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daleks
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daleks
3,770 posts
Chamberlain

This thread is just what the title says it is. It is all about Math. You can post math jokes(the jokes don't have to be good), math questions, what you like about Math, or even why you hate Math.

My math joke: Resistance is not futile. It is voltage divided by amps.

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1235813
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1235813
2 posts
Peasant

Ok here is a raelly bad joke that our teacher told us: A dead parrot is a ___________. A polygon!!!!!!!!!!!! or a polly gone. (as in the parrot name)

aknerd
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aknerd
1,416 posts
Peasant

So: another question. What is a circle? What does it look like? Why do you think your definition of a circle necessitates that a circle looks like what you think it looks like?

In other words, why is a circle a circle, but a square not a circle?

daleks
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daleks
3,770 posts
Chamberlain

What is a circle?

A shape that has no sides. An equation of a circle: X^2 + Y^2 = R^2
What does it look like?

It looks like a circle.
Why do you think your definition of a circle necessitates that a circle looks like what you think it looks like?

Prove me wrong.
aknerd
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aknerd
1,416 posts
Peasant

A shape that has no sides. An equation of a circle: X^2 + Y^2 = R^2
Prove me wrong.


1) What do you mean by sides?
2) What if R is an imaginary number?
3) What if our axis are not perpendicular (and yes, such coordinate systems exist)?
4) What if our axis do not use the same scale?
5) What about circle that are not centered at 0,0? Are you saying that these are not circles? Because they don't fit your equation.
6) what about circles in higher dimensions?
7) What if R is some weird irrational number? How do we know R^2 even exists, and how do we know X^2 and Y^2 exist such that they can add up to R^2? Intuitively, such numbers exist, but how do you actually know?
8) What if....
lightcrux
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lightcrux
622 posts
Peasant

1) What do you mean by sides?...


Too many what ifs. In Euclidean geometry it is the locus of a set of points at a ficed distance from a fixed point. That is how I always define it.
aknerd
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aknerd
1,416 posts
Peasant

Too many what ifs. In Euclidean geometry it is the locus of a set of points at a ficed distance from a fixed point. That is how I always define it.


I actually prefer Daleks' definition to this one. All of the problems with his definition can be fixed by just being a little more careful. The "fixed distance" definition, however...

So, What do you mean by distance? There are actually many (infinitely many, to be exact) ways to define distance. There is daleks' way, where the distance betwen two points is the square root of the sum of their squares.

Ie the distance between Y,X is (Y^2 + X^2)^1/2

In his equation, he sets this equal to a constant, R, and then squares both sides. Note that in this case, x+y is denoting a single point (x,y), so this equation is giving the distance from this point to the origin. So, he is basically saying a circle is the set of all points R away from the center. This comes basically from the Pythagorean theorem.

But, what if I used a different way to define distance? What if I did something like this:
If d(X,Y) is the distance from the the point (x1,x2) to (y1,y2),
then let d(X,Y)= |y1-x1| + |y2-x2|

Where |z| is the absolute value of z.

Then, this definition has all the properties of distance you would want: its always positive, the distance from Y to X is equal o the distance from X to Y, and the distance from X to Z plus Z to Y is great than or equal to the distance going from X straight to Y.

In this picture, the distance between the two points is equal to the length of the yellow, blue, or red lines.
http://upload.wikimedia.org/wikipedia/commons/0/08/Manhattan_distance.svg

BUT (and some careful thought will prove this to you) a circle using this definition is actually a square!

This distance is informally called the taxicab distance, since the distance is measured as you travel between the two point in a zig zag, like taxi would in a large city.

Here is what a "circle" looks like:
http://binary-services.sciencedirect.com/content/image/1-s2.0-S0002929707636179-gr6.sml
It has a radius of two.
daleks
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daleks
3,770 posts
Chamberlain

1) What do you mean by sides?
2) What if R is an imaginary number?
3) What if our axis are not perpendicular (and yes, such coordinate systems exist)?
4) What if our axis do not use the same scale?
5) What about circle that are not centered at 0,0? Are you saying that these are not circles? Because they don't fit your equation.
6) what about circles in higher dimensions?
7) What if R is some weird irrational number? How do we know R^2 even exists, and how do we know X^2 and Y^2 exist such that they can add up to R^2? Intuitively, such numbers exist, but how do you actually know?
8) What if....

