The real and complex number systems

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Chapter 1: The real and complex number systems Contents
Contents
 1.  Introduction
 2.  Ordered sets
 3.  Fields
 4.  The real field
 5.  The extended real number system
 6.  The complex field
 7.  Euclidean spaces

1. Introduction A satisfactory discussion of the main concepts of analysis must be based on an accurately defined number concept. We shall not, however, enter into any discussion of the axioms that govern the arithmetic of the integers, but assume familiarity with the rational numbers. Thus we have \mathbb{N}, \mathbb{Z}, \mathbb{Q}. The rational number system is inadequate for many purposes. This leads to the introduction of so-called "irrational numbers" which are often written as infinite decimal expansions. We may say the sequence 1,1.4,1.41,1.414,\dots "tends to \sqrt{2}". But unless the irrational number \sqrt{2} has been clearly defined, the question must arise: Just what is it that this sequence "tends to"? This sort of question can be answered as soon as the so-called "real number system" is constructed.

We can show that there is no rational p such that p^2=2. Now examine this situation a little more closely. Let A be the set of all positive rationals p such that p^2<2 and let B consist of all positive rationals p such that p^2>2. We shall show that A contains no largest number and B contains no smallest. More explicitly, for every p in A we can find a rational q in A such that p<q, and for every p in B we can find a rational q in B such that q<p. In fact, we can let
q=p-\frac{p^2-2}{p+2}=\frac{2p+2}{p+2}.

It can be easily checked that this q satisfies the conditions we want.

The purpose of the above discussion has been to show that the rational number system has certain gaps, in spite of the fact that between any two rationals there is another: If r<s then r<(r+s)/2<s. The real number system fills these gaps. This is the principal reason for the fundamental role which it plays in analysis.

In order to elucidate its structure, as well as that of the complex numbers, we start with a brief discussion of the general concepts of ordered set and field.
2. Ordered sets

The statement "x<y" may be read as "x is less than y" or "x is smaller than y" or "x precedes y".

It is often convenient to write y > x in place of x<y. The notation x \leqslant y indicates that x<y or x = y, without specifying which of these two is to hold. In other words, x \leqslant y is the negation of x > y.

For example, \mathbb{Q} is an ordered set if r <s is defined to mean that s - r is a positive rational number.

Lower bounds are defined in the same way.

\mathbb{Q} does not have the least-upper-bound property.

There is a close relation between greatest lower bounds and least upper bounds, and that every ordered set with the least-upper bound property also has the greatest-lower-bound property.

Proof. L is nonempty. Every x\in B is an upper bound of L. Thus L is bounded above, and L has a supremum \alpha in S. \alpha\in L, so \alpha is a lower bound of B, but \beta is not if \beta>\alpha. This means that \alpha= \inf B.

3. Fields

The field axioms clearly hold in \mathbb Q if addition and multiplication have their customary meaning. Thus \mathbb Q is a field.

Statement (1) is a cancellation law. Note that (2) asserts the uniqueness of the element whose existence is assumed in (A4), and that (3) does the same for (A5).

For example, \mathbb{Q} is an ordered field.

All the familiar rules for working with inequalities apply in every ordered field: Multiplication by positive [negative] quantities preserves [reverses] inequalities, no square is negative, etc. The following proposition lists some of these.

We now state the existence theorem which is the core of this chapter.

The second statement means that \mathbb{Q} \subset \mathbb{R} and that the operations of addition and multiplication in \mathbb{R}, when applied to members of \mathbb{Q}, coincide with the usual operations on rational numbers; also, the positive rational numbers are positive elements of \mathbb{R}.

The members of \mathbb{R} are called real numbers.

The proof of this theorem is rather long and a bit tedious. The proof actually constructs \mathbb{R} from \mathbb{Q}.

The next theorem could be extracted from this construction with very little extra effort. However, we prefer to derive it from Theorem theorem141 since this provides a good illustration of what one can do with the least-upper-bound property.

Part (1) is usually referred to as the archimedean property of \mathbb{R}. Part (2) may be stated by saying that \mathbb{Q} is dense in \mathbb R: Between any two real numbers there is a rational one.

