What are the real numbers?
From a geometric point of view (and a historical one as well) real numbers are quantities, i.e., lengths of segments, areas of surfaces,
volumes of solids, etc.
For example, once we have settled on a unit of length,
i.e., a segment whose length we call 1, we can, using a compass
and straightedge, construct segments of any rational length k/n.k/n.
In some obvious sense then, the rational numbers
are real numbers. Apparently it was an intellectual shock to
the Pythagoreans to discover that there are some other real
numbers, the so-called irrational ones.
Indeed, the square root of 2 is a real number, since we
can construct a segment the square of whose length is 2
by making a right triangle each of whose legs
has length 1.
(By the Pythagorean Theorem of plane geometry, the square of the hypotenuse of this triangle must equal 2.)
And, Pythagoras proved that there is no rational number
whose square is 2, thereby establishing that there are real numbers tha are not rational.
See part (c) of Exercise 5.
Similarly, the area of a circle of radius 1 should be a real number;
i.e., ππ should be a real number.
It wasn't until the late 1800's that Hermite showed that ππ is not a rational number.
One difficulty is that to define ππ as the
area of a circle of radius 1
we must first define what is meant by the “ area" of a circle, and this
turns out to be no easy task.
In fact, this naive, geometric approach to the definition
of the real numbers turns out to be unsatisfactory in the sense that we are not able to prove or derive from these
first principles certain intuitively obvious arithmetic results.
For instance, how can we multiply or divide an area by a volume?
How can we construct a segment of length the cube root of 2?
And, what about negative numbers?
Let us begin by presenting two properties we expect any set
that we call the real numbers ought to possess.
We should be able to add, multiply, divide, etc., real numbers.
In short, we require the set of real numbers to be a field.
The second aspect of any set we think of as the real numbers is that it has some notion
of direction, some notion of positivity.
It is this aspect that will allow us to “compare” numbers, e.g., one number is larger than another.
The mathematically precise way to discuss this notion is the following.
- Definition 1:
A field FF is called an ordered field if there exists a subset P⊆FP⊆F that satisfies the following two properties:
- If x,y∈P,x,y∈P, then x+yx+y and xyxy are in P.P.
- If x∈F,x∈F, then one and only one of
the following three statements is true.
- x∈P,x∈P,
- -x∈P,-x∈P, and
- x=0.x=0.
(This property is known as the law of tricotomy.)
The elements of the set PP are called positive elements of F,F,
and the elements xx for which -x-x belong to PP are called negative elements of F.F.
As a consequence of these properties of P,P, we may introduce in FF a notion of order.
- Definition 2:
If FF is an ordered field, and xx and yy are elements of F,F, we say that x<yx<y if y-x∈P.y-x∈P. We say that x≤yx≤y if either x<yx<y or x=y.x=y.
We say that x>yx>y if y<x,y<x, and x≥yx≥y if y≤x.y≤x.
An ordered field satisfies the familiar laws of inequalities. They are consequences of the two properties of the set P.P.
Using the positivity properties above for an ordered field F,F, together with the axioms for a field, derive the familiar laws of inequalities:
- (Transitivity) If x<yx<y and y<z,y<z, then x<z.x<z.
- (Adding like inequalities) If x<yx<y and z<w,z<w, then x+z<y+w.x+z<y+w.
- If x<yx<y and a>0,a>0, then ax<ay.ax<ay.
- If x<yx<y and a<0,a<0, then ay<ax.ay<ax.
- If 0<a<b0<a<b and 0<c<d,0<c<d, then
ac<bd.ac<bd.
- Verify parts (a) through (e) with << replaced by ≤.≤.
- If xx and yy are elements of F,F, show that one and only one of the following three relations can hold: (i) x<y,x<y, (ii) x>y,x>y, (iii) x=y.x=y.
- Suppose xx and yy are elements of F,F, and assume that
x≤yx≤y and y≤x.y≤x. Prove that x=y.x=y.
