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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

Please how can I create an application?
Eric Reply
How do we create a software
Florence Reply
give a more direct question
Louis Reply
what r the computer codes
Kawuba Reply
what is software
Manobendra Reply
how can I create a software
Mary Reply
how to trubulshoot is configuration ?
priya Reply
hello
Julius
hello
abeer
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dawood Reply
What is a client engineer
buthayna Reply
A Customer Engineer (CE) is a worker whose primary job scope is to provide a service to customers who have signed a contract with the company.
Fokuo
Thank you
buthayna
Hi
ahly
hello,how are you doing?
Fokuo
nothing
Hemant
it's a pleasure meeting you.
Fokuo
tq
Hemant
Nice to meet you too
Hemant
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Hemant
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Asongafac
hello
Eric
please I want to more about this group
Asongafac
pravinkumar
pravin
hello
pravin
I want to create software
Michael
name suraj kumae
Suraj
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Micheal Reply
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abang
may be it has a problem
abang
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gabriel
what is 404 error
Hemant Reply
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Hemant
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Sahil Reply
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shittu
wht teacher u need?
HANAN
hi
RITESH
is software
RITESH
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sarbajeet
good morning all
Hemant
heyy
Ultimate
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HANAN
hehahahahahha
amjad
what is cmmi
Mulukutla Reply
what are the benefits of been a software engineer
Amissah
hi
Sunday
gud morning oo
Sunday
kk
Sunday
the benefits are life changing
Louis
can have your own ai
Louis
an ai system
Louis
is studying software engineer needs to be math expertise?
Carmela
depends on what your doing
Louis
grasp the whole field by being a problem solver
Louis
mathematics is the language of computing
Louis
but also its the language of the universe
Louis
before you start typing code please plan on pen and paper first with a goal in mind.
Louis
Thank you !!
Carmela
Are you a software engineer already?
Carmela
mathematics is not compulsory for coding
Nadday
just needed basic Nexus calculations
Nadday
what programming language ?
Louis
@Carnela Villegas : Your welcome
Louis
engineering in process student of software engineering
Nadday
@Naddy Naddy: What programming language are you using if i may ask ?
Louis
couple of languages are there not bounded specifically to single one
Nadday
python, Java, Java script, php, swift, go, c# and many more are trending in
Nadday
Java is now superseding his popularity but still the one of the most used coding language
Nadday
I want to be a software engineer too ...
Carmela
best of luck
Nadday
python is now
Louis
python is used by nasa and for data science aswell for live coding and hacking
Louis
it is the fastest growing language to date
Louis
but being efficient at math is a general stand point if your not language specific
Louis
but c++ can't be beat
Louis
especially with the new c++20 coming
Louis
swift for apple
Louis
swift is made by c code
Louis
c++ is > c
Louis
yes pathon is invented in 1991 by guido van rusom it's popular now a days but it's literally used now as much a Java and Java script php but still most popular python
Nadday
c made java
Louis
java is still convenient though
Louis
c made by bell laboratories one of the basic coding languages
Nadday
windows also used it
Nadday
it just depends on what goal you have in mind to you have in mind to use programming language as a conversion
Louis
mathematics > any programming language
Louis
😊
Louis
sorry I've accidentally placed a question mark lol
Louis
😂
Louis
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Nadday
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Louis
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Louis
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Nadday
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Louis
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Nadday
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Louis
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Nadday
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Louis
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Louis
im a good guy 😊
Louis
nice to hear
Nadday
coding phenomena practice practice and practice
Nadday
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Louis
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Louis
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Louis
I like to be software engineer
Toheeb
what is the leading programming language in the world?
HANAN
I have mentioned the names above
Nadday
@HANAN
Louis
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HANAN
java ri8 🤞 @Nadday Nadday
HANAN
yes Java is also but not we could limitize the premises
Nadday
there are many others also
Nadday
i mean i ask the best top 1 ..which language is that @Nadday Nadday
HANAN
@HANAN THE PROGRAMMER IS THE LEADING PROGRAMMING LANGUAGE OF THE WORLD IG YOU KNOW WHAT I MEAN
Louis
@HANAN whatever language your most comfortable with
Louis
ohhh thank u @Louis johnson
HANAN
i love to be a software engineer but i dont know how to learn😔
HANAN
go for formal education
Nadday
now i m doing Advance level + HND in Computing and System Develipment... this time i have to finish my assignment but still i didn't finish ... i like to be a software engineer ....but i don't know how to
HANAN
your previous education level @hanan?
Nadday
@HANAN as an advanced human just build your own learning strategy in which is the best way anyway through trial and error or books self learn teach output to others who are willing to learn from ground up from the organization you've built and spread
Louis
@louis johnson ok i will try😊👍...anyway thank u soo much guys @Nadday and @louis
HANAN
@HANAN any errors that need to be fixed just fix along the way in future from your atomic learning process
Louis
learn the difference in micro and macro put it to use as a handle in real reality objective analysis
Louis
thank u soo much bro @louis ...are u a software engineer?
HANAN
your major sujects in hnd
Nadday
I'm a Human lol
Louis
make your basic concepts through c and C ++
Nadday
but yes im a core engineer
Louis
you can say Engineering scientist
Louis
it is the sole centre and Nexus of coding
Nadday
hi
Md
heyy
Ultimate
hi guys
mubarek
hi
Louis
how to start making a software?
Jerry Reply
books or teacher
Louis
both of
Hemant
both
ben
i prefer books and a reference teacher on side im self taught self learn everything
Louis
time is your best friend
Louis
I want learn more as well
Ogunrinde
well honestly it would be too large to explain everything in chat but, pick the programming language of choice your most comfortable with, learn that language, make it yours, use it as a primary language to learn in parallel any other language of object oriented programming,
Louis
also it wouldn't hurt to have a basic background in math
Louis
but be a scientist behind the machine(Computer)
Louis
have math be your way of translation of human to computer interaction because you can solve any problem in the universe with mathematics
Louis
good to know discrete topics embedded systems, electrical engineering
Louis
doesn't hurt to have a buddy to team up on project while your asleep to keep progressing
Louis
(Important!..) have real data to work on, the reason is because there have to be some goal to be achieved at some point.
Louis
Be alittle greedy to learn any and all about any language you use.
Louis
Another important thing is not to rush through learning
Louis
once you learn a concept try teaching the concept to someone else who is as interested as you are, that way. the concept will be stuck concretely in your mind.
Louis
slow but sure please guys.education is not like food.
Emmanuel
if you like graphics and games you can use a language that have graphics libraries such as C++, Python, or Java as an example
Louis
who can tell Why software is more important in computer science
Emmanuel
don't just learn alittle want the whole thing and want to know everything at your finger tips
Louis
grasp
Louis
because the hardware and software is useful depends on how useful your goal is
Louis
graphics is not about database management system.c++ is about programming.
Emmanuel
be an android who's drug is "addicted to information" related to the field.
Louis
and in need of the most complex algorithm update
Louis
learn console applications
Louis
how does your languge relate to databases
Louis
and web programming
Louis
i know just throwing some things your way to open mind
Louis
im certi at c++
Louis
it is a general programming language used to build many infrastructure
Louis
including database web and many other soft ware frameworks
Louis
excellent
Ogunrinde
Thanks
Louis
Suppose we are faced with developing a system that we expect to have 100000LOC. Compute the effort and the effort development time for each of the three development modes i.e. Organic, semi - detached and embedded .
yukta Reply
c++ or python or java
Louis
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Source:  OpenStax, Intro to logic. OpenStax CNX. Jan 29, 2008 Download for free at http://cnx.org/content/col10154/1.20
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