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{{trope}}
{{quote|''"Math is delicious!"''|'''[[Robot Buddy|Winslow]]''', ''[[Questionable Content]]'' }}
* According to a theorem called the
** This paradox was the subject of an
** Also seen expressed thuslywise: "What's an anagram of 'Banach-Tarski'? 'Banach-Tarski Banach-Tarski'."
** Another catch is that this stems from
*** However, that is underselling its use. While the axiom of choice (which is the axiom that is vital in the construction) is used all over the place in analysis, and used in a couple mind-numbingly ''crucial'' places in algebra. Almost all ordinary mathematicians (read: non-FOMers) accept choice as true, and see the paradoxical results as unfortunate things that are true, and see the benefit from the more normal use of the axiom.
*** Or you can accept the fact that mathematics is simply a lot less intuitive than we'd guess at first glance, and that there's nothing unfortunate or wrong about Banach-Tarski's paradox... in fact, in hindsight I would be much more surprised if such things ''were not'' possible, since there's no reason why splitting a sphere in non-measurable sets would need to preserve any intuitive notion of volume. Poor Axiom of Choice, always wronged for no real reason.
** [[Irregular Webcomic|You'd think they could do it in just four pieces though.]]
* And then there's Whitehead and Russell's ''{{w|Principia Mathematica}}'', a multi-volume opus intended to construct all of mathematics from the most basic of axioms (such as "every statement is either true or false"). After 379 pages of incomprehensibly dense notation, it succeeds in proving that
** [[Understatement|"The above proposition is occasionally useful"]] was their comment when the 1+1=2 proof was finally complete (presumably after arithmetical addition had been defined) in the second volume.
** It bears pointing out that Gödel's Incompleteness Theorem essentially says that no axiomatic system can be both complete and consistent, ''i.e.'', your system can be complete or consistent but not both. Given that, most mathematicians choose the latter, accepting an incomplete axiom system whose incompleteness is tempered by consistency (so that, ''e.g.'', 1+1 does indeed equal 2).
* A {{w|googol}} is equal to 10^100, or a 1 followed by 100 zeroes. A {{w|googolplex}} is equal to
**
*
▲* [http://en.wikipedia.org/wiki/Grahams_number Graham's number]. A number so large that it could not even be written with conventional scientific notation, and a new form of mathematical notation had to be developed ''just so it could be expressed''. It is often regarded as the largest finite number that pure mathematics actually takes seriously. What's worse? The answer to the problem it was created to solve might actually be as low as 13.
** Graham's number is so ridiculously, mindbogglingly huge that there aren't enough elementary particles in the ''whole universe'' to write out the ''number of digits of the size of the power tower (3^3^3^3^3^3...3^3^3^3^3 E.T.C.)" in the ''first'' element of the generating series, of which each number is obtained by performing the operation on the previous number that got that first number from ''four'', and Graham's number is the ''sixty-fourth''.
**
*
** If you cut a Möbius strip in half lengthwise it stays in one piece. If you give it not one but three half-twists and cut it in half lengthwise, not only does it stay in one piece, it ''twists itself into a knot''. If you cut a Möbius strip in ''thirds'' lengthwise (possibly by making the guide lines on the strip of paper beforehand), you'll finish cutting it in one go, and end up with a regular loop with another Möbius strip attached.
** Möbius strips can be made using legos - if you use the thin treads that have the chain link, give them a twist and connect them. the fun part, is if you run them with a gear, they still have full mobility.
*** Incidentally, that's how some machines actually work- they wind the leather motor belts into Möbius strips between wheels to ensure the belt wears out evenly.
** Not just Legos either, you can also make them out of an odd amount of Bucky balls, you just need to stagger them next to each other in a line of 2, then when you have a complete line, twist one end and connect them.
**
** [http://www.kleinbottle.com/ Here's] someone making them in glass.
** And from the same site, [http://www.kleinbottle.com/klein_bottle_hats.htm Klein Hats.]
** In a similar vein, how about some [http://theiff.org/oexhibits/oe1e.html crochet models] of [[Alien Geometries|hyperbolic space]]?
** New use for the Mobius strip: the world's smallest-denomination polyhedral die. (OK, monohedral die...is that even a word?)
*
** First, from the accepted decimal expansions of some fractions, such as:
{{quote|
(1/3)*3{{=}}(0.333...)*3
1{{=}}0.999... }}
** Second, by simple algebraic manipulation:
{{quote|
9.999...{{=}} 10x
9.999... - 0.999... {{=}} 10x - x
9 {{=}} 9x
1 {{=}} x }}
** Third: implicitly, 0.999... represents an infinite geometric sum of its digits, 9/10 + 9/100 + 9/1000 + ..., for which we can use the standard formula for the sum of a geometric series<ref>which, incidentally, is proven using a construction similar to the second proof</ref>:
{{quote|
{{=}} 9/10 + 9/100 + 9/1000 + ...
