<< Chapter < Page Chapter >> Page >
How non-standard interpretations can provide insight into tough problems.

Prime factorization

Note that there are other possible interpretations of prime . For example, since one can multiply integer matrices,there might be a useful concept ofprime matrices.

For example: Consider only the numbers F 1 5 9 13 that is, F k k 4 k 1 . It's easy to verify that multiplying two of these numbers still resultsin a number of the form 4 k 1 . Thus it makes sense to talk of factoring such numbers:We'd say that 45 factors into 5 9 , but 9 is consideredprime since it doesn't factor into smaller elements of F .

Interestingly, within F , we lose unique factorization: 441 9 49 21 21 , where each of 9, 21, and 49 are prime, relative to F ! (Mathematicians will then go and look for exactly whatproperty of a multiplication function are needed, to guarantee unique factorization.)

The point is, that all relations in logical formula need to be interpreted. Usually, for numbers, we usea standard interpretation, but one can consider those formulas in different, non-standard interpretations!

The poincarDisc

A long outstanding problem was that of Euclid's parallel postulate: Given a line and a point not on the line,how many lines parallel to the first go through that point? Euclid took this as an axiom(unable to prove that it followed from his other axioms). Non-Euclidean geometries of Lobachevsky and Riemann took differentpostulates, and got different geometries. However, it was not clear whether these geometrieswere sound whether one could derive two different results that were inconsistent with each other.

Henri Poincardeveloped an ingenious method for showing that certain non-Euclidean geometries are consistentor at least, as consistent as Euclidean geometry.Remember that in Euclidean geometry, the conceptspointandlineare left undefined, and axioms are built on top of them ( e.g. ,two different lines have at most one point in common). While it's usually left to common sense to interpretpoint,line, anda point is on a line, any interpretation which satisfies the axiomsmeans that all theorems of geometry will hold.

The Poincardisc is one such interpretation:pointis taken to meana point in the interior of the unit disc, andlineis taken to meana circular arc which meets the unit disc at right angles. So a statement liketwo points determine a linecan be interpreted as

[*] For any two pointsinside the disc, there is exactly one circular arc which meets the disc at right angles.
Indeed, this interpretation preserves all of Euclid's postulates except for the parallel postulate. You can see thatfor a given line and a point not on it, there are an infinite number of parallel (that is, non-intersecting) lines.

Some lines in the Poincardisc, including several lines parallel to a line L through a point p.

(Note that the distance function is very different within the Poincardisc; in fact the perimeter of the disc is off at infinity.Angles, however, do happen to be preserved.)

Questions & Answers

are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Intro to logic. OpenStax CNX. Jan 29, 2008 Download for free at http://cnx.org/content/col10154/1.20
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Intro to logic' conversation and receive update notifications?

Ask
Jesenia Wofford
Start Quiz
Vanessa Soledad
Start Exam
Madison Christian
Start Exam
Brooke Delaney
Start Exam
Mike Wolf
Start Exam