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The "miraculous" in math

T'ai Chi

Penultimate Amazing
T
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May 20, 2003
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Can anyone give examples of things that are "miraculous" in mathematics? By "miraculous", I guess I mean things that are generally impossible in the real world, but possible in the mathematical world.

Here are my examples:

--one can fit an object with any length or height in the diagonal of a unit hypercube (say a cube with length of 1ft fot all its sides) in the nth dimension, as long as n is large enough.

--objects exists with infinite surface area but finite volume
 
Some of you may have seen this before, but I like it:

a = b + c

multiply both sides by (a-b)

a(a-b) = (a-b)(b+c)
a^2 - ab = ab + ac - b^2 - bc

take ac from both sides

a^2 - ab - ac = ab - b^2 - bc

factor out a-b-c from both sides

a(a-b-c) = b(a-b-c)

take a-b-c from both sides

a = b


Eh? Substitute any numbers you like for a, b and c (making sure a=b+c of course) and it works. Prove that 3=2. The reason is dodgy working at some point, but can you spot where?
 
T'ai Chi said:
Can anyone give examples of things that are "miraculous" in mathematics? By "miraculous", I guess I mean things that are generally impossible in the real world, but possible in the mathematical world.
Click to expand...
Well, none of the geometric objects (point, line, circle, square etc) exist in our universe since they are either defined as being infinitesimally small or they require a euclidian space in which to exist. Many areas of mathematics such as imaginary numbers, infinities, and clock arithmetic were invented to solve specific mathematical problems and have no relation to the real world. I think you're going to end up with a very long list indeed...
 
Ah, I'll add this to my list too:


--Banach-Tarski paradox
(dissect a ball into six pieces (or 5 or 4) which can be reassembled by rigid motions to form two balls of the same size as the original)
 
Another "mistake" proof:

Many of you are probably familiar with Euler's famous formula:

e^(pi i) + 1 = 0

Where i is sqrt(-1).

Move the one:

e^(pi i) = -1

Square both sides:

e^(2pi i) = 1

Take the natural log of both sides. log 1 is zero in any base, and natural log is the inverse of the exponential, so,

2 pi i = 0

Oops! Those are all constants! What happened?
 
--Banach-Tarski paradox
(dissect a ball into six pieces (or 5 or 4) which can be reassembled by rigid motions to form two balls of the same size as the original)
Click to expand...
There's another form of the Banach-Tarski paradox, as well. Take a ball of any size; you can cut it up into a finite number of pieces, then reassemble the pieces to form a ball of any other size. So, for example, a BB can be cut into finitely many pieces, then reassembled into a ball the size of the sun!
 
Zombified said:
Another "mistake" proof:

Many of you are probably familiar with Euler's famous formula:

e^(pi i) + 1 = 0

Where i is sqrt(-1).

Move the one:

e^(pi i) = -1

Square both sides:

e^(2pi i) = 1

Take the natural log of both sides. log 1 is zero in any base, and natural log is the inverse of the exponential, so,

2 pi i = 0

Oops! Those are all constants! What happened?
Click to expand...

ln(e^(2pi i)) = ln (1) = 0.
 
Pantastic said:
Some of you may have seen this before, but I like it:

a = b + c

multiply both sides by (a-b)

a(a-b) = (a-b)(b+c)
a^2 - ab = ab + ac - b^2 - bc

take ac from both sides

<snip>
Click to expand...


Here's your dodgy working. There is no ac on both sides.

Eh? Substitute any numbers you like for a, b and c (making sure a=b+c of course) and it works. Prove that 3=2. The reason is dodgy working at some point, but can you spot where?
Click to expand...
 
Zombified said:
True... but the point is, what's wrong with ln(e^z)=z...
Click to expand...

I suppose it is taking the log of a negative number.

ln(e^2ipi) = 2.ln(e^ipi) = 2.ln(-1), which is indefined.
 
Infinitesimals are kind of miraculous in their own little way as things which don't make obvious sense physically.

A quantity smaller than any finite quantity but greater than zero. :D

It kind of makes sense until you think about it. ;)

As I understand it, the term gets used loosely and incorrectly most of the time just to mean "something very small" and perhaps even something which can be thrown away and ignored, but it's technically incorrect to use it that way mathematically.