1) No angles.
2) Use an imaginary plane?
3) Using Cartesian(hopefully spelled right) plane.
4) Then make them the same scale to make your life easy.
5) Sorry, (X-h)^2 + (Y-k)^2 = R^2
6) Drawing in second dimension. They are spheres in the 3rd.
7) Example please.
aknerd
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aknerd
1,416 posts
Peasant

6) Drawing in second dimension. They are spheres in the 3rd.
7) Example please.


6) Like this: http://www.mathworks.com/matlabcentral/fx_files/11186/1/circle3.jpg

7) Let r=e. How do you even know that a number like e^2 exists? How can you square an irrational number? I mean, this really isn't that big of a deal. But, it is something interesting to think about. Intuitively, you know that squares of irrational numbers exist, but it is much more difficult to say why.
master565
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master565
4,107 posts
Nomad

So, i mentioned writing a math paper in the beginning of the thread, and i found out i made it into the second round of the statewide competition. I'm scheduled to give a live 10 presentation to a panel of judges sometime in March, I can't remember the exact date. Also apparently I also need to give a trial presentation in front of the high honors freshman math class. Why did it have to be freshmen

aknerd
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aknerd
1,416 posts
Peasant

Also apparently I also need to give a trial presentation in front of the high honors freshman math class. Why did it have to be freshmen

Congrats on getting into the second round, but... man... freshmen. That's brutal. Depending on how advanced your paper is, I predict many, many blank stares in your future. ANYWAY:

Forget the circle nonsense, I found something much more fun to play with.

So, remember that infinite sum from a few pages ago? Lets return to that.

Let r =1-1+1-1+1-1+1.... and on and on.
Then, 1-r= 1 - (1-1+1-1+1-1+1...) = 1-1+1-1+1-1+1-1+1-1+1....=r
Then, 1-r=r
then 1=2r
Then r=1/2
So, 1+1-1+1-1+1-1+1-1....=1/2

But wait! There is more!

Let S= 1+2+3+4+5+6.... ie the sum of all the natural numbers.

Then -3s = (1-4)s = s-4s=
1+2+3+4+5.... -2(2+4+6+8....)
=1+2-2-2+3+4-4-4+5+6-6-6....
=1-2+3-4+5.... (=-3s remember this for later)
=1-(2-3+4-5...)
=1-(1+1-2-1+3+1-4-1...)
=1-(1-2+3-4+5...)-(1-1+1-1+1...)
=1-(-3s)-r

Then, -3s= 1-1/2+3s
Then -6s = 1/2
Then, s = -1/12

Therefore, the sum of all the positive integers is -1/12. and don't let anyone tell you otherwise!
daleks
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daleks
3,770 posts
Chamberlain

So, i mentioned writing a math paper in the beginning of the thread, and i found out i made it into the second round of the statewide competition.

Congrats. I bet you will do well.
Also apparently I also need to give a trial presentation in front of the high honors freshman math class.

Some freshman a really smart. Like those d*mn ones last year that would always ruin the chance for a curve.
aknerd
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aknerd
1,416 posts
Peasant

Since my &quotroof" that the sum of all natural numbers was, quite unexpectedly, -1/12 failed to impress anyone, here is (hopefully) a more relatable topic:

Okay, we've probably all had good math teachers and horrible math teachers, but has anyone ever used a math textbook that they really enjoyed? I mean, even at lower levels, other subjects can have really interesting textbooks, so why not math? Also on that note, as anyone ever used a math textbook that was just impossible to use? How so? What do you want to see in a math textbook?

nichodemus
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nichodemus
14,988 posts
Grand Duke

Okay, we've probably all had good math teachers and horrible math teachers, but has anyone ever used a math textbook that they really enjoyed? I mean, even at lower levels, other subjects can have really interesting textbooks, so why not math? Also on that note, as anyone ever used a math textbook that was just impossible to use? How so? What do you want to see in a math textbook?


I liked the pictures in my textbooks until I was 8, then I lost interest until I was 13 when I next used a textbook. Lost interest since then, I'd rather practice rather than read up on the concepts that I've learnt. Don't really see the point of my math textbooks now apart from some exercises.
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