The proof of theorem theorem143 shows how the difficulty pointed out in the Introduction (irrationality of \sqrt{2}) can be handled in \mathbb R.

{Decimals}
We conclude this section by pointing out the relation between real numbers and decimals.

Let x > 0 be real. Let n_0 be the largest integer such that n_0 \leqslant x. (Note that the existence of n_0 depends on the archimedean property of \mathbb R.) Having chosen n_0 , n_1, \dots , n_{k-1}, let n_k be the largest integer such that
n_0+\frac{n_1}{10}+\dots+\frac{n_k}{10^k}\leqslant x.

Let E be the set of these numbers
\begin{equation}
n_0+\frac{n_1}{10}+\dots+\frac{n_k}{10^k}\quad (k=0,1,\dots)
\end{equation}
Then x=\sup E. The decimal expansion of x is
\begin{equation}
n_0.n_1n_2n_3\cdots
\end{equation}

Conversely, for any infinite decimal (12) the set E of numbers (11) is bounded above, and (12) is the decimal expansion of \sup E.

Since we shall never use decimals, we do not enter into a detailed discussion.

{The construction of \mathbb{R}}
Theorem theorem141 will be proved here briefly by constructing \mathbb R from \mathbb Q. The "cuts" in \mathbb Q which we used later were invented by Dedekind. In another construction, each real number is defined to be an equivalence class of Cauchy sequences of rational numbers. The construction of \mathbb R from \mathbb Q by means of Cauchy sequences is due to Cantor. Both Cantor and Dedekind published their constructions in 1872.

We shall divide the construction into several steps.

Step 1.

The members of \mathbb R will be certain subsets of \mathbb Q, called cuts. A cut is, by definition, any set \alpha \subset \mathbb Q with the following three properties.

1. \alpha is not empty, and \alpha\neq\mathbb Q.
   
2. If p\in \alpha,\, q\in\mathbb{Q} and q<p, then q\in\alpha.
   
3. If p\in\alpha, then p<r for some r\in\alpha.

The letters p, q, r, \dots will always denote rational numbers, and \alpha, \beta, \gamma, \dots will denote cuts.

Note that (3) simply says that \alpha has no largest member; (2) implies two facts which will be used:

If p\in\alpha and q\notin\alpha then p<q. If r\notin \alpha and r<s then s\notin\alpha.

Step 2.

Define "\alpha<\beta" to mean: \alpha is a proper subset of \beta.

It can be checked that this meets the requirements of Definitiondef11. Thus \mathbb R is now an ordered set.

Step 3.

The ordered set \mathbb R has the least-upper-bound property.

Step 4.

If \alpha\in\mathbb R and \beta\in\mathbb R we define \alpha+\beta to be the set of all sums r + s, where r \in\alpha and s\in\beta.

We define 0^\ast to be the set of all negative rational numbers. It is clear that 0^\ast is a cut. We can verify that the axioms for addition hold in \mathbb R, with 0^\ast playing the role of 0.

Step 5.

Having proved that the addition defined in Step 4 satisfies Axioms (A)
of Definition def16, it follows that Proposition prop131 is valid in \mathbb R, and we can prove one of the requirements of Definition def17: If \alpha,\beta,\gamma\in\mathbb R and \beta<\gamma, then \alpha+\beta<\alpha+\gamma.

It also follows that \alpha > 0^\ast if and only if -\alpha<0^\ast.

Step 6.

Multiplication is a little more bothersome than addition in the present context, since products of negative rationals are positive. For this reason we confine ourselves first to \mathbb R^+, the set of all \alpha\in\mathbb R with \alpha> 0^\ast.

If \alpha,\beta\in\mathbb{R}^+, we define \alpha\beta to be the set of all p s.t. p\leqslant rs for some choice of r\in\alpha,\, s\in\beta,\, r>0,\, s>0.

We define 1^\ast to be the set of all q<1.

Then the axioms (M), (D) of Definition def16 hold, with \mathbb R^+ in place of \mathbb F, and with 1^\ast in the role of 1.

Step 7.