- If FF is an ordered field, show that 1∈P;1∈P; i.e., that 0<1.0<1.
HINT: By the law of tricotomy,
only one of the three possibilities holds for 1.1.
Rule out the last two.
- Show that F7F7 of (Reference)
is not an ordered field; i.e., there is no subset P⊆F7P⊆F7
such that the two positivity properties can hold.
HINT: Use part (a) and positivity property (1).
- Prove that QQ is an ordered field,
where the set PP is taken to be the usual set of positive rational numbers.
That is, PP consists of those rational numbers a/ba/b for which both aa and bb
are natural numbers.
- Suppose FF is an ordered field and that xx is a nonzero element of F.F.
Show that for all natural numbers nnnx≠0.nx≠0.
- (e) Show that, in an ordered field, every nonzero square is positive;
i.e., if x≠0,x≠0, then x2∈P.x2∈P.
We remarked earlier that there are many different examples of fields,
and many of these are also ordered fields.
Some fields, though technically different from each other, are really
indistinguishable from the algebraic point of view, and
we make this mathematically precise with the following definition.
- Definition 3:
Let F1F1 and F2F2 be two ordered fields,
and write P1P1 and P2P2 for the set of positive elements in F1F1 and F2F2 respectively. A 1-1 correspondence JJ between F1F1 and F2F2 is called an isomorphism if
- J(x+y)=J(x)+J(y)J(x+y)=J(x)+J(y)
for all x,y∈F1.x,y∈F1.
- J(xy)=J(x)J(y)J(xy)=J(x)J(y)
for all x,y∈F1.x,y∈F1.
- x∈P1x∈P1 if and only if J(x)∈P2.J(x)∈P2.
REMARK. In general, if A1A1 and A2A2 are two algebraic systems, then a 1-1 correspondence between A1A1 and A2A2 is called an isomorphism
if it converts the algebraic structure on A1A1 into the
corresponding algebraic structure on A2.A2.
- Let FF be an ordered field. Define a function J:N→FJ:N→F by J(n)=n·1.J(n)=n·1. Prove that JJ is an isomorphism of NN onto a subset N˜N˜ of F.F.
That is, show that this correspondence is one-to-one
and converts addition and multiplication
in NN into addition and multiplication in F.F.
Give an example to show that this result is not true
if FF is merely a field and not an ordered field.
- Let FF be an ordered field.
Define a function J:Q→FJ:Q→F by J(k/n)=k·1×(n·1)-1.J(k/n)=k·1×(n·1)-1.
Prove that JJ is an isomorphism of the ordered field QQ
onto a subset Q˜Q˜ of the ordered field F.F.
Conclude that every ordered field FF contains a subset
that is isomorphic to the ordered field Q.Q.
REMARK. Part (b) of Item 23 shows that the ordered
field QQ is the smallest possible ordered field, in the sense that every other ordered field contains an isomorphic copy of Q.Q. However, as mentioned earlier, the ordered field QQ cannot suffice as the set of real numbers. There is no rational number whose square is 2, and we want the square root of 2 to be a real number. See Exercise 5 below. What extra property is there about an ordered field FF that will allow us to prove that numbers like 2,2,π,π, and so on are elements of F?F? It turns out that the extra property we need is related to a quite subtle point concerning upper and lower bounds of sets. It gives us some initial indication that the known-to-be subtle concept of a limit may be fundamental to our very notion of what the real numbers are.
- Definition 4:
If SS is a subset of an ordered field F,F, then
an element x∈Fx∈F is called an upper bound for SS if x≥yx≥y for every y∈S.y∈S. An element zz is called a lower bound
for SS if z≤yz≤y for every y∈S.y∈S.
A subset SS of an ordered field FF is called
bounded above if it has an upper bound;
it is called bounded below if it has a lower bound;
and it is called bounded if it has both an upper bound
and a lower bound.
