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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

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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 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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