We complete the definition of multiplication by setting \alpha0^\ast = 0^\ast\alpha = 0^\ast, and by setting
\alpha\beta=\begin{cases}
(-\alpha)(-\beta),& \text{if }\alpha<0^\ast,\, \beta<0^\ast,\\
-((-\alpha)\beta),& \text{if }\alpha<0^\ast,\, \beta>0^\ast,\\
-(\alpha\cdot(-\beta)),& \text{if }\alpha>0^\ast,\, \beta<0^\ast.
\end{cases}

The products on the right were defined in Step 6.

\mathbb{R} now is an ordered field with the least-upper-bound property.

Step 8.

We associate with each r \in\mathbb Q the set r^\ast which consists of all p \in\mathbb Q such that p<r. It is clear that each r^\ast is a cut; that is, r^\ast \in\mathbb R. These cuts satisfy the following relations:

• r^\ast+s^\ast=(r+s)^\ast,
  
• r^\ast s^\ast=(rs)^\ast,
  
• r^\ast<s^\ast if and only if r<s.

Step 9.

We saw in Step 8 that the replacement of the rational numbers r by the corresponding "rational cuts" r^\ast \in\mathbb R preserves sums, products, and order. This fact may be expressed by saying that the ordered field \mathbb Q is isomorphic to the ordered field \mathbb Q^\ast whose elements are the rational cuts. Of course, r^\ast is by no means the same as r, but the properties we are concerned with (arithmetic and order) are the same in the two fields.

It is this identification of \mathbb Q with \mathbb Q^\ast which allows us to regard \mathbb Q as a subfield of \mathbb R.

The second part of Theorem theorem141 is to be understood in terms of this identification. Note that the same phenomenon occurs when the real numbers are regarded as a subfield of the complex field, and it also occurs at a much more elementary level, when the integers are identified with a certain subset of \mathbb Q.

It is a fact, which we will not prove here, that any two ordered fields with the least-upper-bound property are isomorphic. The first part of Theorem theorem141 therefore characterizes the real field \mathbb R completely.
5. The extended real number system

It is then clear that +\infty is an upper bound of every subset of the extended real number system, and that every nonempty subset has a least upper bound. If, for example, E is a nonempty set of real numbers which is not bounded above in \mathbb R, then \sup E = +\infty in the extended real number system.

Exactly the same remarks apply to lower bounds.

The extended real number system does not form a field, but it is customary to make the following conventions: (a) If x is real then x+\infty=\infty,\, x-\infty=-\infty,x/(\pm\infty)=0; (b) If x>0 then x\cdot(+\infty)=+\infty,\, x\cdot(-\infty)=-\infty; (c) If x<0 then x\cdot(+\infty)=-\infty,\, x\cdot(-\infty)=+\infty.

When it is desired to make the distinction between real numbers on the one hand and the symbols +\infty and -\infty on the other quite explicit, the former are called finite.
6. The complex field

Let x =(a, b),\, y = (c, d) be two complex numbers. We write x = y if and only if a= c and b =d. (Note that this definition is not entirely superfluous; think of equality of rational numbers, represented as quotients of integers.) We
define x+y=(a+c,b+d),xy=(ac-bd,ad+bc).

Theorem theorem162 shows that the complex numbers of the form (a, 0) have the same arithmetic properties as the corresponding real numbers a. We can therefore identify (a, 0) with a. This identification gives us the real field as a subfield of the complex field.

The reader may have noticed that we have defined the complex numbers without any reference to the mysterious square root of -1. We now show that the notation (a, b) is equivalent to the more customary a+ bi.

We shall occasionally write a=\operatorname{Re}(z),\, b=\operatorname{Im}(z).

The existence (and uniqueness) of |z| follows from Theorem theorem143 and part four of Theorem theorem165. Note that when x is real, then \bar x = x, hence |x| =\sqrt{x^2}. Thus |x| = x if x>0, |x| = -x if x<0.

For the last inequality, note that |z+w|^2=(z+w)(\bar z+\bar w)\leqslant|z|^2+2|z\bar w|+|w|^2=(|z|+|w|)^2.

This theorem (1),(2) and (6) will allow us to regard \mathbb R^k as a metric space.