Also seen expressed thuslywise: "What's an anagram of 'Banach-Tarski'? 'Banach-Tarski Banach-Tarski'."
Another catch is that this stems from an axiom (namely, the axiom of choice) that's commonly omitted from set theory, being totally consistent with and independent of the others, but frequently yielding paradoxical results like these. You can choose to use it or not use it, similar to the parallel postulate.
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.
And then there's Whitehead and Russell's 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 1+1=2◊. And then Gödel's Incompleteness Theorem undermined the whole thing using a real-life Logic Bomb.
"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 googol is 10^100, or 1 with 100 zeroes after it. This is bigger than the number of elementary particles in the known universe (10^80). A googolplex (10^(10^100), or 1 followed by a googol zeroes) is bigger than the number of elementary particles in the known universe (10^80) and the number of planck times since the big bang (8* 10^60)...times each other.
And then there is the googolplexian (10^(10^(10^100))). It has a googolplex of zeroes after the first digit. But even the googolplexian is nothing compared to...
The Chihiro numbers, abbreviated C(n), are a series of numbers that grow ridiculously large. Named after mathematician Chihiro Kagachi, the name is amusing to many anime fans. They follow as the extension of 1+1, 2*2, and 3^3 (2, 4, 27...). The fourth term is equal to 4^(4^(4^(4)), which is already larger than a googolplex. This can be even more ridiculous by looking at C(C(C(n))), but you'd need about 63 iterations of the function to approach...
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.
TREE(3). Graham's number, an upper bound is dwarfed by a lower bound!
Möbius strips. It's one thing to have a ridiculous, mind-bending shape like that on paper, but you can make them in real life. It's the weirdest feeling in the world to hold one.
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 Lego pieces. 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 Lego either, you can also make them out of an odd amount of Buckyballs magnets 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.
Klein bottles take it a step further by doing away with those pesky edges. And you can cut them in half into a pair of Möbius strips and it's possible to cut a Klein bottle into a single Möbius strip. Unfortunately, unlike their Möbian cousin, they can't actually exist in our reality. The glass bottles, hats, and the like that one sometimes sees in museums and online stores are actually 3D projections of the true 4D Klein bottle, and don't really have its bizarre properties.
New use for the Mobius strip: the world's smallest-denomination polyhedral die. (OK, monohedral die...is that even a word?)
You could just use a marble, except a marble has about as many sides as atoms on the surface, which would make a marble the largest-denomination polyhedral die.
Here's a nifty little poem about Mobius strips:
A mathmetician confided
That a Mobius strip is one-sided.
And you'll get quite a laugh
If you cut one in half,
For it stays in one piece while divided!
0.999..., which is a decimal with an infinite number of nines behind it, is exactly equal to 1. Not almost, not "so close there's no practical difference," exactly equal. Here are three proofs of this:
First, from the accepted decimal expansions of some fractions, such as:
1/3=0.333... (1/3)*3=(0.333...)*3 1=0.999...
Second, by simple algebraic manipulation:
Let 0.999...= x 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^{note }which, incidentally, is proven using a construction similar to the second proof:
There are the same number of odd numbers (1, 3, 5, ...) as there are natural numbers (1, 2, 3, ...). (It's ridiculously easy to prove it, too: just multiply all the natural numbers by 2 and subtract 1.) What's even more bizarre is that there are the same number of 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—"computable numbers"—fit into a countable set. More on this here.
Which brings up that for almost all numbers, there is literally no process of computation that doesn't have some finite limit to its precision, and never can be.
This can be extraordinarily confusing, and tends to 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 Hilbert's paradox of the Grand Hotel: in a hotel with an infinite number of rooms, the propositions "every room is occupied" and "no room for more people" are not equivalent.
Knowledge of algorithms, induction, counting, what real numbers are, and how you tell two real numbers apart seems to be all you need to know to understand Cantor's Diagonalization Argument.
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.
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 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? Gabriel's Horn is such a figure. And then you get the Koch Snowflake, which has an infinite perimeter but a finite area. Similarly, the Menger Sponge has infinite surface area and finite volume, but unlike Gabriel's Horn, it can fit within a finite bounding box. Sierpiński's Carpet has an infinite perimeter but zero area.
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.
The reason you can do this sort of thing is that any finite-dimensional vector space V over a given field F is isomorphic to F × F × … × F for dim(V) factors F. In other words, no matter what your vector space is, any vector in it can be represented as a tuple (i.e., a field-valued vector in the classic sense) as long as it is finite-dimensional.
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.
