Mathematics marathon

[blah]

Two plane figures are defined to be similar ...

I do see your argument/justification. I'm not trying to be obnoxious or anything, but imho there's too much vagueness in "rotation", "enlargement/reduction", "translation", etc. I prefer a more analytic definition, such as the area of a unit circle, or more precisely, \( \pi=2\int_{-1}^1 \sqrt{1-x^2}\,dx \) (and the trig functions could be defined from here). Or if the trig functions have been defined by series or other means, then something like \( \pi=\inf\{x\in\mathbb{R}: x>0,\sin x=0\} \) would do. I'm aware that such definitions are not very accessible to most people but I'm not trying to be a douche or anything, it's just that the ratio definition has never settled well with me when I was in high school because of it's heavy reliance on intuition. I'm just bringing this up for people to realize that there are other more "solid" definitions lying around.

 

[qqwref]

How, exactly, do you plan to prove pi is a constant using such advanced techniques? (And if you ask me, intuition is fine in geometry, even though it should often be avoided in analysis.)

 

[blah]

How, exactly, do you plan to prove pi is a constant using such advanced techniques? (And if you ask me, intuition is fine in geometry, even though it should often be avoided in analysis.)

I was simply defining pi as an integral, which is obviously a constant. To show that the ratio A/r^2 is a constant (which happens to be pi) is then (almost) trivial.

First, \( x=ru\Rightarrow dx=r\,du \). Then \( \frac{A}{r^2}=\frac{2}{r^2}\int_{-r}^r \sqrt{r^2-x^2}\,dx=\frac{2}{r^2}\int_{-1}^1 \sqrt{r^2-r^2u^2}\cdot r\,du=2\int_{-1}^1 \sqrt{1-u^2}\,du=\pi \).

 

[qqwref]

I'm a lot more interested in the C/2r ratio than the A/r^2 ratio, but I suppose the derivation would be more or less similar.

(And sure, while you technically could define pi as the area of a unit circle as expressed by an integral, or as the sum of some infinite series, etc., it is a LOT less interesting and useful to do so - and you then have to explain why that integral or series also happens to be the circumference/diameter ratio, and half the period of a sine, and so on.)

 

[blah]

Intuition is what makes math interesting. It also happens to make math less rigorous/more "questionable".

 

[qqwref]

You can always make intuition rigorous if you try, but you can't always make rigor intuitive.

 

[keemy]

I was simply defining pi as an integral, which is obviously a constant.

Pi is defined as the ratio of the circumference to the diameter (it can also be defined as the ratio of the area to the radius squared). If anything you were taking the definition and then using it to find and analytic expression of pi. It would be like if I defined \( \binom nk \) as \( \frac{n!}{(n-k)!k!} \) when it's definition is the number of ways to choose k elements from a set of n elements or the coefficient of of the \( x^k \) term in the binomial expansion of \( (x+1)^n \)

 

[malcolm]

Here's a cool one:

Find a polynomial p(x,y) so that, when x and y are integers, p gives a unique value for each possible value of x and y. So, if you know x and y are integers, and you are given a value of p, you should be able to figure out the single possible (x,y) pair that gives that value.

\( P(x,y)=(2x^2+x+1)^2((2y^2+y+1)^2+1) \) works.
Note that \( Q(x)=2x^2+x+1 \) is distinct for each integer \( x \) and is strictly positive, so \( P(x,y) \) is always a positive integer.
Then if \( P(x,y)=p=a^2b \) where \( a,b >0 \) and \( b \) is squarefree, we must have \( a^2=(2x^2+x+1)^2, b=(2y^2+y+1)^2+1 \) and since \( Q(x) \) is distinct for each integer \( x \), we can work out \( x \) and \( y \) from here.

 

[qqwref]

\( P(x,y)=(2x^2+x+1)^2((2y^2+y+1)^2+1) \) works.
Note that \( Q(x)=2x^2+x+1 \) is distinct for each integer \( x \) and is strictly positive, so \( P(x,y) \) is always a positive integer.
Then if \( P(x,y)=p=a^2b \) where \( a,b >0 \) and \( b \) is squarefree, we must have \( a^2=(2x^2+x+1)^2, b=(2y^2+y+1)^2+1 \) and since \( Q(x) \) is distinct for each integer \( x \), we can work out \( x \) and \( y \) from here.

