# 3.3 Nonstandard interpretations (optional)

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How non-standard interpretations can provide insight into tough problems.

## Prime factorization

Note that there are other possible interpretations of $\mathrm{prime}$ . 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, $F=\{k\colon k\in \mathbb{N}\}$ . It's easy to verify that multiplying two of these numbers still resultsin a number of the form $4k+1$ . Thus it makes sense to talk of factoring such numbers:We'd say that 45 factors into $59$ , but 9 is consideredprime since it doesn't factor into smaller elements of $F$ .

Interestingly, within $F$ , we lose unique factorization: $441=9\times 49=21\times 21$ , 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.

(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.)

Please how can I create an application?
How do we create a software
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what r the computer codes
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Julius
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abeer
explain coco model
What is a client engineer
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
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Hemant
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Fokuo
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Hemant
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Hemant
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Hemant
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Asongafac
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pravin
I want to create software
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abang
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abang
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Hemant
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shittu
wht teacher u need?
HANAN
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RITESH
is software
RITESH
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sarbajeet
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Hemant
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HANAN
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Amissah
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Sunday
the benefits are life changing
Louis
Louis
an ai system
Louis
is studying software engineer needs to be math expertise?
Carmela
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
just needed basic Nexus calculations
what programming language ?
Louis
Louis
engineering in process student of software engineering
Louis
couple of languages are there not bounded specifically to single one
python, Java, Java script, php, swift, go, c# and many more are trending in
Java is now superseding his popularity but still the one of the most used coding language
I want to be a software engineer too ...
Carmela
best of luck
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
Louis
java is still convenient though
Louis
c made by bell laboratories one of the basic coding languages
windows also used it
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
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Louis
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Louis
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Louis
Louis
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Louis
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Louis
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Louis
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Louis
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Louis
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Louis
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Louis
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Louis
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Toheeb
what is the leading programming language in the world?
HANAN
I have mentioned the names above
@HANAN
Louis
HANAN
HANAN
yes Java is also but not we could limitize the premises
there are many others also
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
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
@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
I'm a Human lol
Louis
make your basic concepts through c and C ++
but yes im a core engineer
Louis
you can say Engineering scientist
Louis
it is the sole centre and Nexus of coding
hi
Md
heyy
Ultimate
hi guys
mubarek
hi
Louis
how to start making a software?
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
Louis
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
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 .
c++ or python or java
Louis
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