{{=}} 9 * (1/10)<sup>1</sup> + 9 * (1/10)<sup>2</sup> + 9 * (1/10)<sup>3</sup> + ...
{{=}} 9 * (1/10)/(1 - 1/10) {{=}} 9 * 1/(10-1) {{=}} 9 * 1/9 {{=}} 1 }}
* There are the same number of odd numbers (1, 3, 5, ...) as there are natural numbers (1, 2, 3, ...). What's even more bizarre is that there are the same number of [http://www.homeschoolmath.net/teaching/rational-numbers-countable.php fractions] as there are natural numbers. What's more bizarre even than that is that this is not a trivial fact. There are multiple types of infinity (an infinite number, actually). The set of real numbers (everything that can be plotted on a continuous number line) is a larger type of infinity. (The set of all possible subsets of the real numbers is larger still.) However, most numbers that we care about--"
** As do any numbers recognized by a
* Which brings up that ''for
* [[Animal Farm|All infinities are infinite, but some are more infinite than others]]. The study of infinite sets is so fantastically weird that when the mathematician Georg Cantor first came up with it, he was accused of ''[[Rage Against the Heavens|challenging God]]''.
** This can be extraordinarily confusing, and tends to [[Flame Bait|prompt discussion]] when it shows up, likely because it involves things like "all odd numbers" which are widely understood, while the more difficult formal ideas (like cardinality) aren't expressly stated. In short, two sets are defined as having the same cardinality ("size") if you can match up their members in some one-to-one fashion. This is perfectly intuitive for finite sets, but a bit confusing for infinite ones. It ''is'' consistent, though, and more sensible than any other definition. Possibly the best example is
** Is there a set with cardinality (size) that is strictly larger than the set of natural numbers but smaller than the set of real numbers? There is no answer (at least in ZFC), this is possible because of Gödel's incompleteness.
*** If you buy the
** Even trippier? All ''n''-dimentional real spaces have the same cardinality. That is, the number line (''x''), the Cartesian plane (''x'',''y''), three-dimensional space (''x'',''y'',''z''), etc.
*** What's more, the closed interval [0,1] over the reals, and the corresponding open interval (0,1), both have the same cardinality as as the entire real line itself. In other words, there are the same number of real numbers ''[[Gratuitous Latin|in toto]]'' as there are real numbers between zero and one, inclusively OR noninclusively.
* How about a geometric figure that has an infinite surface area, but a finite volume?
** Create your own Menger Sponges with [[Magic
* By using the concept of "inner product spaces," you can do absolutely absurd things with anything that can be considered a vector space. Like calculating the angle between two matrices. Or finding the distance between two polynomials.
** That is basically what is going on in modern mathematics, and has been for over a century. Mathematicians have been trying to find the correct generalizations, and see where it gives useful ideas. There is a germ of usefulness in the latter one that you gave for sure.
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* Topology has some fun ones, but there is no way to explain why they're fun unless you know a little topology yourself. The simplest one is that there is a notion of "closeness" on the natural numbers so that the sequence 1, 2, 3, ... is close to every natural number. Moreover, this notion of closeness doesn't seem all that unintuitive before you start digging.
* [http://www.xkcd.com/179/ e^(iπ) = -1].
** And the graph of that
** Slightly cooler (it's the generalization) is that e^(i* x)=cos(x)+i* sin(x).
** What's even cooler is that (i^i)^i=-i
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** Some sources say the number is 616, though...
* Probability can be a [[You Fail Statistics Forever|very strange thing to behold]] at times. For example: You can have a situation where the probability of any given event is 0, yet events still happen (thanks to our good old friend "continuum").
** [[Don't Explain the Joke|That's]] defined more precisely as
** If you have a 50% chance of winning one cent, a 25% chance if winning two, etc., the expected winnings is infinite, despite the fact that it's impossible to win an infinite amount of money.
*** However, if the winnings grow linearly rather than exponentially (1/8 chance to get 3 cents, 1/16 to win 4), then that expected winnings is both finite and readily calculable: {{spoiler|two cents}}.
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** There's also the Pidgeonhole Principal, complete with proof, on explaining that if you have ''x+1'' elements and ''x'' slots, one slot will have at least two elements.
* "A point is that which has no part"-Euclid, ''The Elements''.
* For the entirely useless, but interesting spectrum of math we have the weird number bases. If you know decimal (the normal base) and get your head around binary or hexadecimal, the jump to the the other natural bases is not so great. However, consider a number base of -10. Or
** Quiz: What is 1+1 in base φ? {{spoiler|10.01}}
* There are infinitely many primes, which the above-mentioned Gödel's theorem implicitly relies on, along with integers having unique factorizations.
** And the proof is ridiculously easy: Suppose there were finitely many primes. If you multiply them all together you get a number that can be divided by any prime. If you add one you then get a number that cannot be divided by any prime. Therefore this big number is either prime and not a member of the set of all primes, or has prime factors that aren't members of the set of all primes. That's absurd. Hence there aren't finitely many primes.