Careful. \( (2y^2+y+1)^2+1 \) isn't necessarily squarefree - if y = -2, it gives 50 (= 2*5*5). I'm not sure if this leads to paired x and y values or not, but it means the approach you used isn't quite perfect.

 

[blah]

Pi is defined as the ratio of the circumference to the diameter (it can also be defined as the ratio of the area to the radius squared). If anything you were taking the definition and then using it to find and analytic expression of pi.

Are you serious? :fp There's a "correct" definition now? If two statements are equivalent, what's wrong with using one as the definition and then proving its equivalence with the other?

I personally like how integrals are rigorously defined from the ground up in calculus. Further manipulation convinces me that the ratios C/2r and A/r^2 are indeed constants. Simple as that.

On the other hand, terms like "rotation" and "translation" don't feel rigorous enough to me, so the fact that all circles are similar is questionable if I were skeptical enough. The implication that ratios of one-dimensional objects should be constant when two shapes are similar, without rigorous justification, is also questionable for an extreme skeptic.

I would like to mention that I *am* perfectly fine with the ratio definition. After all, what's intuition there for?

When someone keeps asking why, people generally label him/her as being insufferable. But the impenetrable truth is what characterizes mathematics. I don't know about you, but if someone asks me what pi is and I give them the ratio definition and they asked the very same questions that I asked, sure I'd be annoyed, but at the same time I'd also feel inadequate, like I don't know my stuff well enough. How else do you think people like Hilbert and Weierstrass laid down the foundations of modern math? Keep asking questions and finding answers, until the most annoying person in the world has no more questions to ask, that's mathematics, that's impenetrable truth.

tl;dr: Giving an answer that you're prepared to defend to the end is imo the better approach to mathematics.

 

[y3k9]

Ok, I am going to ask a question, that is not really a problem but I still like an answer and explanation:

Spoiler

How do you find a complex cube root of a number?

 

[blah]

Ok, I am going to ask a question, that is not really a problem but I still like an answer and explanation:

Spoiler

How do you find a complex cube root of a number?

What's with the spoiler? First you draw a circle. Find the cube root of its radius. Trisect the angle. Voila.

 

[einstein00]

Intuition is what makes math interesting. It also happens to make math less rigorous/more "questionable".

look here please. Not everything in the world can be proven rigorously. Try proving Angle-Angle triangle similarity to see what I mean.

Are you serious? :fp There's a "correct" definition now? If two statements are equivalent, what's wrong with using one as the definition and then proving its equivalence with the other?

This is a very serious logic flaw. Pi was first defined as the ratio C/D. Your "equivalent" statement is not a definition. You only know that pi is the area of a unit circle because a unit circle has a radius 1, and thus its area is pi * 1^2 = pi. But here, you are using a previous definition of pi. Without the first definition of pi, the area formula would not exist, and thus your "equivalent" statement cannot stand alone as a definition.

Edit:

If we define A and use it to prove B, we cannot use B to prove A. We can only assume B is true because it is derived from A. If we wish to use B to prove A, then we must first assume that A is not necessarily true (or else it's circular logic). However, since A is not necessarily true, then B is not necessarily true, either, and the proof breaks down immediately because of a false initial assumption.

This is because logic is like a building - you start with your definitions as the foundation, and then you build upward. You can't use the second floor to "prove" the first floor because the second floor exists only because the first floor exists already.

I'm not sure how else I can explain this, so I hope you understand this analogy.

Edit 2:

Unless you're stating that this is your new definition of pi, not some formula devised depending on the existing historical definition of pi. At this point, I guess you can define anything you want, as long as it all makes sense. No one can argue with a definition. I'm just sticking to the historically first definition of pi being C/D.

 

[blah]

look here please. Not everything in the world can be proven rigorously. Try proving Angle-Angle triangle similarity to see what I mean.

Are you making the assumption that I don't know what axioms are? If you are, then I'm offended.

