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Algebraic and positivity probabilities of real numbers. Ordered fields, isomorphism, bounded below, bounded above, supremum, infimum, and complete are defined.

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., π should be a real number. It wasn't until the late 1800's that Hermite showed that π is not a rational number. One difficulty is that to define π 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

We should be able to add, multiply, divide, etc., real numbers. In short, we require the set of real numbers to be a field.

Positivity properties

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.

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

  1. If x , y P , then x + y and x y are in P .
  2. If x F , then one and only one of the following three statements is true.
    1. x P ,
    2. - x 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 P . We say that x y if either x < y or x = y .

We say that x > y if y < x , and x y if y x .

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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