why, when using the Pythagorean theorem or other equations, is it necessary to square the numbers in order for it to work? Why squaring specifically, and not cubing or something else?

why, when using the Pythagorean theorem or other equations, is it necessary to square the numbers in order for it to work? Why squaring specifically, and not cubing or something else?
because it works?
thank you, exactly the answer i did not want. Yes, because when we square it, it works, but is there a particular mathematical reason why squaring it causes it to work as opposed to performing some other operation on it?Originally Posted by Nevyn
This site may help answer that question:
http://mathforum.org/isaac/problems/pythagthm.html
Note that the above is just one such proof of the Pythagorean Theorem, because there are many. A google search will help you with this. In this particular proof, a smaller square with sides c is inscribed in a large square. The area of the larger square is (a+b)^2 and the smaller one is c^2. However, if you decide to represent the area of the larger square as a sum, you end up with the Pythagorean Theorem.
Another one would be this; Imagine that the sides of a right triangle are the sides of squares, like so:
Now then, no matter what the size or length of the triangle, you will find that the area of the squares adjacent to the sides that form the right angle, when added up, will equal the area of the square adjacent to the hypotenuse. Just note that you cannot use any arbitrary value for the length of the sides because the angles have to be equal to 180 degrees and still remain a right triangle in proportion.
For precisely the same reason that you have to multiply your age in years (plus any fraction) by 12 in order to get your age in months.Originally Posted by Chemboy
All formulas work the way they do (using those specific terms) in order to reach the right answer. In many cases, it took years to discover how to actually formulate them in order to get them to work every single time they are applied. And as has been said, often times there are other alternative formulas  and some that apply only in special cases, too.
OMG! One of the most common misunderstandings about math is that people seem to believe that mathematical formulas are something that people randomly came up with. THIS IS NOT THE CASE!
People did not fudge them to get an answer, they were derived from a given set of axioms and then proven over and over again. It is the same with various well known ratios and constants such as pi and e. They are not random values.
As for the Pythagorean Theorem, this was derived by what I demonstrated earlier.
As for measurement of time, that was not random either. In early times, calendars were based phases of the moon, and on the position of the stars and the sun at various times of the year. The second, minute, and hour is influenced from the Babylons, who used a base 60 numerical system.
The fact that it doesn't work for cubes or something else is actually fermat's last theorem Chemboy  so there is a 100 page answer to that involving modular forms and elliptic curves etc.Originally Posted by Chemboy
I think I know what you are looking for.
You are looking for a type of reasoning that explains the FACT of the matter of Pythagoras' theorum, not some "find", but the reasoning that transcends experiment. I think you are looking for a reason why there is no other "way" such a law can be a fact, and thus perhaps why spacetime operates in such a linedrawn manner.
I actually asked the same question years ago and went in search of a theory that explained the absolute way points and lines can relate with one another. Subsequently, I have been able to arrive at a new theory of space and time that explains how it is possible to construct a "squarerule" constructed circle, and how.........well, you can find it on page 335 of a download of the work I have (www below: but, understandably, it's a big read). My point is, I know what you are looking for, namely a theory that has "all angles covered" such that there is no other way of relating Pythagoras' theorum using extra dimensions.
Your question is valid.
Nice question.
Huh?Originally Posted by streamSystems
Sorry, give me a chance to explain myself.
When one asks the question, "but why", it can go on and on and on and on and on..........until you need a great body of work to explain that everything else has been tried, "attempted", as a theorem.
Yet, that does not eliminate the possibility that something can be so "obvious" as to make perfect sense as a reason of proof, something that for instance our perception is "hardwired" to accept as a "DaaAA".
A "DaaAA" is one of those things we have, as mathematicians or scientists, yet to fully comprehend: we have left it to schoolgirls. It is something so obvious that we fail to see the forest for the trees. Science and mathematics sometimes encourages that in we investigators of the truth of reality.........we know the earth is round, yet we try to explain it with rulers and measurements, as though ultimately trying to tatoo our very awareness with the perfect algorithm. Asking therefore why Pythagoras commands the squarehypotenuse rule is taking a good look at a sizeable piece of lumbar in trying actually to be a little more "fundamental" on the "DaaAA" level.
I have actually written a 300 or so page "DaaAA" testimony to the forest which assists my greater digestion of cutting edge tendancies of scientists and mathematicians.........but I know that's no way through this airport of ideas we call a chatroom. If any of us are going to take off or land with a new idea, without gloating about our status of degree achievement, we are going to involve one another in interesting "debate".
Sure, but let me ask you this, out of curiosity. Assuming Euclid's 5th axiom to hold, for any R<sup>n</sup>, n countable, the Pythagorean is relatively easily shown. But, suppose we ditch the 5th axiom, as Riemann did, or rather, supposing the axiom not to hold in general, do we take the Pythagorean itself as an axiom, or is it still derivable?Originally Posted by river_rat
Like, the Riemannian line element assumes the Pythagorean (with differential coefficients), but I've never seen it derived.
That depends on (what course is) the greater reference of utility to other equations relevant to better understanding spacetime, which we obviously know is what we use mathematics for, namely to explain spacetime more than anything else.
If I can be more helpful with that answer, when deciding to take upon a line of thought of something being an "axiom", we are taking upon a type of "theme" to arrive at.......what?
We are trying to arrive at something mathematics has thus far failed to ACHIEVE explaining that our senses already accept to be something we register daily, something we assume, if not moment to moment, as a phenomena.
Mathematics, ultimately, will calculate our worth as humans.........hence the modern "apriori" "applied mathematics" use of mathematicians in the "reinsurance" industry.
streamSystems: I've apologized to you once for being discourteous, but now I intend to make no such apology.
So. Shut the eff up, you are gibbering, totally, nobody has any idea what you are babbling about. Please take your ravings elsewhere.
No worries.
Diving deep into UNKNOWNS has the "bends" effect.
The "Big Blue"........if mathematicians could only do that on the level of debate.
If I was travelling out into space right now, if we were, I would suggest I take a full medical, 2001 style.
"gibbering", "babling", "raving": you would at least understand where I am coming from, though?
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