And the graph of that looks like this◊. Makes a nice T-shirt.
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
The principal value — in fact, every value — of i^i is a real number.
By reducing x of e^(xπi) to only integers you have a function with an imaginary part but can only take in and put out real numbers. Specifically it alternates between 1 and -1 for an eternity.
10^2+11^2+12^2=13^2+14^2=365. Looks like someone had fun screwing with our neighborhood.
If you want proof that Evil Is Cool, play around with the Number of the Beast. 666 is not only the sum of the squares of the first seven primes, but it's also a palindrome. 1^3+2^3+3^3+4^3+5^3+6^3+5^3+4^3+3^3+2^3+1^3 is another palindrome, which equals 666. And that middle number, 6^3? That's three sixes, of course. Cool, right?
Want another fun game with 666? You'll discover that it is the sum of all cardinal numbers from 1 to 36. Since 36 = 6^2, that means you can fit it into a Magic Square. Let he who hath understanding reckon the number of the beast.
Some sources say the number is 616, though that's still a palindrome.
Probability can be a very strange thing to behold at times. For example: You can have a situation where the probability of any given outcome is 0, yet outcomes still happen (thanks to our good old friend "continuum").
That's defined more precisely as probability density. Instead of probability of specific values, you calculate the probability of landing within selected boundaries by finding a slice of area under the density function (the total area under it being 1 = 100%).
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: two cents.
You can justify it by saying that if we pick a real number at random, we can never be sure what exactly we picked. Even if we could somehow express arbitrary irrational numbers in the real world, calculating them requires an infinite number of steps—in practice, we stop somewhere and declare margins of error. And we can't even express them in the physical world—the circumference of a real physical circle with a diameter of 1 cm would never be exactly π cm (and on that matter, we cannot expect the diameter to be exactly 1 cm either). If we choose numbers by, for example, marking a point on a ruler, then not only does the ruler only have a finite number of atoms, but the mark would have finite thickness, specifying a whole range of numbers (presumably much wider than the size of an atom), and ranges have nonzero probability.
Another weird probability distribution is the Cauchy Distribution, which does not have a well-defined mean because the integral does not converge.
Here's a strange one, the minimum number of people you need so that there's a 50% chance of two having the same birthday is only 23. (Perhaps not so strange if you remember that with 23 people, there are 253 ways to choose two...)
There's also the Pidgeonhole Principle, complete with proof, on explaining that if you have x+1 elements and x slots, one slot will have at least two elements.
For the entirely useless^{note }say for binary, hexadecimal, etc.… famous for being used in computers. Some have argued the base 12 system would make it easier to teach everyday math but we probably won't switch to it for the same reason the US won't switch to the metric system. This process is also used in extending the rationals into the p-adic numbers instead of the real numbers., 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 2i. Or even the base of the golden ratio.
Quiz: What is the number 2 (base 10) when written in base φ? Hint: it has four digits, plus a decimal point. Answer: 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.
New Math. Yes folks, the brilliant folks in charge of our education systems tried to throw matrix ops, set theory, number bases, and abstract algebra at elementary schoolers. Many of the elementary teachers ordered to do the throwing didn't know the material themselves and wouldn't have been able to teach it to high school and college students of appropriate level and background.
And a truly trivial piece of trivia: you know prime numbers, the numbers which can't be divided by any integer but 1 and themselves? One such number is ... eight six seven five three oh ni-i-ine!
e to the power of π is a strange number. When you take π from it, it's ridiculously close to 20.
Combine this with Euler's Identity and you have (20 + π)^i ~ -1 (The imaginary part is a bit off, though.
The fraction 355/113 is a far better approximation to π than any of us will ever need (seven-figure accuracy). Two fives, two threes, two ones, in a really simple fraction.
But a far more fun approximation is 42/1337 is approximately π/100, in other words π% of 1337 almost exactly 42 (actually 42.003...).
Multiply nine by any "natural" number. Any at all. Now add up the digits. If it leaves you with multiple digits, keep adding up digits (from that and so on) until you're left with only one. That remaining digit will be nine, every time, no matter what number you picked for the multiplication. This works because 9=10-1. If we used a different base, say, trinary, then all even numbers would have a digital sum of two. The side-effect of all multiples of three having a digital sum that is also a multiple of three (the only three of which are less than ten, thus being possible values for digital sums^{note }digital sums can only result in a one-digit number. If it has more than one digit, it's not the digital sum, are 3, 6, and 9) is because nine is a multiple of three.
