Standard Properties (Axioms) of the number system
Natural numbers
The basis of all number systems is the natural numbers Þ = {1, 2, 3, 4, ... }. This is also the
simplest example of an infinite set.
The formal construction of this set according to axioms of set theory is the province of
mathematics and logic (but note we need a special axiom to guarantee the existence of an infinite
set).
Once we have the set sensibly defined it is quite easy to construct the usual operations: +, * and
verify they satisfy the standard properties:
a+b=b+a
a*b = b*a
æ a, b 0 Þ. (commutative)
æ a, b 0 Þ. (commutative)
(a + b) + c = a + (b + c)
(a*b)*c = a*(b*c)
æ a, b, c 0 Þ. (associative)
æ a, b, c 0 Þ. (associative)
a*(b + c) = a*b + a*c
æ a, b, c 0 Þ. (distributive)
Ordering. There exists an order relation < on Þ which satisfies
x < y or x = y or x> y
æ x, y 0 Þ
One important property of natural numbers which can be derived from the axioms of set theory
is
Well ordering. Every non-empty subset S of Þ has a minimum element which is in S.
This may seem fairly obvious but it is not true for the rationals for example. The well ordering
property is closely related to the
Principle of induction. If P is a proposition about natural numbers such that P(1) is true
and P(k) implies P(k+1) for all k 0 Þ then P(n) is true æ n 0 Þ.
Example. Let P(n) be the proposition that the sum of the first n natural numbers is ½n(n+1).
Then
1. The proposition is true for n = 1, because 1 = ½.1.2.
2. If 1 + 2 + ...+ k = ½k(k+1) then (by adding k+1 to both sides)
1
1 + 2 + ... k + (k+1) = ½k(k+1) + (k+1)
and ½k(k+1) + (k+1) = ½(k+1)(k+2), which is the formula with n = k+1.
Hence P(k) imples P(k+1). Hence, by the principle of induction, P(n) is true æ n 0 Þ. Exercise (Binomial theorem). Using the result
n
r
%
n
r&1
'
n%1
r
prove by induction on n that
n n&1
a b %
1
(a % b)n ' a n %
%
n n&2 2
a b % ... % b n
2
for any real a, b and natural number n.
Properties of natural numbers
The additive identity 1 can be used to generate Þ additively, 1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 4, etc.
But note that we can't do this for multiplication.
Subtraction. If a, b 0 Þ and a > b then ý a unique x 0 Þ such that b + x = a, but no such
x exists if a # b.
Divisibilty. We say a divides b (often written a * b) if ý c 0 Þ such that b = a*c.
Note. 1 divides every a 0 Þ since one can show from the axioms that a = 1*a æ a 0 Þ. For the
same reason every natural number divides itself.
Primes. A natural number p 0 Þ, p > 1, is called prime if the only natural numbers
dividing p are 1 and p. E.g. 2, 3, 5, 7, 11, 13, 17, 19, ... are primes.
Theorem. Every natural number n 0 Þ is uniquely representable as a product of primes
e
e
n ' p1 1 p2 2 ... ph
eh
(3)
The remainder theorem
We mention in passing the following useful theorem regarding natural numbers (and more
generally integers)
Theorem. For any pair of natural numbers x, y with 1 # x # *y* there exist a unique pair
(m, r) such that y = m x + r, and 0 # r < x.
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We can use the remainder theorem to construct a simple algorithm for constructing the highest
common factor of two integers (Euclid’s algorithm).
[See the Mathematica™ file euclid.ma]
Demonstration Euclid’s algorithm
To find the HCF of 10946 and 6765.
10946 = 1*6765 + 4181
6765 = 1*4181 + 2584
4181 = 1*2584 + 1597
2584 = 1*1597 + 987
1597 = 1*987 + 610
987 = 1*610 + 377
610 = 1*377 + 233
377 = 1*233 + 144
233 = 1*144 + 89
144 = 1*89 + 55
89 = 1*55 + 34
55 = 1*34 + 21
34 = 1*21 + 13
21 = 1*13 + 8
13 = 1*8 + 5
8 = 1*5 + 3
5 = 1*3 + 2
3 = 1*2 + 1
2 = 2*1 + 0
Conclusion: the HCF is 1.
