Finding a fraction of a number

[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?