not to assume commutativity
54
S3 is nonabelian
• When a group is commutative, it is said to
be abelian
• When a group is noncommutative, it is said
to be nonabelian
• The term abelian is derived from the name
of the Norwegian mathematician, Niels
Henrik Abel (1802-1829), who did
significant mathematics before dying at a
young age of tuberculosis
55
Is S3 really a new group?
• We have now seen three groups of order
six: 6, H, and S3
• We saw that 6 and H are isomorphic, so
they are essentially the same
• What about S3?
• Is S3 isomorphic to 6 or H?
• If S3 is isomorphic to one of 6 or H, it's
isomorphic to the other also
56
Let's try a correspondence
If S3 is isomorphic to 6 then there is a
correspondence like this one
0 ↔ R0
1 ↔ R1
2 ↔ R2
3 ↔ F0
4 ↔ F1
5 ↔ F2
57
It doesn't work out like we want
• All true statements in one group map to true
statements in the other group
• We know 1 + 2 = 3 in 6
• The corresponding statement in S3 is
R 1 + R2 = F0
• But this is false because
R 1 + R2 = R 0
And R0 corresponds to 0 in 6, not to 3
58
Are we done yet?
• Does this mean S3 is not isomorphic to 6?
• No, not at all; we tried one correspondence
between elements of S3 and 6 and it didn't
work out, but there are a bunch more
possible correspondences to try
• How many?
• There are 6! = 6∙5∙4∙3∙2∙1 = 720 ways to
order six things, so we only have 719 to go
59
Flips are their own inverses



Let's not use main strength and awkwardness
Elements of S3 which are their own inverses:
R0 * R0 = R 0
F0 * F 0 = R 0
F1 * F1 = R 0
F2 * F2 = R 0
Elements of 6 which are their own inverses:
0+0=0
60
S3   6
• So S3 has four elements which are their
own inverses, and 6 has only one such
element
• Thus there is no possible correspondence
between elements of the two groups which
will preserve true statements
• S3 and 6 are NOT isomorphic
61
How many groups are there?
• So, now we have two essentially different
groups of order six: S3 and 6
• Are there other groups of order six?
• No, there are only these two
• This raises an interesting question: How
many groups are there of
order 1?
order 2?
...
62
No. of groups of a given order
Ord
1
2
3
4
5
6
7
#
1
1
1
2
1
2
1
Ord
8
9
10
11
12
13
14
#
5
2
2
1
5
1
2
Ord #
15
1
16 14
17
1
18
5
19
1
20
5
21
2
63
How many groups did you say?
• For every positive integer n there is at least
one group, namely the cyclic group n
• If p is prime there is only one group of order
p, namely, p
• The general case is very complex:
n = 510 = 2∙3∙5∙17:
8 groups
n = 511 = 7∙73:
1 group
n = 512 = ?:
10,494,213 groups
n = 513 = 3∙3∙3∙19:
15 groups
64
What about multiplication?
• A group has only one operation, usually
called addition (but, as we have seen, it is
not necessarily the usual addition we are
used to)
• In many cases of interest there are two
operations: addition and multiplication
• Let's return to our favorite structure, 6,
and see if we can define multiplication for it
65
Multiplication made easy
• Denote multiplication by *
• For x and y in , we define x * y like this:
• Multiply x times y in the usual way as if they
were ordinary whole numbers to get a
product z
If z < 6, then z is the product
If z ≥ 6, then subtract 6 from z
repeatedly until a number < 6
results; it is the product
66
Examples of multiplication
2*2=4
2 * 3 = 6, so 6 - 6 = 0 is the product
3 * 3 = 9, so 9 - 6 = 3 is the product
3 * 4 = 12, so 12 - 6 - 6 = 0 is the product
1*5=5
4 * 4 = 16, so 16 - 6 - 6 = 4 is the product
5 * 5 = 25, so 25 - 6 - 6 - 6 - 6 = 1 is the
product
67
Multiplication table for 6
*|
0|
1|
2|
3|
4|
5|
0
0
0
0
0
0
0
1
0
1
2
3
4
5
2
0
2
4
0
2
4
3
0
3
0
3
0
3
4
0
4
2
0
4
2
5
0
5
4
3
2
1
68
1 is a unity


0*1=0
1*1=1
2*1=2
3*1=3
4*1=4
5*1=5
So x * 1 = x for every x
We say 1 is a unity for *
69
Keep on associating
• How to compute 5 * 4 * 2 ?
(5 * 4) * 2 = (2) * 2 = 4
5 * (4 * 2) = 5 * (2) = 4
• For any three elements x, y, z of 6 we
have
(x * y) * z = x * (y * z)
• Another way to say this: Multiplication is
associative (we have not proved this)
70
Better commute too
• How do 5 * 4 and 4 * 5 relate?
5*4=2
4*5=2
• For any two elements x, y of 6 we have
x*y=y*x
• Another way to say this: Multiplication is
commutative (we have not proved this)
71
The distributive law
• How to compute 5 * (4 + 3) ?