Integers The integers Z = { ..., -3, -2, -1, 0, 1, 2, 3, ...} are constructed as an extension of the
natural numbers Þ so that:
æ a, b 0 Z ý x 0 Z such that b + x = a
Note. Given the natural numbers the formal construction of the integers is quite straight forward.
This idea of embedding one number system into a bigger one which has nicer properties is one
we shall see again when embedding the real numbers within the complex numbers. However,
it can only be taken so far.
The invention of zero (i.e. that 0 should be considered as a number) was a huge conceptual leap.
It is called the additive identity because 0 + a = a æ a 0 Z.
We identify that subset of the constructed set of integers which corresponds to the natural
numbers with the natural numbers. So that it is OK to write
ÞfZ
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Ordering. We can do this extension from Þ to Z in such a way that the natural ordering
on Þ extends nicely to Z. So, as you might expect ...-2 < -1 < 0 < 1 < 2 etc.
Identities. In Z we have addition and multiplication. We also have
0 + a = a æ a 0 Z and 1*a = a æ a 0 Z
Thus Z has both an additive identity and a multiplicative identity.
Rings
The set of integers form a simple example of a ring. A ring R is an algebraic structure which has:
Commutative and associative addition +.
Commutative and associative multiplication *.
Multiplication is distributive over addition.
ý 0 0 R such that a + 0 = a æ a 0 R.
ý 1 0 R such that a*1 = a æ a 0 R.
æ a 0 R ý a’ 0 R such that a + a’ = 0. This says there always exists an additive inverse.
Note. In a ring there is no guarantee of a multiplicative inverse. i.e. we cannot always solve a*x
= b for some x 0 R.
Representation of integers.
We generally use the decimal system to represent integers (and numbers in general) so for
example
345 = 3*100 + 4*10 + 5 = 3*102 + 4*101 + 5*100
The reason we use 10 as the base to which our numbers are represented is probably because we
have 10 fingers, but we can represent numbers using any natural number base.
In particular for computers it is particularly convenient to represent integers (numbers) to base
2, i.e. in binary.
0, 1, 10, 11, 100, 101, ...
Note. An integer has a unique logical identity which is the same no matter how it is represented.
Thus the set
Þ = {1, 2, 3, ... } = {1, 10, 11, ...}
where it is understood that the first description is in decimal and the second is in binary but that
the object being described is the same in both cases.
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Rational numbers.
It would be nice to have multiplicative inverses. We extend the ring Z to a set of numbers which
allows us to solve q*x = p, when q 3 0. The idea is to give a solution x = p/q. The technical
problem is that p/q does not yet have a meaning.
We define Q (the set of rationals) as the set of all ordered pairs (p, q) with q 0 Þ (notice
that this means q > 0) and p 0 Z.
Technically this is not difficult, and so we arrive at the rational numbers (fractions) which form
a field, which is a ring with the additional property that every element other than the additive
identity (zero) has a unique multiplicative inverse, i.e. q*x = p is soluble in Q for any q 3 0.
Equality. We define (p, q) –
= (r, s) if and only if p*s = r*q 0 Z.
Ordering. We define (p, q) –
< (r, s) if and only if p*s < q*r.
Addition. We define addition in Q by (p, q) £ (r, s) –
= (p*s + r*q, q*s) 0 Q
Multiplication. We define multiplication in Q by (p, q) ¡ (r, s) –
= (p*r, q*s)
Question. What are the additive and multiplicative identities in Q?
Properties of rational numbers.
With these definitions we can then check:
Exercise. Show that addition £ and multiplication ¡ in Q are commutative, associative and that
¡ is distributive over £.
Once these definitions have been checked we can make a natural embedding of a 0 Z <-> (a, 1)
0 Q.
The ordering on Q naturally extends the ordering of Z.
If we then identify Z with its identical image Z´ = {(a, 1) 0 Q : a 0 Z} we can then sensibly
abuse language and write Z f Q. So that
ÞfZfQ
Once all these details are taken care of we can drop the circles around addition, multiplication
and equals.
Cardinality
One way to tell if two sets have the same number of elements is to ‘pair elements off’ i.e. set up
a 1-1 correspondence. If we can do this we say the two sets have the same cardinality.
In fact this is the most primitive way to count.
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If two finite sets A, B have the same cardinality then they have the same number of elements,
n say. We can write Card(A) = Card(B) = n 0 Þ.
Theorem. If A and B are finite sets, if A f B and Card(A) = Card(B) then A = B.
