Mathematics

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Mathematics is the study of the relationships between numbers (and sometimes concepts). Branches of mathematics include algebra, geometry, trigonometry, calculus, graph theory, game theory, probability, and many more. Mathematics is a fundumental truth, and when done correctly is impossible — for rational people — to deny, which is to say that mathematics is also one of the few aspects of life where absolute proof is considered possible. For example, the only way to deny that 1 + 1 = 2 is to use a definition of "1", "2", "+", or "=" that is not commonly accepted.

Mathematics is a mainstay of science (but surprisingly is not considered a science itself), and important in everday life as a whole.

Despite being incredibly important, no Nobel prize is awarded in the field of mathematics (some claim that Nobel's wife ran off with a mathematician - though since Nobel was never married, this seems unlikely). The poor mathematicians have to be content with the much less prestigious (to non-mathematicians) Fields Medal.

A few fundamental statements in mathematics, known as axioms, are not proven and are instead assumed to be true. The most basic of these is the reflexive property, which states: For all $a$ , $a = a$ . For any sort of logic to exist, we must assume this to be true. One thing about axioms in mathematics (and other logical endeavors) is that they are always stated up front. It is always interesting to see what structures can be built by discarding some of them, such as in non-Euclidean geometry.

1 Cool (but incorrect) proof that 1 = 2
2 Cool (but incorrect) proof that 0 = -1 (or 1 = 2 if you prefer)
3 Cool (but incorrect) proof that 1 = -1
4 Cool (but incorrect) proof that an elephant and a mosquito have the same mass
5 Cool (but incorrect) proof that I am the Pope
6 Footnotes

[edit] Cool (but incorrect) proof that 1 = 2

assume:
$A = B$
multiply both sides by "A":
$A A = A B$
subtract $B 2$ from both sides:
$A 2 - B 2 = A B - B 2$
factor both sides:
$(A - B)(A + B) = B (A - B)$
divide both sides by (A-B):
$A + B = B$
as A and B are equal, substitute all "A"s with "B"s:
$B + B = B$
continuing:
$2 B = B$
$2 = 1$
Q.E.D.

[edit] Cool (but incorrect) proof that 0 = -1 (or 1 = 2 if you prefer)

$\int\tan x\;dx = \int\tan x\;dx$
substitute "tan x":
$\int\tan x\;dx = \int\sin x\sec x\;dx$
Integrate by parts,^[1] assume u = sec x and dv = sin x dx:
$\int\tan x\;dx = -secx cosx + \int\cos x\tan x\sec x\;dx$
but cos x * sec x = 1 so:
$\int\tan x\;dx = - 1 + \int\tan x\;dx$
we substract both sides by ʃtan x dx:
$\int\tan x\;dx - \int\tan x\;dx = - 1 + \int\tan x\;dx - \int\tan x\;dx$
then:
$0 = - 1$

[edit] Cool (but incorrect) proof that 1 = -1

assume:
$\mathbf{-1} = \mathbf{-1}$
rewrite -1 two different ways:
$\frac{1}{ -1} = \frac{-1}{1}$
take the square root of both sides:
$\sqrt{\frac{1}{-1}} = \sqrt{\frac{-1}{1}}$
using laws of square roots, rewrite both sides:
$\frac{\sqrt{1}}{\sqrt{-1}} = \frac{\sqrt{-1}}{\sqrt{1}}$
multiply both sides by $\sqrt{1}\sqrt{-1}$ and reduce:
$\sqrt{1}\sqrt{1} = \sqrt{-1}\sqrt{-1}$
the square root of a number squared equals the number itself, so:
$\mathbf{1} = \mathbf{-1}$

[edit] Cool (but incorrect) proof that an elephant and a mosquito have the same mass

Let a = mass of elephant in kg
Let x = mass of mosquito in kg
Let y = their combined mass in kg
Then:
$a + x = y$
$a = y - x$
$a - y = - x$
multiplying the two latter equations:
$a 2 - a y = x 2 - x y$
adding $(\frac{y}{2})^2$ to both sides:
$a^2 - ay + (\frac{y}{2})^2 = x^2 - xy + (\frac{y}{2})^2$
which can be rewritten:
$(a - \frac{y}{2})^2 = (x - \frac{y}{2})^2$
from which derives:
$a - \frac{y}{2} = x - \frac{y}{2}$
and finally:
$a = x$
that is, mass of elephant = mass of mosquito.

[edit] Cool (but incorrect) proof that I am the Pope

This is a classic. We don't know who invented it.

The Pope and I are two. [That is, two people.]

By the previous theorem, 2 = 1.

Therefore, the Pope and I are one.

[edit] Footnotes

↑ For a more complete discussion of this tactic, see Wikipedia. Here is a quick explanation of what is being done here:
In the traditional calculus curriculum, this rule is often stated using indefinite integrals in the form
$\int f(x) g'(x)\,dx = f(x) g(x) - \int f'(x) g(x)\,dx,$

or in an even shorter form, if we let u = f(x), v = g(x) and the differentials du = f ′(x) dx and dv = g′(x) dx, then it is in the form in which it is most often seen:

$\int u\,dv=uv-\int v\,du.$

Mathematics Articles on RationalWiki
	- Cauchy Sequence - Complex numbers - Conservapedian mathematics - Probability (Conservapedia) - Fermat's last theorem - Fibonacci sequence - 40gon (Fun) - Phli (fun) - Golden Ratio - Gödel's incompleteness theorems - Irrational number - limit - Metric system - Pentagon - Pyramid - Recursion - Rene Descartes - Statistics - TeX - Zero -