*
** I believe that our [[Viewers Are Geniuses|youth can handle this!]]
{{quote|
''2 is less than 3''
''so you look at the 4 in the tens place''
''Now that's really 4 tens''
''so you make it 3 tens''
''and regroup and change a 10 to 10 ones''
''and add them to 2 and that's twelve''
''and you take away 3 and leaves... 9''! }}
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** Take any natural number. Divide it by 9 and write down the remainder. Then do the trick of adding the digits until you get a single number. You'll get the remainder unless the original number was a multiple of 9 (in which case you'll get a sum of 9 and a remainder of 0).
*** 204/9 = 22 with remainder 6. 2+0+4=6.
*** {{spoiler|1=For those who can do
* Along the same vein, if you sum the digits of any multiple of three, you get a multiple of three.
* This is probably really obvious but write 0123456789 down a paper and write 9876543210 next to it (also down). Hey look, it's the 9 times table!
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** The trick? {{spoiler|1=(x-5)(x+5) = x^2 - 25}}.
*** More generally: {{spoiler|1=(x-y)(x+y) = x^2 - y^2}}.
* Contrary to what most over-zealous math teachers might suggest, it is possible to divide by zero, provided you're in a
** So if x-x is not zero, and you add x to both sides, x=x+ (some non-zero number)...?
** Wait, the [http://www.mathwords.com/r/reflexive_property.htm reflexive property] has a point?
*** It does. So does
** The overzealous math teachers to whom you refer can fall back on the fact that essentially all math that's taught from kindergarten through high school is done in
* Here's a fun one: 1+2+3+4+5+6... etc. can in some sense be shown to be equal to -1/12.
* Behold, [http://www.smbc-comics.com/index.php?db=comics&id=1914 Polish hand magic.] A visual method for multiplying any two integers without a calculator. And it even comes with an explanation.
** This troper tried the 7*8 example and got 53... primarily because the troper's right hand is not like the other. (Google image Brachysyndactyly, I dare you.) Then again, "imagine negative fingers and extra fingers!" [[Writers Cannot Do Math]] anyways, so...
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*** There exists a well-ordering of the reals in ZFC. However, the usual ordering—''i.e.'', '''<'''—is not a ''well''-ordering on the reals. Furthermore, the complex numbers do ''not'' form an ordered field, so well-ordering them is even harder.
* Most people know about 5 number systems: Natural Numbers, Integers, Rational Numbers, Real Numbers, and Complex Numbers. However some other interesting ones are:
** the Algebraic Numbers, which consist of numbers that are solutions of polynomials with rational coefficients — ''e.g.'', all rational numbers, ''φ'' (the so-called
** the Quaternions, Octonions, and Sedenions: Complex numbers extended to quadruples, octuples, 16-tuples etc.
** the Hyper-real numbers: includes infinite numbers and infinitesimal numbers
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* Here's a fun one to amuse your friends: 111111111 X 111111111 = 12345678987654321
* So you need to find the square root of a number, but you have one of those cheap calculators that don't have a square root function, or maybe you don't have a calculator. No worries, there's a quick, simple and Babylonian way to solve this problem.
{{quote|
2) Find the average of G and N/G (i.e. (G + N/G)/2 ).
3) Use the average as the new G and repeat step 2. Keep on doing so until G stops changing (due to rounding errors, not enough of digits or laziness).
4) The final G is the square root of N. }}
** Example: What is √65536<ref>Programmers, don't blurt out the answer</ref>? (Uh... I don't know, 376?)
{{quote|
G<sub>1</sub> {{=}} (376 + 65536/376) / 2 {{=}} 275.1489362
G<sub>2</sub> {{=}} (275.1489362 + 65536/275.1489362) / 2 {{=}} 256.6663332
G<sub>3</sub> {{=}} (256.6663332 + 65536/256.6663332) / 2 {{=}} 256.0008649
G<sub>4</sub> {{=}} (256.0008649 + 65536/256.0008649) / 2 {{=}} 256
G<sub>5</sub> {{=}} (256 + 65536/256) / 2 {{=}} 256
'''STOP HERE'''
Therefore, √65536 {{=}} 256 }}
** Fun fact: If your initial guess is a negative number, you'll get the negative root as the answer. That is, if the initial guess for √65536 was -376, the final answer would have been -256.
* Out of all the undefined values (commonly occurring a mix of 0 or ∞), the reason why 1^∞ is undefined, as opposed to 1 (as 1 raised to anything equals 1, right?), is because 1^∞ can equal Euler's constant. Or to be more precise, the naive solution to how to get e, lim x -> ∞ ( 1 + 1/x ) ^ x is 1 ^ ∞. But there's some rules that rearrange the equation, such that you can approach e.
----▼
▲----
<small>...my brain hurts. I think I'll stick to biology.</small>
:<small>Back to [[{{ROOTPAGENAME}}]]</small>
{{reflist}}
[[Category:Math]]
▲[[Category:Trivia]]
▲[[Category:checknamespace]]
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