Note. It is a rather counter intuitive fact that this theorem is false for infinite sets.
Definition. If there is a 1-1 correspondence between a set S and Þ we say that S has cardinality
)0 (aleph zero, aleph is the first letter of the Hebrew alphabet). If S has cardinality )0 we say S
is countable.
E.g. The most primitive of all infinite sets is Þ. Now consider S = {a2 : a 0 Þ} = {1, 4, 9, 16, ...}.
This is an infinite set and because we can set up the 1-1 correspondence a2 0 S <-> a 0 Þ it
follows Card(S) = Card(Þ) = )0. But obviously S 3 Þ.
Theorem. Card(Z) = )0, i.e. the integers are countable.
Proof. Consider the 1-1 correspondence
... &4, &3, &2, &1, 0,
1,
2,
3,
4, ...
...
2,
4,
6,
8, ...
9,
7,
5,
3, 1,
which establishes the assertion. It is a most remarkable and counter intuitive fact that
Theorem (Cantor). Card(Q) = )0, i.e. the rationals are countable.
Proof. Exercise.
It begins to look as if all infinite sets have cardinality )0, i.e. are countable. In fact this is not true
(as was also proved by Cantor); not all infinities are the same! We shall see why later.
Fields
The set of rationals form a simple example of a field. There are other systems which form rings
or fields.
Example. The ring of integers (mod n).
Write Zn = {0, 1, 2, 3, ... , n-1}. Any integer x when divided by n leaves one of these k (say)
numbers as a remainder. Under these circumstances we write x / k (mod n).
We can define addition £ in Zn by writing a £ b = c whenever a, b 0 Zn and a + b / c (mod n),
where c 0 Zn. Of course we need to check commutativity and associativity, but this turns out to
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be easy because these properties are inherited from Z.
Similarly, we can define a ¡ b = c to mean a * b / c (mod n), where c 0 Zn and again check that
the new multiplication is commutative and associative.
We finally have to check that ¡ is distributive over £. Again this poses no problem.
Thus Zn forms a ring.
It is an interesting question to ask:
When does Zn form a field?
I.e. When do we also have multiplicative inverses for all non-zero elements?
Example. Consider Z5 which we shall write as
F5 = {0, 1, 2, 3, 4}
In F5 we have 3 £ 4 = 2 because 3 + 4 = 7 and 7 / 2 (mod 5). Similarly, 3 ¡ 4 = 2 because
3 * 4 = 12 and 12 / 2 (mod 5).
Notice that 2 ¡ 3 = 1, 3 ¡ 2 = 1, and 4 ¡ 4 = 1 in Z5 so that every non-zero element has a
multiplicative inverse.
I.e. F5 is a field.
[See the Mathematica™ file finflds.ma]
Finite Fields.
Theorem. For any prime p 0 Þ we have Fp = Zp = {0, 1, 2, ..., p-1} is a field.
In particular when p = 2 we have the important field F2 = {0, 1}, with multiplication and addition
tables
Table 1 Addition in F2.
Table 2 Multi-plication in F2.
£
0
1
¡
0
1
0
0
1
0
0
0
1
1
0
1
0
1
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Giving a mathematical structure to the set {0, 1} allows us to develop useful ideas in Computer
Science. For example, in order to detect errors in data stored on magnetic media we might use
n + 1 bits to store n bits with the extra bit bn+1 defined in terms of b1, ..., bn as
bn+1 = b1 £ b2 £ ... £ bn
E.g n = 7 = 1110011 could be stored as 11100111 This enables a 1-bit error check on any data.
Real numbers
So far we have encountered sets Þ f Z f Q. It was known to the Greeks that whilst a square of
area 2 was a perfectly reasonable object it’s side length was not rational.
Theorem. %2 is not rational.
Proof. Suppose that %2 is rational, i.e. suppose
2 '
p
q
where p and q > 0 have no common factor other than 1.
Then
2 '
p2
q2
i.e.
p 2 ' 2(q 2
But this means p2 is even. Hence p is even (why?). Thus we can write p = 2*r, where r 0 Þ.
Substituting gives
4r 2 ' 2q 2
i.e.
2r 2 ' q 2
Hence q2 is even, so that q is even. But this is a contradiction against the fact that p and q have
no common factor. Hence the original hypothesis that %2 is rational must be false. %2 is an example of a real number which is not rational but this begs the question: what
is a real number?