An element MM is called the least upper bound or supremum of a set SS if it is an upper bound for SS and if M≤xM≤x for every other upper bound xx of S.S. That is, MM is less than or equal to any other upper bound of S.S.
Similarly, an element mm is called
the greatest lower bound or infimum of SS if
it is a lower bound for SS and if z≤mz≤m for every other lower bound zz of S.S.
That is, mm is greater than or equal to any other lower bound of S.S.
Clearly, the supremum and infimum of a set SS are unique.
For instance, if MM and M'M' are both least upper bounds of a set S,S,
then they are both upper bounds of S.S.
We would then have M≤M'M≤M' and M'≤M.M'≤M.
Therefore, by part (h) of Exercise 1, M=M'.M=M'.
It is important to keep in mind that an upper bound of a set SS need not be an element of S,S, and in particular, the least upper bound of SS may or may not actually belong to S.S.
If MM is the supremum of a set S,S, we denote MM by supS.supS. If mm is the infimum of a set S,S, we denote it by infS.infS.
- Suppose SS is a nonempty subset of an ordered field FF and that xx is an element of F.F.
What does it mean to say that “xx is not an upper bound for S?''S?''
- Let FF be an ordered field, and let SS
be the empty set, thought of as a subset of F.F.
Prove that every element x∈Fx∈F is an upper bound for SS
and that every element y∈Fy∈F is a lower bound for S.S.
HINT: If not, then what?
- If S=∅,S=∅, show that SS has no least upper bound and no greatest lower bound.
REMARK. The preceding exercise shows that peculiar things about upper and lower bounds happen when SS is the empty set.
One point is that just because a set has an upper bound
does not mean it has to have a least upper bound.
That is, no matter which
upper bound we choose, there is always another one that is strictly smaller.
This is a very subtle point, and
it is in fact quite difficult to give a simple concrete example of this phenomenon.
See the remark following (Reference).
However, part (d) of Exercise 5 contains the seed of an example.
A natural number aa is called even if
there exists a natural number cc such that a=2c,a=2c,
and aa is called odd if there
exists a natural number cc such that a=2c+1.a=2c+1.
- Prove by induction that every natural number is either odd or even.
- Prove that a natural number aa is even if and only if a2=a×aa2=a×a is even.
- Prove that there is no element xx of QQ
whose square is 2.
That is, the square root of 2 is not a rational number.
HINT: Argue by contradiction. Suppose there is a rational number
k/nk/n for which k2/n2=2,k2/n2=2,
and assume, as we may, that the natural numbers kk and nn have no common factor.
Observe that kk must be even, and then observe that nn also must be even.
- Let SS be the set of all positive rational numbers
xx for which x2=x×x<2.x2=x×x<2.
Prove that SS has an upper bound and a lower bound.
Can you determine whether or not SS has a least upper bound?
The existence of least upper bounds and greatest lower bounds of bounded sets
turns out to be the critical idea in defining the real numbers.
It is precisely the existence of such suprema and infimas that enables us to define as real numbers
quantities such as 2,2,π,π,e,e, and so on.
- Definition 5:
An ordered field FF is called complete
if every nonempty subset SS of FF that has an upper bound
has a least upper bound.
REMARK.
Although QQ is an
ordered field, we will see that it is not a complete ordered field. In fact, the answer to part (d) of Exercise 5 is no.
The set described there, though bounded above,
has no least upper bound.
In fact, it was one of nineteenth century mathematicians' major achievements to prove the following theorem.
There exists a complete ordered field.
We leave the proof of this theorem to the appendix.
Perhaps the most reassuring result along these lines is the following companion theorem,
whose proof we also leave to the appendix.
If F1F1 and F2F2 are two complete ordered fields,
then they are isomorphic.
Taken together, the content of the two preceding theorems is
that, up to isomorphism, there exists one and only one complete ordered field. For no other reason that that, this special field should be an important object in mathematics. Our definition of the real numbers is then the following:
- Definition 6:
By the set RR of real numbers we mean the (unique) complete ordered field.