You need to be using hyperreal numbers instead of real numbers to be talking of infinitesimals in a way which is technically correct.
 
Zombified wrote:
Oops! Those are all constants! What happened?
Click to expand...
The imaginary exponentials are basically the same as trig functions, which don't have unique inverses. It's true that e^(2pi i) equals one, but it's also true that e^0 equals one. Just taking the simple ln of one gives zero, but if you're talking complex numbers, there are other answers as well. This would basically be equivalent to:

sin(2pi) = sin(0)

take the arcsin of both sides to give the astonishing result that 2pi = 0!
 
T'ai Chi said:
Can anyone give examples of things that are "miraculous" in mathematics? By "miraculous", I guess I mean things that are generally impossible in the real world, but possible in the mathematical world.
Click to expand...

I think there's an important difference between things that are miraculous and true (like the BT paradox) and things that appear miraculous but are actually false (like the 1=0 proofs). Blurring this difference is dangerous....

My suggestion for the list of miraculous but true would include :

* Infinity comes in lots of different sizes.
* There are sentences that can be proved to be unprovable.
 
wipeout said:
You need to be using hyperreal numbers instead of real numbers to be talking of infinitesimals in a way which is technically correct.
Click to expand...
What about when you do an integration with a lower bound of 0+ or an upper bound of 0-? Or is that something different again?
 
EHocking said:


Here's your dodgy working. There is no ac on both sides.
Click to expand...

No no. If you take ac from the side without an ac in it, you get -ac, have another look.
Drooper got the right answer. a-b-c=0, so you have ax0=bx0, then you remove the zeros to get a=b, which obviously is where the dodgy working comes in.
 
Drooper said:
I suppose it is taking the log of a negative number.

ln(e^2ipi) = 2.ln(e^ipi) = 2.ln(-1), which is indefined.
Click to expand...
It's not undefined, exactly, but it requires imaginary numbers.

2 ln -1 = ln (e^2i pi) = 2i pi

ln -1 = i pi

Since complex ln is multivalued, you get to add arbitrary multiples of 2i pi as well, so in general,

ln -1 = i pi + 2i pi n

for any integer n.
 
wipeout said:
Infinitesimals are kind of miraculous in their own little way as things which don't make obvious sense physically.

A quantity smaller than any finite quantity but greater than zero. :D

It kind of makes sense until you think about it. ;)

As I understand it, the term gets used loosely and incorrectly most of the time just to mean "something very small" and perhaps even something which can be thrown away and ignored, but it's technically incorrect to use it that way mathematically.
Click to expand...
Kinda like the word "infinitely". You hear it used carelessly all the time, as in "Eating a piece of apple pie is infinitely better than having your eye gouged out with a rusty spoon." If that's the case, then nothing is better than apple pie, not even apple pie with ice cream on top, or apple pie with ice cream on top and winning the lottery and having your worst enemy drop dead of envy.

Betcha ten bucks that the next time you hear someone say "infinitely", it will be misused.
 
Iconoclast said:
What about when you do an integration with a lower bound of 0+ or an upper bound of 0-? Or is that something different again?
Click to expand...

If you're using real numbers, then there are no infinitesimals and that would apply to all things in calculus which use the real numbers as well.

The "dx" of calculus often gets called an infinitesimal but my understanding is that it just means "if you take this smaller and smaller" and it always refers to finite quantities.

I'd guess that's along the same lines as numbers heading towards zero from above or below.

I was having a look at this page today:

http://en.wikipedia.org/wiki/Infinitesimal

An infinitesimal is only a notional quantity - there exists no infinitesimal real number. This can be shown using the least upper bound axiom of the real numbers: consider whether the least upper bound c of the set of all infinitesimals is or is not an infinitesimal. If it is, then so is 2c, contradicting the fact that c is an upper bound. If it is not, then neither is c/2, contradicting the fact that among all upper bounds, c is the least.
Click to expand...

That page gives a definition for infinitesimals as well.

I learned a bit about infinitesimals when I decided to try and find out what the "summa" integral sign, dx, dx/dy and other calculus things really mean.
 
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