Up to now we have extended the number system by essentially saying something like..
"Within this number system we cannot solve a certain type of equation, so let us try to
extend the system to a bigger one which includes the solution to the equations that are
not soluble within the present number system."
This approach is called building an algebraic extension.
Thus we might try to add the ‘solution’ to x2 = 2 to the rationals and get an extension, say Q(%2)
which covered this case. But there are several problems with this approach.
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E.g. %3, %5, ... are not rational and are not in Q(%2) (whatever that is exactly). We should
need to do this kind of extension an infinite number of times!
Even if we did that unfortunately, we still cannot get the reals from the rationals by using an
approach like this.
Dedekind was the first to perform a careful construction of the reals from the rationals (he was
teaching an introductory undergraduate course and realised that no one had ever done it
convincingly!).The construction is not terribly difficult but it is a bit finicky and more the kind
of thing that Mathematics students have to do for the good of their soul.
The basic idea is to define any ‘point gap’ in the ‘real line’ to be an equivalence class of
sequences of rationals which ‘pile up’ at the gap.
r1
r2
r3
r4
r5 r6 q6 q5 q4
q3
q2
q1
ε
Figure 1 Defining real numbers in terms of sequences of rationals.
The essential requirement for two sequences of rationals r1, r2, ... and q1, q2, ... to be equivalent
is that given any number > 0 we can make the difference *rn - qm* < simply by making both
n and m sufficiently large.
We denote the set of real numbers by ß.
The glb/lub principle for real numbers
The essential property possessed by the reals (but not by the rationals) can be summed up as
follows:
Let S be a non-empty bounded subset of the reals. We define the greatest lower bound
(glb) of S to be the real number L such that
(i) L # x æ x 0 S (L is a lower bound)
(ii) If y > L then ý x 0 S such that x < y (L is the greatest possible lower bound).
In a similar way we can define the least upper bound (lub).
Theorem. For every non-empty set S of real numbers lub(S) and glb(S) are real numbers.
This theorem (which we shall not attempt to prove) says essentially that ‘the reals haven't got
any holes’. Since that was how we were trying to construct the reals this is a reassuring result.
This is a very useful property which one can use in proving other important theorems.
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With some careful construction we can define reals with addition, multiplication and ordering
so that:
Addition and multiplication are commutative and associative.
Multiplication is distributive over addition.
The reals form a field.
There is a natural embedding of Q in ß, which is consistent with addition and
multiplication in Q, and the ordering of ß preserves the ordering of Q.
Thus we can sensibly abuse language by writing
ÞfZfQfß
Notice that, unlike the algebraic constructions of the integers and the rationals, the construction
of the reals involved a notion of nearness (a so called topological notion).
The power of the formal approach can be used to prove things from the axioms
that would otherwise have to be taken as obvious - sometimes that which may
seem obvious is actually wrong!
Theorem. 0*x = 0 æ x 0 ß.
Proof. By definition of zero 0 + x = 0 for all x. In particular 0 + 0 = 0. Now multiplying both
sides by x we have 0*x + 0*x = 0*x and
0(x % 0(x ' 1((0(x) % 1((0(x)
(multiplicative identity)
' (1 % 1)((0(x)
(distributive)
' 2((0(x)
(definition of 2)
Thus 0*x = 2*(0*x) and so adding the additive inverse of 0*x to both sides we obtain 0 = 0*x.
Theorem. (-1)*(-1) = 1.
Proof. We start from the observation that because -1 is by definition the additive inverse of 1
(which is the multiplicative identity) we have 1 + (-1) = 0. Now we have for any x 0 ß
0(x ' (1 % (&1))(x
(definition of 0)
' x((1 % (&1))
(multiplication is commutative)
' x(1 % x((&1)
(distributive)
' x % x((&1) ' 0 (previous theorem)
Hence x*(-1) must be the additive inverse of x i.e. x*(-1) = -x for any x 0 ß.
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But -(-x) + (-x) = 0 by definition of additive inverse. Hence (adding x to both sides) -(-x) = x æ
0 ß. Thus (putting x = 1) we have -(-1) = 1.
Now putting x = -1 in x*(-1) = -x we get (-1)*(-1) = -(-1) = 1, as required. Representation of real numbers
The thing to grasp about real numbers is that in general any one of them requires an infinite
number of natural numbers (or integers, or rationals) to specify it. In fact )0 bits are required.