The best part is, no other numbers have this property. if a number's digits don't add up to 9, you can know it isn't a multiple of nine.
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.
For those who can do modular arithmetic: 10000a*+1000b+100c+10d+e mod 9 = 9999a+999b+99c+9d+(a+b+c+d+e) mod 9 = a+b+c+d+e mod 9. And it works with any base too, like with F in hexadecimal or 3 in quaternary.
This also means that not only does every multiple of 3 have a digital sum that is also a multiple of 3, but they repeat in the pattern 3,6,9,3,6,9... Try it yourself: Take any multiple of 3 and divide it by 3, then take the result and divide by 3 again, and write down the remainder. Then find the digital sum of the original number, and divide that by 3. You'll get the remainder unless the original number was a multiple of 9 (in this case, since the sum was 9, dividing it by 3 equals 3).
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 oriented downwards on a paper and write 9876543210 next to it (also oriented downwards). Hey look, it's the 9 times table!
The sum of any two consecutive, positive, whole numbers is equal to the difference of their squares. E.g.
More simply, imagine a square. To get to the next square, you add a row on the side, making it one wider. Then add a row on the top, and its length is the side of the original square plus one for the row just added on the side. So you add a side (the square root), and then you add another side +1.
Or more intuitively:
3*3 = 9
3*4 = 9+3
4*4 = 9+3+4
4*4 - 3*3 = 3+4
Like the above trick, the product of two integers two away from each other is one less than the square of the integer between them: for example, 2^2=1*3+1. 5^2=4x6+1. 126^2=125*127+1.
Why? By factorizing, x^2-1 = (x+1)(x-1). This actually comes in all the time when you do algebra in middle and high school, but I bet you didn't realize its implications.
Relating to the above two points. The product of ANY two numbers(including decimals and negatives) is equal to the square of the mean of these two numbers minus the square of the difference to the mean.
4x6=5^2-1^2=24
5x9=7^2-2^2=45
This rule can be useful for larger multiplications without the need for a calculator. Eg. 28x32=30^2(900)-2^2(4)=896
The third step of the above proof suggests an alternate version, one that actually helped early computers: to multiply any two numbers, take the sum and difference between them and square them, then subtract the square of the difference from the square of the sum, and divide the result by 4.
Using a trick with factorizing, you can make it seem like 2=1.
A=B
A^2=AB
A^2-B^2=AB-B^2
(A-B)(A+B)=B(A-B)
A+B=B
B+B=B
2B=B
2=1
The trick is that since A = B, A - B is equal to zero; dividing by zero is not allowed.
If you were to allow division by 0 (calculus voodoo), you'd still end up with something undefined, as you should get A = B = ∞. But since ∞/0 is undefined...
You can play a similar trick using integral calculus — in this case there's no division by zero and the error is more subtle. It works like so:
INTEGRAL 1/x dx = INTEGRAL u dv, where u = 1/x, v = x.
So, integrating by parts, INTEGRAL 1/x dx = uv - INTEGRAL v du = (1/x) x - INTEGRAL x*(-1/x^2) dx.
Simplifying this gives INTEGRAL 1/x dx = 1 + INTEGRAL 1/x dx, so 1=0.^{note }Note, however, that these are indefinite integrals — that is, they're only defined up to a constant of integration anyway. Essentially, you just have to pick the constant on one side of the equation to be 1 greater than the constant on the other side, and then everything works out nicely.
Using Euler's Formula:
e^2πi=1
ln(e^2πi)=ln1
2πi*lne=ln1
2πi=0
2πi/2πi=0/2πi
1=0
The trick is the ln(c) where c is any nonzero complex number will have infinitely many solutions due to trigonometry and euler's formula.
2+2=5 is actually very simple by abusing definitions of rounding (that is, 0.4 and below is rounded low and 0.5 and above is rounded up). For 2.4=2, 2.4+2.4=4.8, which can be rounded off to 5, so 2+2=5 for rounded off numbers.
2+2=5 for sufficiently large values of 2.
I've always liked this for 1=-1
-1=-1
1/-1=-1/1
1/i=i/1
1=-1
In this case, the trick is that the third line attempts to use the property that sqrt(a/b)=sqrt(a)/sqrt(b) — which is only true when a and b are both positive real numbers. If you allow a or b to take on negative or complex values (like, say, -1), this is no longer a valid manipulation.
Wait, -1x-1=1, the root of 1 is 1, .: -1=1.
Want to find the square of a number ending in 5? Multiply the numbers 5 greater and 5 fewer than it, and then add 25, e.g. 35^2 = (40 * 30) + 25 = 1225. From here, you can move on to squares of other numbers, or just numbers that are relatively close in value.
