Finding a fraction of a number

[mtanti]

OK then what is math without physical meaning? Why would it exist if there was no link to real problems?

 

[FrogPad]

OK then what is math without physical meaning? Why would it exist if there was no link to real problems?

Read the about the http://en.wikipedia.org/wiki/Complex_number#History"of the complex number.

http://en.wikipedia.org/wiki/Complex_number#History

then read about its applications

 

[mtanti]

Wasn't there a time when integration was proved to be wrong because the theory and practice (hydrolics) did not match? And it was thus reexamined? What do you call that?

 

[FrogPad]

Wasn't there a time when integration was proved to be wrong because the theory and practice (hydrolics) did not match? And it was thus reexamined? What do you call that?

gibberish

cite a source

 

[mtanti]

It was told by my maths teacher. Appearantly the area under a curve was found by using the y-intercept. Anyway the story is not important. Aren't theory and practise mutual to each other? If one doesn't imply the other than at least one of them is considered wrong and restudied. Isn't that right?

 

[arildno]

Wasn't there a time when integration was proved to be wrong because the theory and practice (hydrolics) did not match? And it was thus reexamined? What do you call that?

Not at all. The matematical discipline known as "hydrodynamics of the ideal fluid" gave results inconsistent with those fluid phenomena in which friction could not be neglected.
For engineers, in the design of pipes for example, the ideal fluid approximation was (and is!) fairly useless, their discipline was called "hydraulics".

The ideal fluid approx. remains fairly important, in particular when studying&predicting the propagation of waves (and other cases).


This, however, has nothing whatsoever to do with "proving integration wrong".
I've never heard a more idiotic claim before.

 

[arildno]

It was told by my maths teacher. Appearantly the area under a curve was found by using the y-intercept. Anyway the story is not important. Aren't theory and practise mutual to each other? If one doesn't imply the other than at least one of them is considered wrong and restudied. Isn't that right?

This time, it seems that you have been told that Lebesgue integration "disproves" Riemann integration. It doesn't, and never will.
This is the second most idiotic claim I've ever heard.

 

[Data]

OK then what is math without physical meaning? Why would it exist if there was no link to real problems?

Mathematics is independent of reality. Sometimes we're lucky and we find that in some model and approximation, reality seems to obey the same axioms we're using to do some mathematics. Then we can approximate the real world using mathematics. There's no reason to believe we'll ever have a perfect description of reality in terms of mathematics, since we have no idea what the real axioms that things in the universe obey are (if they exist at all!).

So asking why half of half of an apple is a quarter of an apple is a philosophical question. We happen to have some nice axioms and definitions which let us have similar behaviour mathematically, so we can model things that way. Maybe sometime someone will cut an apple in half and they'll get two whole apples! There's nothing that says this is impossible; the universe can do as it likes.

 

[CRGreathouse]

This time, it seems that you have been told that Lebesgue integration "disproves" Riemann integration. It doesn't, and never will.
This is the second most idiotic claim I've ever heard.

What was the first?

 

[FrogPad]

Mathematics is independent of reality. Sometimes we're lucky and we find that in some model and approximation, reality seems to obey the same axioms we're using to do some mathematics. Then we can approximate the real world using mathematics. There's no reason to believe we'll ever have a perfect description of reality in terms of mathematics, since we have no idea what the real axioms that things in the universe obey are (if they exist at all!).

So asking why half of half of an apple is a quarter of an apple is a philosophical question. We happen to have some nice axioms and definitions which let us have similar behaviour mathematically, so we can model things that way. Maybe sometime someone will cut an apple in half and they'll get two whole apples! There's nothing that says this is impossible; the universe can do as it likes.

Awesome post man.

 

[arildno]

What was the first?

See previous post.

 

[HallsofIvy]

Wasn't there a time when integration was proved to be wrong because the theory and practice (hydrolics) did not match? And it was thus reexamined? What do you call that?

