Why Is Math So Fascinating?

[ShawnD]

Let's share some stories about math things that are interesting enough to tell somebody about, even if it's just to sound smart at parties.

My head exploded when I learned that dropping the compound period for growth to 0 increased growth by a factor of "e". Up until then I had no idea why anybody cared about e, e^x, or natural log.

 

[siddharth]

http://xkcd.com/c179.html

Warning: this comic contains strong language (which may be unsuitable for children), unusual humor (which may be unsuitable for adults), and advanced mathematics (which may be unsuitable for liberal-arts majors).


That was a wow moment when I first learned it in school.

 

[Gelsamel Epsilon]

http://xkcd.com/c179.html

Warning: this comic contains strong language (which may be unsuitable for children), unusual humor (which may be unsuitable for adults), and advanced mathematics (which may be unsuitable for liberal-arts majors).


That was a wow moment when I first learned it in school.

Yeh, that screwed with me too.

(XKCD is the best. I run around the forums a lot).

 

[Manchot]

Linear algebra. Seriously.

 

[twisting_edge]

e^2*pi*i=1 always gets people who haven't gone through the Taylor expansion to get Euler's relation. When you first look at it, 'e', 'pi' and 'i' have absolutely nothing to do with each other, and all come from different areas of math.

My first reaction upon getting handed Euler's relation (having just spent a week just working with the theoretical underpinnings of the Taylor expansion in another class, so it was sort of inviolate) was, "That's just sick."

Of course, it was also over two decades ago, so I may be misremembering the name of "Euler's relation". I mean by it the mapping of e^x*i onto the unit circle in the complex plane.

After you work with it constantly for so many years (no way to avoid it in theoretical physics, for those who aren't or are just beginning), you forget just how twisted the relation actually is in terms of the underlying concepts. It's the rest of the world that looks weird in contrast.

 

[dontdisturbmycircles]

Was teaching my brother about exponential functions yesterday and he was pretty surprised to hear that if you fold a piece of paper 50 times it would be 50 million miles thick (assuming 0.1mm thickness).

 

[Gelsamel Epsilon]

Actually what just screwed me over was that e^(pi*i) gives a negative answer, yet e is positive.

 

[Ivan Seeking]

Cantor's Theorem.

An infinity of infinities...

 

[BobG]

http://xkcd.com/c179.html

Warning: this comic contains strong language (which may be unsuitable for children), unusual humor (which may be unsuitable for adults), and advanced mathematics (which may be unsuitable for liberal-arts majors).


That was a wow moment when I first learned it in school.

More surprising to me was the natural log of a negative number always resulted in a complex number with pi as the imaginary part.

(Maybe more surprising was that my math teacher said there was a reason for that, but he couldn't recall exactly what it was at the moment. Once a person recalled Euler's identity, the result was more of a 'Duh!' realization than a shocking result. The subject came up when a student asked why integrals used the natural log of the absolute value of some variable.)

 

[selfAdjoint]

I LOVED the Cantor Set. c points, totally disconnected, everywhere dense, what's not to like? Antoine's necklace is even better.

 

[BobG]

Was teaching my brother about exponential functions yesterday and he was pretty surprised to hear that if you fold a piece of paper 50 times it would be 50 million miles thick (assuming 0.1mm thickness).

You can't fold a piece of paper 50 times. You can only fold it twelve times.

 

[Office_Shredder]

And that's the reason you can't

 

[Ivan Seeking]

I LOVED the Cantor Set. c points, totally disconnected, everywhere dense, what's not to like? Antoine's necklace is even better.

Also, not only did we have the greatest professor for that class, there were only about six of us in it. It was a real treat.

 

[Ivan Seeking]

There is very simple brain teaser that caught me off gaurd.

If you wrap a string around the Earth at the equator and pull it tight, and then add one foot to the length of the string, what would be the resulting gap?

What is the radius of a circle having a circumference of one foot?

 

[EL]

If you wrap a string around the Earth at the equator and pull it tight, and then add one foot to the length of the string, what would be the resulting gap?

What is the radius of a circle having a circumference of one foot?

Wow, that's a good one!

 

[chroot]

If you wrap a string around the Earth at the equator and pull it tight, and then add one foot to the length of the string, what would be the resulting gap?

I don't even get it... what gap? Gap in what?

- Warren

 

[Ivan Seeking]

...the gap formed between the string and the Earth by adding one foot to the length of the string.

 

[chroot]

...the gap formed between the string and the Earth by adding one foot to the length of the string.

Oh, I see.. so the ends of the string are brought together and pulled off the surface of the earth.

- Warren

 

[Ivan Seeking]

Implicity we assume a uniform gap.

 

[chroot]

Implicity we assume a uniform gap.