The trick? (x-5)(x+5) = x^2 - 25.
More generally: (x-y)(x+y) = x^2 - y^2. That means the differences of the squares of any two numbers is equal to the product of the sum and difference of the two numbers, and the difference of any two numbers is equal to the product of the sum and difference of their square roots. The thing about the product of two numbers two apart above is another special case.
A different trick for squaring numbers ending in 5: (1) Take the numeral n formed by the left digits up to the tens place. (2) Multiply n by n+1. n*(n+1) are the leftmost digits of the answer. (3) Multiply n*(n+1) by 100 to ensure that it's in the right spot. (4) Add 25. Done. Example: find the square of 155. (1) n = 15. (2) 15*16 = 240. (3) 240*100 = 24000. (4) 24000+25 = 24025. Done. The trick? (10x + 5)^2 = 100x^2 + 100x + 25 = 100x(x+1) + 25.
Contrary to what most over-zealous math teachers might suggest, it is possible to divide by zero, provided you're in a wheel. The consequences though? x - x is not always zero^{note }x-x=0x^2, 0 * x is not always zero, and x / x is not always 1^{note }x/x=1+0x/x.
So if x-x is not zero, and you add x to both sides, x=x+ (some non-zero number)...?
Specifically, x=x+0x^2, and if 0x=/=0, then 0x^2=/=0.
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 rings and fields, wherein division by zero is not defined (assuming that division is even possible).
Here's why you can't divide by zero: Suppose that x is a real number (for argument's sake). Now 0 = 0 + 0, so 0 × x = (0 + 0) × x = (0 × x) + (0 × x), so 0 × x = (0 × x) + (0 × x), i.e., 0 = 0 × x. Now, suppose that it is possible to divide by zero. Then there is a unique real number, say z, such that 0 × z = 1. Then the equation 0 × x = y has a unique solution, namely, x = z × y. However, we just showed that 0 × x = 0 for any x, so y = 0. Hence x = z × 0. We assumed that z × 0 = 1, so x = 1. However, by our above argument, x = 0, so x = 0 = 1, a contradiction. Therefore it must not be possible to divide by zero.
The Riemann Sphere and Real Projective Line allow division by zero and the former has useful applications in science and engineering. However these number systems still leave some arithematic operations undefined such as infinity + infinity or infinity * zero.
Here's a fun one: 1+2+3+4+5+6... etc. can in some sense be shown to be equal to -1/12. In two different ways.
Behold, Polish hand magic. A visual method for multiplying any two integers without a calculator. And it even comes with an explanation.
There's a joke proof that states that there are no uninteresting numbers. Here it is: Assume that there is some set of uninteresting numbers. Of this set, one of these numbers must be the smallest one. Thus, it would be the smallest uninteresting number, and therefore interesting, which would remove it from the set. Then the next smallest uninteresting number would become the smallest, which would be interesting, and so on, until the set of uninteresting numbers is empty. Therefore, all numbers must be in some way interesting!
In fact, there is a definition of "interesting" numbers (actually called definable numbers), but one has to venture outside of set theory to state it.
Also, you can only pick the smallest if the set of all numbers you're considering is well ordered, which is true if it's the set of integers. As for real or complex numbers well....
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. In fact, it is known that there is no well-ordering on the complex numbers that is compatible with the usual arithmetic.
Believe it or not, this isn't really that far off from the proof of Godel's incompleteness theorem, or from Russell's paradox, which basically involves trying to answer the question "should This Index Is Not an Example contain itself?"
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 golden ratio), √2, etc. Those numbers that aren't algebraic are called Transcendental Numbers, and include e and pi.
the Quaternions, Octonions, and Sedenions: Complex numbers extended to quadruples, octuples, 16-tuples etc. Basically, take the coordinate-plane number line you learned when you learned complex numbers, and extend it through four, eight, sixteen, or any power of 2 dimensions.
the Hyper-real numbers: includes infinite numbers and infinitesimal numbers
the Surreal numbers: which is the largest possible totally ordered class allowed by set theory
the surcomplex numbers take the form a + bi where a and b are surreal numbers.
Here's a fun one to amuse your friends: 111111111 X 111111111 = 12345678987654321
Better, you can show it's true, by hand, as early as elementary school!
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.
1) Take a wild guess at what the square root of the number might be. Let's call the guess G and the original number N.
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^{note }Programmers, don't blurt out the answer? (Uh... I don't know, 376?)
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.