This is a very serious logic flaw. Pi was first defined as the ratio C/D. Your "equivalent" statement is not a definition. You only know that pi is the area of a unit circle because a unit circle has a radius 1, and thus its area is pi * 1^2 = pi. But here, you are using a previous definition of pi. Without the first definition of pi, the area formula would not exist, and thus your "equivalent" statement cannot stand alone as a definition. <--- lolololololololololol

Now this whole chunk of text is a very serious logical flaw.

In response to your additional comments that you just edited in, thank you very much for your explanation of a circular argument. I think I know pretty well what that is. But there is nothing circular about defining pi to be the area of a unit circle. I don't know you but from all your explanations and arguments, I'm going to make the assumption that you're still in high school. Is this true? If you're gonna ask why this matters, it's because in short, they lie to you a lot in high school.

 

[keemy]

look here please. Not everything in the world can be proven rigorously. Try proving Angle-Angle triangle similarity to see what I mean.

I am loling pretty hard right now.

 

[einstein00]

Are you making the assumption that I don't know what axioms are? If you are, then I'm offended.

Hey hey, no one's trying to offend anyone here. I certainly don't want this to be a flame war. If you already know something I've pointed out, it's wise to simply ignore it.

After thinking some more, I edited the post (a lot). I hope you understand what I'm trying to say. And let's not start a flame war, that just makes life miserable.

But there is nothing circular about defining pi to be the area of a unit circle.

That's cleared up in my edit 2. problem solved

 

[blah]

I'm not trying to start anything. I'm just saying that you should be more open-minded. Real math isn't what they teach you in high school. Real math isn't even what they teach you in the calculus sequence in college. Just accept that.

 

[qqwref]

On the other hand, terms like "rotation" and "translation" don't feel rigorous enough to me,

In what way are they less rigorous than integrals? In a 2-dimensional system, they are transformations on functions, relations, sets, etc. of points:
Rotation (by angle t): (x,y) -> (x cos(t) - y sin(t), x sin(t) + y cos(t))
Translation (by (a,b)): (x,y) -> (x + a, y + b)
Dilation (by factor r): (x,y) -> (rx, ry)
And I'm pretty sure you can prove invariance of straight and arc lengths by rotation/translation, and length ratios and angles by all rotation/translation/dilation, if you just define them in a mathematically meaningful way.

I don't know about you, but if someone asks me what pi is and I give them the ratio definition and they asked the very same questions that I asked, sure I'd be annoyed, but at the same time I'd also feel inadequate, like I don't know my stuff well enough.

But if your problem is that you don't know your stuff well enough, you'd learn a lot more (and end up with a better answer) by finding a way to rigorously fix the intuitive issues you saw, as opposed to defining pi as something totally different, then using advanced techniques to prove the invariance you wanted to start with.


einstein00: blah was defining pi to be "the area of a unit circle", or more specifically "this integral", which clearly exists, so that wasn't a problem. But it IS awkward to redefine something as the value of an integral just to solve a basic issue you had, especially since historically people were asking "I wonder what is this constant C/d ratio" a huge number of years earlier than people were asking "what is the value of this integral".

Real math isn't what they teach you in high school. Real math isn't even what they teach you in the calculus sequence in college. Just accept that.

Too bad real math is often boring :'(

 

[blah]

I actually sat down and did everything you just did there a couple of hours ago. My issue is that the OP didn't.

Of course, I failed to define an integral and a bunch of other stuff as well, but that's a different story...

As for your second point, yes, that is what I would do. But for now, I don't know enough about euclidean geometry to defend anything; however, I do know enough about calculus/analysis to defend my definition of pi as an integral, hence the preference.

 

[einstein00]

I'm just saying that you should be more open-minded.

I actually am a very open minded person. I just didn't immediately realize that you wanted to use that formula as your definition of pi. My bad. I understand and agree that no one can argue with a definition, because by definition, a definition is defined to be true. Of course, that also means you have to "reprove" everything else based off of this new definition... but that's totally besides the point.

And yes, I know the maps that qqwref posted. I just (perhaps wrongly) felt it was unnecessary to post it because it's one of the first things learned in middle school math.