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Elephant in the Refrigerator

Analysis:

  1. Differentiate it and put into the refrig. Then integrate it in the refrig.
  2. Redefine the measure on the referigerator (or the elephant).
  3. Apply the Banach-Tarsky theorem.

Number theory:

  1. First factorize, second multiply.
  2. Use induction. You can always squeeze a bit more in.

Algebra:

  1. Step 1. Show that the parts of it can be put into the refrig. Step 2. Show that the refrig. is closed under the addition.
  2. Take the appropriate universal refrigerator and get a surjection from refrigerator to elephant.

Topology:

  1. Have it swallow the refrig. and turn inside out.
  2. Make a refrig. with the Klein bottle.
  3. The elephant is homeomorphic to a smaller elephant.
  4. The elephant is compact, so it can be put into a finite collection of refrigerators. That's usually good enough.
  5. The property of being inside the referigerator is hereditary. So, take the elephant's mother, cremate it, and show that the ashes fit inside the refrigerator.
  6. For those who object to method 3 because it's cruel to animals. Put the elephant's BABY in the refrigerator.

Algebraic topology:
Replace the interior of the refrigerator by its universal cover, R^3.

Linear algebra:

  1. Put just its basis and span it in the refrig.
  2. Show that 1% of the elephant will fit inside the refrigerator. By linearity, x% will fit for any x.

Affine geometry:
There is an affine transformation putting the elephant into the refrigerator.

Set theory:

  1. It's very easy! Refrigerator = { elephant }
  2. The elephant and the interior of the refrigerator both have cardinality c.

Geometry:
Declare the following:
Axiom 1. An elephant can be put into a refrigerator.

Complex analysis:
Put the refrig. at the origin and the elephant outside the unit circle. Then get the image under the inversion.

Numerical analysis:

  1. Put just its trunk and refer the rest to the error term.
  2. Work it out using the Pentium.

Statistics:

  1. Bright statistician. Put its tail as a sample and say "Done."
  2. Dull statistician. Repeat the experiment pushing the elephant to the refrig.
  3. Our NEW study shows that you CAN'T put the elephant in the refrigerator.

Geometry Drugs

"Psst, c'mere," said the shifty-eyed man wearing a long black trenchcoat, as he beckoned me off the rainy street into a damp dark alley. I followed.

- What are you selling?

- Geometrical algebra drugs.

- Huh!?

- Geometry drugs. Ya got your uppers, your downers, your sidewaysers, your inside-outers...

"Stop right there," I interrupted. "I've never heard of inside-outers."

- Oh, man, you'll love 'em. Makes you feel like M.C. ever-lovin' Escher on a particularly weird day.

- Go on...

"OK, your inside-outers, your arbitrary bilinear mappers, and here, heh, here are the best ones," he said, pulling out a large clear bottle of orange pills.

- What are those, then?

- Givens transformers. They'll rotate you about more planes than you even knew existed.

- Sounds gross. What about those bilinear mappers?

- There's a whole variety of them. Here's one you'll love -- they call it 'One Over Z' on the street. Take one of these little bad boys and you'll be on speaking terms with the Point at Infinity.

The Long Green Crocodile

Prove that the crocodile is longer than it is wide.

Lemma 1. The crocodile is longer than it is green: Let's look at the crocodile. It is long on the top and on the bottom, but it is green only on the top. Therefore, the crocodile is longer than it is green.

Lemma 2. The crocodile is greener than it is wide: Let's look at the crocodile. It is green along its length and width, but it is wide only along its width. Therefore, the crocodile is greener than it is wide.

From Lemma 1 and Lemma 2 we conclude that the crocodile is longer than it is wide.

One Equals One Half

Theorem: 1 = 1/2

Proof:
We can re-write the infinite series 1/(1*3) + 1/(3*5) + 1/(5*7) + 1/(7*9)+... as 1/2((1/1 - 1/3) + (1/3 - 1/5) + (1/5 - 1/7) + (1/7 - 1/9) + ... ).

All terms after 1/1 cancel, so that the sum is 1/2.

We can also re-write the series as (1/1 - 2/3) + (2/3 - 3/5) + (3/5 - 4/7)+ (4/7 - 5/9) + ...

All terms after 1/1 cancel, so that the sum is 1.

Thus 1/2 = 1.

Log of Negative One

Theorem: log(-1) = 0

Proof:
a. log[(-1)^2] = 2 * log(-1)
On the other hand:
b. log[(-1)^2] = log(1) = 0
Combining a) and b) gives:
2* log(-1) = 0
Divide both sides by 2:
log(-1) = 0

One Equals Negative One

Theorem: 1 = -1

Proof:
1 = sqrt(1) = sqrt(-1 * -1) = sqrt(-1) * sqrt(-1) = 1^ = -1

Also one can disprove the axiom that things equal to the same thing are equal to each other.
1 = sqrt(1)
and -1 = sqrt(1)
Therefore 1 = -1

As an alternative method for solving:

Theorem: 1 = -1

Proof:
x=1
x^2=x
x^2-1=x-1
(x+1)(x-1)=(x-1)
(x+1)=(x-1)/(x-1)
x+1=1
x=0
0=1
=> 0/0=1/1=1

All Numbers Are Equal

Theorem: All numbers are equal.

Proof: Choose arbitrary a and b, and let t = a + b. Then
a + b = t
(a + b)(a - b) = t(a - b)
a^2 - b^2 = ta - tb
a^2 - ta = b^2 - tb
a^2 - ta + (t^2)/4 = b^2 - tb + (t^2)/4
(a - t/2)^2 = (b - t/2)^2
a - t/2 = b - t/2
a = b
So all numbers are the same, and math is pointless.

N Equals N Plus One

Theorem: n=n+1
Proof:
(n+1)^2 = n^2 + 2*n + 1
Bring 2n+1 to the left:
(n+1)^2 - (2n+1) = n^2
Substract n(2n+1) from both sides and factoring, we have:
(n+1)^2 - (n+1)(2n+1) = n^2 - n(2n+1)
Adding 1/4(2n+1)^2 to both sides yields:
(n+1)^2 - (n+1)(2n+1) + 1/4(2n+1)^2 = n^2 - n(2n+1) + 1/4(2n+1)^2
This may be written:
[ (n+1) - 1/2(2n+1) ]^2 = [ n - 1/2(2n+1) ]^2
Taking the square roots of both sides:
(n+1) - 1/2(2n+1) = n - 1/2(2n+1)
Add 1/2(2n+1) to both sides:
n+1 = n