Finding a fraction of a number

[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)?

 

[Data]

...unless the axioms are stupidly simple.

indeed

 

[Data]

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?

Well, that's more of a question about history, not about mathematics. It's true that the axiomatic way of handling mathematics has not been around forever. The systems of arithmetic that we use, for example, on integers and rational numbers today are motivated by the fact that the real world also seems to behave like that, at least in some limit. But the correctness of the mathematics is independent of reality.

The basic problem is that, as I hinted at in a previous post, we just don't know what rules things in the universe follow, so we can't prove anything at all about the universe. For now, all we know is that for some reason, the universe seems to be modeled pretty closely by certain mathematical constructions.

For example, as Greathouse mentioned, in everyday life things in the real world seem to be in a Euclidean geometry. That's why you learn Euclidean geometry in high school. But we know now that space isn't really Euclidean (via general relativity) at all, and it just appears to be so for slow-moving small masses.

The geometry of spacetime isn't Euclidean, but that doesn't mean that the mathematics behind Euclidean geometry is wrong - just that it doesn't describe nature perfectly (it still does a pretty good job for everything most people do on a daily basis, though!).

 

[mtanti]

Is mathematics the only subject which builds on axioms? Can philosophy be so as well? (Just answer this and I'll start a thread in the philosophy section afterwards)

 

[arildno]

Is mathematics the only subject which builds on axioms? Can philosophy be so as well? (Just answer this and I'll start a thread in the philosophy section afterwards)

Check up on Spinoza!

The main problem with axiomatic philosophy is that the philosophers seem not to see the trivial truth that precisely because we CHOOSE which axioms to use, ALL truths gained from this can only be relative truths, i.e, statements following from the arbitrarily chosen "first truths" (the axioms).

 

[mtanti]

Isn't that the same with mathematics? Aren't all axioms which supposedly represent reality based on concurrent logic?

 

[arildno]

Not at all. Mathematicians have never said that they are speaking about "reality". That is a much too deep issue; instead mathematicians content themselves to talk about things they have invented themselves, like numbers.

 

[mtanti]

And numbers don't represent physical quantities?

Anyway, the original question was about why is it that multiplying a quantity by a fraction gives the fraction of that quantity. I have come to think that maybe that is just how fractions are defined, an extension of integer multiplication. Instead of using numbers to always add up during multiplication, denomenators are there to vary this process by instead, by definition, divide. This means that there is no actual logic as to what is happening when you actually find the fraction of the quantity, it's just what you're supposed to do when you multiply by the denomenator, divide. How that happens is another process known as division. Is this correct?

I need to relearn mathematics using euclide's method of starting off from the axioms and building up from there. Can anyone suggest a good site or book please?

 

[arildno]

And numbers don't represent physical quantities?

No, they don't, but they may!

 

[xAxis]

Well, a fraction is defined to be a division of quantity into equal quantities.
A third of something means dividing it into three equal parts.
Now, what is two thirds? It is two such parts. Now as explained earlier, it is easy to find a fraction of a number. Divide by the denominator (in order to get one part) and multiply by the number of parts wanted, which is the same as multiplying with fraction, as these two operations are of the same precedence.
You sad you undersood this when whole number is involved, but not when fraction of a fraction is to be found.
Well, why should it be any different? There are two things involved here. A fraction (F) and a quantity (Q) whose fraction we want to find. We always do the same:
Fraction of a quantity = F x Q
whether Q is whole number or quantity.You can't say there is no logic in it.

And numbers don't represent physical quantities?

Not in maths, but we can and we do represent them as physical quantities as they can help us solving many problems, lika this one.

... I have come to think that maybe that is just how fractions are defined, an extension of integer multiplication. Instead of using numbers to always add up during multiplication, denomenators are there to vary this process by instead, by definition, divide. This means that there is no actual logic as to what is happening when you actually find the fraction of the quantity, it's just what you're supposed to do when you multiply by the denomenator, divide. How that happens is another process known as division. Is this correct?

Fraction means division. You can always exchange the fraction line with division. So for example, when we say "two times bigger", then we mean
Q * 2. , and "two times smaller" is Q/2 or Q halves.
You can think of Denominator as the name or classifier of a quantity. For instance, 2/3 and 2/5 are boat "equal" two, but of diferrent kinds. Multiplying thirds will alwas result in thirds, multiplying fifths will alwaus result in fifths etc. The same is with dividing them.
Looking that way, it's easy to see why multiplying of fractions gives fractions of fractions.
One fifth of a 2 is: 1/5 * 2 = 2/5

One fifth of a 2 quarters is the same, but result is in quarters
(remember, multiplying quarters always result in quarters, ):
1/5 * 2/4 = 2/5 quarters
if you want to have a beter view how much it is, just use the same logic, ie find one fifth of a quarter (which is one twentieth) and multiply by two, which is two twentieth, or one tenth.
I think this is quite logical

 

[mtanti]

Fine, I think this question has been finally concluded thanks to everyone's patience and effort to explain. I think there are many mathematical processes which can be satisfactorily explained through history of mathematics.

Can anyone suggest a really good book to learn the order of how mathematics was developed?