What does a "uniform gap" mean? You seem to have put all kinds of assumptions into this "brain teaser," to the point where it doesn't even make any sense to me what quantity I should try to find. Sorry.

- Warren

 

[Ivan Seeking]

Funny, in the fifteen years or so that I've been telling this to people, I've never had to explain it before.

 

[Jimmy Snyder]

what quantity I should try to find.

The difference between the radius of a circle whose circumference is that of the Earth, and the radius of a circle whose circumference is one foot longer than that of the Earth.

 

[chroot]

The difference between the radius of a circle whose circumference is that of the Earth, and the radius of a circle whose circumference is one foot longer than that of the Earth.

4.9 centimeters. How is this a brain teaser?

It's just a calculation..

http://www.google.com/search?hl=en&...)+/+(2+*+pi))+-+(radius+of+earth)&btnG=Search

- Warren

 

[dontdisturbmycircles]

Chroot,

What is the radius of a circle having a circumference of one foot?

 

[chroot]

dontdisturbmycircles,

Yeah, it's obvious that the radius of the Earth vanishes and you end up with 1 / 2 pi.

<br /> \begin{gathered}<br /> \frac{{2\pi R + 1}}<br /> {{2\pi }} - R \hfill \\<br /> = \frac{{2\pi R + 1 - 2\pi R}}<br /> {{2\pi }} \hfill \\<br /> = \frac{1}<br /> {{2\pi }} \hfill \\ <br /> \end{gathered} <br />

That's great and all, but it's so straightforward that I don't understand what makes it a good brain teaser. It actually seemed to so straightforward that I figured that I must have been interpreting it wrong, so I started asking Ivan questions about it.

After all, if you imagined that the string with one foot of excess length was pulled up off the Earth's surface as far as possible, letting the rest of it rest on the Earth's surface... that would be a brain-teaser (or a brain-imploder, I guess). I tried solving that for a while, but you end up with a mess that I don't believe has any analytic closed-form solution.

- Warren

 

[backdoorstudent]

4.9 centimeters. How is this a brain teaser?

It's just a calculation..

http://www.google.com/search?hl=en&...)+/+(2+*+pi))+-+(radius+of+earth)&btnG=Search

- Warren

And so is this:
http://www.google.com/search?hl=en&lr=&q=(1+foot)+/+(2+*+pi))&btnG=Search

 

[chroot]

And so is this:
http://www.google.com/search?hl=en&lr=&q=(1+foot)+/+(2+*+pi))&btnG=Search

And... ?

- Warren

 

[backdoorstudent]

Oops. I was too late with the post.

 

[backdoorstudent]

And... ?

- Warren

And so you noticed what the orginal poster thought was mind-blowing at first glance.

 

[chroot]

Anyway, sorry for the off-topic noise.

Back on-topic... I remember the following things as mathematical epiphanies:

1) Figuring out how solving simultaneous equations via subtracting equations is equivalent to solving simultaneous equations via matrix reduction.

2) Discovering why e is special.

3) Discovering how the formulas for the areas and volumes of 2D and 3D objects can be found via calculus.

4) Finally grasping the Taylor series and why it works.

5) Finally grasping Fourier decomposition and why it works.

6) Figuring out why the multi-valued nature of the log(z) function makes the Riemann surface a ramp.

7) Figuring out what div and curl really do.

8) Finally understanding how the wedge product is a generalization of the cross-product.

9) Path integrals and their relationship to geometric topology.

10) Stoke's theorem and its relationship to Green's theorem and the fundamental theorem of calculus.

11) The coverage of SO(3) by SU(2).

The list goes on...

- Warren

 

[morphism]

The axiom of choice and everything equivalent to it, and everything it can do.

 

[dontdisturbmycircles]

Thanks for posting your work chroot, I didn't have time to think about why it works, but I suppose it's fairly simple. I thought it was kind of neat myself, although I do admit, it is simple (as it was supposed to be I guess.)

I look at your list of math that amazed/intrigued you, and I can't wait to figure some of the stuff out. I know that I will be studying taylor series very shortly and have already had small doses of information as to why e is so special (basically just that it's slope at x=0 is 1).

 

[chroot]

(basically just that it's slope at x=0 is 1).

e is a contant. Its slope is everywhere zero.

- Warren

 

[dontdisturbmycircles]

Woops sorry, yes, e^x. The slope of y=e^x at (0,1) is 1. (derivative).

 

[0rthodontist]

Getting my first rough inkling of the incredible size of the field of math.

 

[dontdisturbmycircles]

Sorry to repost, but I meant the slope of the tangent line.

My bad.

 

[Quaoar]

http://en.wikipedia.org/wiki/Graham%27s_number" , which is so big that it has its own special notation.

 

[Office_Shredder]

Holy frick. That's a big number.

