1.3 The real numbers

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 $k/n.$ In some obvious sense then, the rational numbers are real numbers. Apparently it was an intellectual shock tothe 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 2by 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 numberwhose square is 2, thereby establishing that there are real numbers tha are not rational. See part (c) of [link] .

Similarly, the area of a circle of radius 1 should be a real number; i.e., $\pi$ should be a real number. It wasn't until the late 1800's that Hermite showed that $\pi$ is not a rational number. One difficulty is that to define $\pi$ as the area of a circle of radius 1we 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 thesefirst 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.

Algebraic properties

Positivity properties

A field $F$ is called an ordered field if there exists a subset $P\subseteq F$ that satisfies the following two properties:

1. If $x,y\in P,$ then $x+y$ and $xy$ are in $P.$
2. If $x\in F,$ then one and only one of the following three statements is true.
   1. $x\in P,$
   2. $-x\in P,$ and
   3. $x=0.$ (This property is known as the law of tricotomy .)

The elements of the set $P$ are called positive elements of $F,$ and the elements $x$ for which $-x$ belong to $P$ are called negative elements of $F.$

As a consequence of these properties of $P,$ we may introduce in $F$ a notion of order.

If $F$ is an ordered field, and $x$ and $y$ are elements of $F,$ we say that $x<y$ if $y-x\in P.$ We say that $x\le y$ if either $x<y$ or $x=y.$

We say that $x>y$ if $y<x,$ and $x\ge y$ if $y\le x.$