Not "gibberish", not "idiotic", but a confused reference to the relationship between Fourier series and Lebesque integration. It wasn't immediately connected with "hydraulics" but Anton Fourier was an engineer who developed a method for dealing with the heat equation- the Fourier series. He asserted two things: that given a periodic function one could write it as a sine and cosine series, giving formulas for the coefficients as an integral of the function, and the converse- that given such a series, it summed to an integrable function.

It turned out that the second claim is not true- it is easy to find Fourier series that give functions that are not Riemann integrable. It was the fact that Fourier series worked so nicely that led to the development of the Lebesque integral for which it is true!

 

[FrogPad]

Not "gibberish", not "idiotic", but a confused reference to the relationship between Fourier series and Lebesque integration. It wasn't immediately connected with "hydraulics" but Anton Fourier was an engineer who developed a method for dealing with the heat equation- the Fourier series. He asserted two things: that given a periodic function one could write it as a sine and cosine series, giving formulas for the coefficients as an integral of the function, and the converse- that given such a series, it summed to an integrable function.

It turned out that the second claim is not true- it is easy to find Fourier series that give functions that are not Riemann integrable. It was the fact that Fourier series worked so nicely that led to the development of the Lebesque integral for which it is true!

Gibberish... cite a source.

:) totally kidding! Thanks halls, that was interesting... now I have something to read about before doing my homework.

 

[arildno]

I really can't see the difference between a statement being hopelessly confused and being gibberish.

Interesting interpretation, though, HallsofIvy.


I would just add that the poster DID say that "hydraulics" proved integration wrong. I took him on his word.

 

[mtanti]

Right, sorry for the confusion. Perhaps should have been more sure before saying it. However practise still had to do with the change in theory right?

 

[arildno]

Right, sorry for the confusion. Perhaps should have been more sure before saying it. However practise still had to do with the change in theory right?

Experiences from "practice" as you call it, may inspire someone to develop and enrich what you call "theory".

 

[Data]

Right, sorry for the confusion. Perhaps should have been more sure before saying it. However practise still had to do with the change in theory right?

There wasn't a "change in theory," there were just new tools developed to deal with the cases that the old ones weren't suited to. Once you've proved something in mathematics (and assuming you haven't made any errors), it can never be disproved. It may not be useful, but it's still valid, because everything in mathematics is deductive. You take some axioms and deductively prove things using the axioms. Assuming no errors, whatever you prove is forever valid under those axioms.

There just may not be any physical situation that operates with similar "axioms!"

 

[arildno]

mtanti:
Just to give you a picture that may enable you to understand what axioms are in maths:

Consider pastimes, like football:
In, say, soccer, you have a lot of rules laid down that specify how a valid game is to be played. By these rules, a referee may note whether a particular move or event in a soccer is allowed or not.

But, the rules governing soccer aren't at all the rules governing American football!

However, does this mean that American football disproves soccer?
Of course not!
It's a different game, that's all.

 

[DaveC426913]

Question about applications of i.

I'v always had trouble understanding real world applications of imaginary numbers.

It seems odd that imaginary number math, was first ... discovered?, and then an area of science was found that they nicely describe (such as in electrical engineering).

These things that imaginary numbers describe in electrical engineering: could they not have been grappled-with before imaginary numbers came along? Would engineers have said "I see this current amplitude graph (or whatever) but I am unable to describe it mathematically"?

 

[arildno]

No, they would have said we can describe this with the aid of product functions of exponentials and trigonometric functions.

 

[DaveC426913]

No, they would have said we can describe this with the aid of product functions of exponentials and trigonometric functions.

So then what is gained by introducing i?

 

[CRGreathouse]

So then what is gained by introducing i?

Insight into the workings; less cumbersome notation; more & faster discoveries through deepr understanding of the physical principles; better information sharing with the complex number-using mathematical community.

Just off the top of my head, of course.