I'm going to prove tomorrow that g₆₆ > 1. I will utilize g₆₅ in the process, thus breaking the record.

 

[gravenewworld]

The Church-Turing Thesis.

The topology trick with the strip of paper where you twist it 180, 360, and 540 degrees then cut it down the middle to get different results. that was totally mind blowing.

 

[moose]

The topology trick with the strip of paper where you twist it 180, 360, and 540 degrees then cut it down the middle to get different results. that was totally mind blowing.

Im kinda interested to know what you're talking about...

 

[z-component]

The topology trick with the strip of paper where you twist it 180, 360, and 540 degrees then cut it down the middle to get different results. that was totally mind blowing.

Like the Möbius strip?

 

[gravenewworld]

Like the Möbius strip?

Yes

Im kinda interested to know what you're talking about...

Get a piece of regular 8x11 paper. Cut 1" wide strips (so the length of the strip is 11"). Take a strip and turn one end 180 degrees. Tape it to the other end. Get a pair of scissors and cut it down the all the way down the middle until you end where you started cutting(i.e.cut it down the "spine"). What did you get? Repeat the same procedure but take another strip and put a 360 degree twist and a 540 degree twist in it. What did you get? This little grade school trick still completely shatters my mind.

 

[mathwonk]

as a kid, i waS READING A DONALD DUCK COMIC BOOK AND HE haD THE IDEA TO GET RICH from a double your money back offer. a store offered double money back if a hair restorer failed to work. he figured he could not grow hair, being a duck, and remarked that if he doubled a dollar 20 times he'd be a millionaire. i checked and he was right. i was amazed.

of course he lost. he took the stuff back for his first or second refund and the guy ridiculed him and rubbed the hair grow tonic all over his body and kicked him out. a day later he popped out in hair all over and had to pay double to get the hair remover.

 

[DaveC426913]

as a kid, i waS READING A DONALD DUCK COMIC BOOK AND HE haD THE IDEA TO GET RICH from a double your money back offer. a store offered double money back if a hair restorer failed to work. he figured he could not grow hair, being a duck, and remarked that if he doubled a dollar 20 times he'd be a millionaire. i checked and he was right. i was amazed.

of course he lost. he took the stuff back for his first or second refund and the guy ridiculed him and rubbed the hair grow tonic all over his body and kicked him out. a day later he popped out in hair all over and had to pay double to get the hair remover.

That's so ... mind blowing.

 

[Quaoar]

Another thing that blew my mind: Using residues to solve integrals that I couldn't solve before without Mathematica. That was pretty neat.

 

[Gelsamel Epsilon]

The version that 'Add a foot to the circumference' thing that I heard was something like there is a circle with circumference of Some really really big X if you added 1 meter to that circumference then could you fit a person through it or something like that.

 

[FrogPad]

http://en.wikipedia.org/wiki/Graham%27s_number" , which is so big that it has its own special notation.

haha :) that was really interesting

 

[mathwonk]

that is a great list chroot! those are all wonderful examples of math gems.I remember another life changing moment, when in college i noticed in a footnote to courants calculus book, about page 27, the proof of the formula for the sum of the rth powers of the first n positive integers, by induction. i was blown away, as this footnote had more content than the entire book of any course I had taken in high school. i knew I wasn't in kansas any more, to paraphrase dorothy.

another experience was sitting in the univ library at vanderbilt and eading the proof there are infinitely many primes.

another was the day we proved the number of rationals is the same as the number of integers, but less than the number of reals, by cantors diagonal arguments. this still seems mind boggling.

a sad sequel was learning recently that cantor was so far ahead of his time he was ostracized for these fantstic insights to the point where he became depressed and died unhappy. later hilbert is said to have exclaimed "we will never be driven out from this paradise cantor has built for us!"

i myself obtained admittance to an advanced honors calc class taught by john tate, at harvard entirely based on my knowledge of cantors proof of the uncountability if the reals. up to that point in my interview he was politely ushering me out the door as a hopeless dunce.

 

[Alkatran]

sum of 2^n from 0 to i-1 = (2^i) - 1
I realized it when I was reading a short story in middle school about a man playing some game with the devil, and the devil casually remarked that if he drank the next glass he would get 100$ more than all he had already gotten.

It really bugged me, because I remember we had to do that sum once, and we did it term by term! Before long I had figured out how to do it for any base and any starting point, but I still can't believe we had to do that sum!

 

[dontdisturbmycircles]

Just wanted to add that I learned why e is so important today. :P Since when differentiating an exponential function of the form a^x you get ax*(the value of f'(x) at x=0). So since the slope of the tangent line drawn at x=0 for the graph e^x is 1, it's derivative is the simplest of all possible, itself. :) (Chroot said that discovering why e is important was neat, and I agree)