# Math things that blew your mind

• ShawnD
In summary, the conversation revolved around sharing interesting stories and concepts in mathematics, particularly those that may seem complicated or impressive to others. Some topics that came up included the compound period for growth, the concept of e, the natural log, and various mathematical equations and identities. The conversation also touched on the Cantor Set, exponential functions, and a brain teaser involving the circumference of the Earth and its resulting gap when one foot is added to the length of a string wrapped around it.

#### 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.

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.

siddharth said:
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).

Linear algebra. Seriously.

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.

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).

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

Cantor's Theorem.

An infinity of infinities...

siddharth said:
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.)

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

dontdisturbmycircles said:
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.

And that's the reason you can't

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.

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?

Ivan Seeking said:
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!

Ivan Seeking said:
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

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

Ivan Seeking said:
...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

Implicity we assume a uniform gap.

Ivan Seeking said:
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

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

chroot said:
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,

Ivan Seeking said:
What is the radius of a circle having a circumference of one foot?

dontdisturbmycircles,

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

$\begin{gathered} \frac{{2\pi R + 1}} {{2\pi }} - R \hfill \\ = \frac{{2\pi R + 1 - 2\pi R}} {{2\pi }} \hfill \\ = \frac{1} {{2\pi }} \hfill \\ \end{gathered}$

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 said:

And... ?

- Warren

Oops. I was too late with the post.

chroot said:
And... ?

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

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

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

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).

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

e is a contant. Its slope is everywhere zero.

- Warren

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

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