 

[mtanti]

So axioms are independant of practicle and physical events? How come mathematics is so compatible with the natural world then?

 

[arildno]

So axioms are independant of practicle and physical events? How come mathematics is so compatible with the natural world then?

That is a deeply fascinating PHILOSOPHICAL question that no one has got a final answer for.

 

[FrogPad]

So axioms are independant of practicle and physical events? How come mathematics is so compatible with the natural world then?

Mtanti: You should check out this https://www.amazon.com/dp/0465026567/?tag=pfamazon01-20. Judging from the questions you are asking, I think you will like it.

 

[mtanti]

Are there any axioms which are not compatible? In some unorthodox mathemetical theory? Can you make up some mathematical exioms and start a new form of mathematics?

 

[CRGreathouse]

Are there any axioms which are not compatible? In some unorthodox mathemetical theory? Can you make up some mathematical exioms and start a new form of mathematics?

Yes, just like you can invent your own sports.

 

[Data]

Are there any axioms which are not compatible? In some unorthodox mathemetical theory? Can you make up some mathematical exioms and start a new form of mathematics?

You can come up with any set of axioms that you like and start proving things with them.

Godel's famous incompleteness theorems say that every set of axioms is either inconsistent or incomplete - that is, either there are statements that, under those axioms, can be shown to be both true and false (inconsistency), or there are unprovable but true statements (incompleteness).

 

[CRGreathouse]

nit-pick, sorry

Godel's famous incompleteness theorems say that every set of axioms is either inconsistent or incomplete - that is, either there are statements that, under those axioms, can be shown to be both true and false (inconsistency), or there are unprovable but true statements (incompleteness).

...unless the axioms are stupidly simple.

 

[mtanti]

But the orthodox axioms we use are compatible with practicle situations right? Were they chosen to be so or were they just found inductively to be so?

So the more axioms you have, the more complete your mathemetics is?

Damn, if mathemetics was thought by stating this just after teaching numbers and operations, it would be dead simpler to grasp instead of just memorizing everything until high school...

So this is what mathemetics is all about? Applying simple axioms to create complex theorems?

 

[CRGreathouse]

But the orthodox axioms we use are compatible with practicle situations right? Were they chosen to be so or were they just found inductively to be so?

That's hard to say. Euclidian geometry seemed the only one compatible with intuition, real life, and common sense (the only system with appealing properties like "rectangles exist") and yet now we use hyperbolic & elliptic geometries for common tasks.

It's not so easy to know which assumptions to use.

So the more axioms you have, the more complete your mathemetics is?

Mathematicians generally try to reduce the number of axioms needed and used. I wouldn't agree with this at all. Further, it's generally hard to show the new system is consistent if it is at all more powerful.

 

[arildno]

So this is what mathemetics is all about? Applying simple axioms to create complex theorems?

You could say that.

 

[mtanti]

I used to think that mathematics is the application of patterns since all rules of algebra are following a pattern. Is that true?

Also, my teacher once said that if an axiom was to be proved incompatible with reality, all practicle subjects using theorems which use the axiom will collapse (in practise almost all theorems use every axiom I guess). Could the axioms be modified and the orthodox mathematics be discarded in such a case?

 

[CRGreathouse]

Also, my teacher once said that if an axiom was to be proved incompatible with reality, all practicle subjects using theorems which use the axiom will collapse (in practise almost all theorems use every axiom I guess). Could the axioms be modified and the orthodox mathematics be discarded in such a case?

There are tools like the MetaMath Explorer to show what results rely on which axioms, if you're interested.

If axioms were shown to be in conflict with reality, it would likely be in a manner similar to Newtonian vs. Einsteinian mechanics -- not much would practially change.

 

[arildno]

I used to think that mathematics is the application of patterns since all rules of algebra are following a pattern. Is that true?

What else are axioms than the basic strands of a pattern(or rather THE patterns themselves)?