5 * (4 + 3) = (5 * 4) + (5 * 3)
=2+3=5
• For any three elements x, y, z of 6 we
have
x * (y + z) = (x * y) + (x * z)
• Another way to say this: Multiplication
distributes over addition (we have not
proved this)
72
Definition of a ring
• A ring <R,+,*> is a non-empty set R
together with two operations + and *, called
addition and multiplication, such that
• <R,+> is an abelian group
• Multiplication is associative
• Multiplication distributes over addition
• A commutative ring with unity is a ring in
which multiplication is commutative and
there exists a unity element
73
<6,+,*> is a commutative ring with
unity
• We have seen that <6,+> is an abelian
group (remember abelian means + is
commutative)
• * is associative
• * distributes over +
• * is commutative
• 1 is a unity
• <6,+,*> is a commutative ring with unity
74
Examples of Rings
• Let n = { 0, 1, 2, ..., n-1 } for any integer n
> 0, and define + and * in analogy with +
and * for 6
• <n,+,*> is a commutative ring with unity
• So there are an infinite number of such
rings
• There are also infinite rings, e.g.,  with the
usual addition and multiplication
75
Matrices
• Let Mn be the set of all nn matrices of real
numbers
• Here are some typical elements of M2:
1 4
3 2
0 2
2 1
 We can define addition and multiplication of
these elements in such a way that M2 is a
ring with unity
76
More about matrices
• The identity element for addition is
0 0
0 0
 The unity element for multiplication is
1 0
0 1
 Matrix multiplication is NOT commutative
77
What about division?
• In a group we can add (not necessarily the
kind of addition we use in daily life)
• We can also subtract, because every
element has an inverse, and adding the
inverse of x is the same as subtracting x
• In a ring, we can multiply
• But can we divide?
78
Be careful!
• Looking back at the multiplication table for
6, we see that
2*3=0
• Let's divide both sides by 3 to get
2=0
• NO! NO! NO! NO! NO! NO! NO! NO!
• What's the problem?
79
A ring divided ...
• In order to divide by an element x, we must
have an element x' such that x * x' = 1, i.e.,
we need a multiplicative inverse for x
• Then, to divide by x, we simply multiply by
the multiplicative inverse x'
• Looking back at the multiplication table for
6, we see that only 1 and 5 have
multiplicative inverses
• So division in 6 is just not going to work
80
If 6 doesn't work, try 5
• Let's look at <5,+,*>, a commutative ring
with unity consisting of 5 elements 0, 1, 2,
3, and 4
• We can add and multiply in this ring much
like we did in 6: When a value is greater
than or equal to 5, we subtract 5 repeatedly
until we get 0, 1, 2, 3, or 4
• Let's look at the multiplication table for 5
81
Multiplication table for 5
*| 0 1 2 3 4
0| 0 0 0 0 0
1| 0 1 2 3 4
2| 0 2 4 1 3
3| 0 3 1 4 2
4| 0 4 3 2 1
How is this different from the multiplication
table for 6?
82
Unity at last
• In every row of the table except the first, we
see that the multiplicative unity 1 appears
• This means that every element except 0
has a multiplicative inverse:
1 * 1 = 1
2 * 3 = 1
3 * 2 = 1
4 * 4 = 1
83
We can divide in 5
To divide by 1:
To divide by 2:
To divide by 3:
To divide by 4:
Multiply by 1
Multiply by 3
Multiply by 2
Multiply by 4
To divide by 0: Multiply by ???
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Don't divide by nada
• Don't ever, ever, ever divide by zero
• Division is only defined when the divisor is
non-zero
• So dividing by zero is meaningless
• [But can't we define x/0 =  ?]
• [Yes, but that opens up another can of
worms which we are not dealing with today]
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What's your field?
• Given a non-zero element x of a ring, if
there exists an element x-1 such that x * x-1
= 1, then x-1 is the multiplicative inverse
of x
• A field is a commutative ring with unity in
which every non-zero element has a
multiplicative inverse
• In a field we can not only add, subtract, and
multiply, we can also divide
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Examples of fields
• 5 is a field
• p is a field for every prime p, so there are
an infinite number of finite fields
• Is , the set of all integers, a field?
• Is , the set of all rational numbers, a field?
• Is , the set of all real numbers, a field?
• Is , the set of all complex numbers, a
field?
87
A subset of the real numbers
• Consider the set S = { a+b√2 | a,b   }
• Some elements of S: 2+√2, ½ +3√2, 5
• Note that S is bigger than  and smaller
than :  S 
• Define addition and multiplication on S as
the usual operations on the real numbers
• Is S a field?
• What are the requirements on a set in order
for it to be a field?
88
We'll just do the fun ones
• Because  is a field and S is a subset of
, most of the requirements for S to be a
field are easily shown
• There are three things, however, which we
must carefully check:
1. S is closed under +
2. S is closed under *
3. Every non-zero element of S
has a multiplicative inverse
89
You do the third one
• 1. (a+b√2) + (c+d√2) = (a+c)+(b+d)√2
• 2. (a+b√2) * (c+d√2) = ac + ad√2 + bc√2 +
bd√2√2 = (ac+2bd) + (ad+bc)√2
• 3. The multiplicative inverse of a+b√2 is
a/d - (b/d)√2, where d = a2 - 2b2
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Fields, fields, everywhere
•
•
•
•
So S is a subfield of 
Are there other subfields of ?
Yes, bijillions of them
Are there other fields besides the ones
we've talked about?
• Yes, bijillions of them
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What have we seen?
• Groups – structures in which we can add
and subtract
• Rings – structures in which we can add,
subtract, and multiply
• Fields – structures in which we can add,
subtract, multiply, and divide
• Some are finite, some are infinite
• They are abstractions of concrete number
systems like , , , and 
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Who cares?
“… Monsieur Fourier was of the opinion that
the principal aim of Mathematics is to serve
mankind and to explain natural phenomna;
but a philosopher such as he ought to have
known that the sole aim of science is the
fulfillment of the human spirit, and that,
accordingly, a question about numbers has
as much significance as a question about
the workings of the world.”
C.G.J.Jacobi (1804-1882)
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In closing …
“In mathematics you don't understand things.
You just get used to them.”
John von Neumann (1903-1957)
“It's so complicated …”
Mick Jagger (1943-)
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