The most convenient way to represent a real is by using an infinite sequence of rationals. We can
do this in many ways but to avoid confusion it is best to fix on a standard way.
Thus we usually represent a real in [0, 1] by
(0 # ai # 9)
0.a1a2a3...
which means
a1
101
%
a2
102
%
a3
103
% ...
although notice that this might involve an infinite sum(or, as it is called later, an infinite series)
and thus poses a question of interpretation - which we deal with much later. This notation is an
extension (to the right of the decimal point) of the decimal representation for integers.
Of course we can also use base 2 to represent reals.
Rational numbers have a decimal expansion which either terminates or recurs.
E.g.
1/2 = 0.5 or 0.499999...
1/3 = 0.333333....
Facts about real numbers.
The definition of a real number involves an infinite process (of successive approximation). The
set of real numbers is not just all the solutions of various algebraic equations such as
anx n % an&1x n&1 % ... % a0 ' 0
where the ai 0 Z or Q. Any number which satisfies such an equation is called an algebraic
number.
Examples of algebraic numbers are %2 (which is real) and %(-1) (which is not real).
There are also numbers (a huge infinity of them) which are real but not algebraic. Examples of
such numbers are = 3.14159... and e = 2.7182818... (it is actually quite hard to prove that any
particular number is not algebraic).
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Theorem. The algebraic numbers are countable.
Real numbers which are not algebraic are called transcendental.
Transcendental and algebraic numbers have infinite decimal expansions which do not
terminate and do not recur.
E.g.
= 3.14159... or %2 = 1.4142135...
Since there is generally no simple way to predict the decimal expansion of a given real number,
if we want to know one to a given precision we have to develop ways of calculating it (i.e.
algorithms). The algorithm terminates when we have computed the number to a specified
precision.
E.g
e ' 1 %
1
1
1
%
%
% ...
1!
2!
3!
where n! = n*(n-1)*(n-2)*...1 for n 0 Þ
We have the following remarkable theorem.
Theorem. (Cantor). The reals are not countable, i.e. cannot be put into 1-1
correspondence with the natural numbers.
Proof. Suppose the reals between [0, 1] were countable. Let an enumeration of all of them in
decimal notation be
0.a11a12a13a14a15
0.a21a22a23a24a25
0.a31a32a33a34a35
0.a41a42a43a44a45
...
...
...
...
...
where the convention is that any decimal ending in an infinite string of 9’s is replaced by one
ending in an infinite string of 0’s (this is just to avoid ambiguity).
Now consider the number x = 0.b1b2b3...where bi is defined by
bi '
1, if aii ' 0,
0, if 1 # aii # 9
Now we can be sure of three things:
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x is a real number in [0, 1]
x does not end in a string of 9’s
x is not contained in the enumeration
But this is a contradiction against the fact that the enumeration was supposed to include all reals
in [0, 1] whose decimal expansion does not end in an infinite string of 9’s.
Hence there can be no such enumeration. Hence the reals are not countable. We denote Card(ß) by c. We have established that in some sense )0 < c. Thus not all infinities
are the same. In fact there are infinitely many different kinds!
Powers We make the following definition for powers
Definition. For real a > 0 and x 0 ß we write
a x ' Exp[xLoge[a]]
This all looks a bit dodgy because we have not yet defined Exp, Log or e. We shall see later that
Exp is defined by an infinite series, Log is defined as the inverse function of Exp and e is just
Exp[1] . 2.7182818284...
To arrive at the ordinary laws of powers, such as
a x ( a y ' ax % y
we need the basic properties of Exp, in particular we need Exp[a]*Exp[b] = Exp[a + b], so that
a x ( a y ' Exp[x(Log[a]] ( Exp[y(Log[a]]
' Exp[x(Log[a] % y(Log[a]] ' Exp[(x % y)(Log[a]]
' ax % y
How these basic properties of the exponential function follow from its definition as a
power series will be sketched later in Analytic Math II.
At this point we note that there is no problem with positive or negative integral powers - we can
just define these in the ordinary way, e.g. a2 = a*a. (So we shall continue to use these kinds of
powers without comment.) For these cases the ordinary laws of powers obviously hold, and we
note that a0 = 1 for any a > 0.
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