Or to put it another way, if f[a]=(f[a-1] + n/f[a-1])/2 where a is any positive integer, n is a constant, and f[0] is any arbitrary positive number, lim(a->∞) f[a]=√n. By the way, this is pretty close to how calculus works. That's right, the Babylonians were almost doing calculus centuries before Newton.
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 number. 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.
Another cool math “trick”: n^2 = the sum of n consecutive positive odd numbers.
Suppose n = 3. n^2 = 9; 1+3+5 = 9
Suppose n = 5. n^2 = 25; 1+3+5+7+9 = 25
Suppose n = 11. n^2 = 121; 1+3+5+7+9+11+13+15+17+19+21 = 121
General proof, by induction: (n^2)-(n-1)^2 = (n^2)-(n^2 - 2n + 1) = 2n-1, and 1 = 1^2 = 2(1)-1.
For some light (though trippy) relief, try searching for "seahorse valley" on YouTube. You don’t even need to know the maths to appreciate the pretty animations. Be sure to try at least one of the 3D ones.
People use irrational numbers by rounding to the nearest degree of accuracy. In essence they're subtracting a really small part because it's so little as to not make much of a difference. So π can become 3.14 or √2 1.414. If you think about it every number N has an infinite amount of numbers a very small distance away where it would be easier just to round to N. If you had to use 2.00000000000000000000000000000000001^{note }Professionals would need a number closer to 2 than this one but it should get the point across. in an equation would you?
Ever wonder why we have sixty minutes in an hour and sixty seconds in a minute? Blame it on the Ancient Middle East. Way back when, there were two tribes who counted things differently. One tribe used the ten fingers on both hands, so they went from 1-10. The other tribe used one hand and using their thumb counted the segments on the remaining fingers. They would go from 1-12 before starting over (for example 67 would be (5, 7) base 12.) Sixty is the least common multiple between these two numbers.
Within some irrational numbers, a number that never repeat nor do they end, is everything. In the decimal expansion of a "normal" number, there exists ever social security number, every birth date down to the second, the nuclear launch codes, the works of Shakespeare (if one translates numbers into letters and even punctuation), that tenth grade English paper. Everything.
Almost all irrational numbers are normal, and it is conjectured that Pi, e, √2, and Phi are among them. So, when you look at a circle again or some number under the square root sign, remember this. And for some more fun, this page has the first two billion digits of these four irrational numbers and you can search through them for a number. But despite all this empirical evidence, there's no proof that they're normal. As such, it is not certain that every permutation of number will appear in the expansion. In fact within our current understanding of said numbers, it is not out of bounds, albeit highly unlikely, for the digit 2 to never appear again after the 2^2^1299709 - 7th digit.
One irrational number that is not normal is Liouville's constant, which goes something like this: 0.1100010000000000000000001000000000000... You have rapidly widening gap between each one, and then just zeroes. Its irrational, and even better, it's transcendental, and you have no social security numbers there, no works of Shakespeare, nothing but information about factorials.
The Pythagorean Theorem is not just about the relationships of sum of squared numbers. It is about area. It says that if you have the two similar geometric figures (it could be two squares, two triangles, two circles, or even atypical shapes), then you can combine them to create a third geometric figure of similar shape. In fact, that's basically how Pythagoras' own proof of it works.
This ancient theorem has been proved many times over, including then Representative James A. Garfield of Ohio (he would go on to be President) during one session of Congress in 1876. An animation of this proof can be seen here and here. It has been called "really a clever proof" by one mathematics historian on the other Wiki.
Computer Scientists have something known as recursion which is really useful if you want say a list of the first million fibonacci numbers because you don't have to manually go through and say the first number is 1, the second number is 1, and so on. What's really interesting is that basic stuff in math has this same recursion property. For instance if you wanted a list of the natural numbers. You only have:
Set the zeroth term in your list equal to 1
Set any subsequent term equal to the last term plus one.
This is important because it allows mathematicians to base their theorems on as few assumptions as possible and nature comes equiped with recursion for the simplest of things.
Tired of using sixteenth of an inches. No problem. Just invent a unit of length that is equal to 1/16 of an inch. Tired of multiplying all those four digit numbers together. Invent a unit of length that is equal to 100 smaller units of length. The metric and si systems are designed to take advantage of the ability to do this. You could invent a unit of length equal to a graham's number of smaller units. In one sense that would be a really large unit of length but in another sense it would be really small.
Don't get too caught up in being able to do this. Mathematicians have a funny story about a man named Pythagoras who did that. Some distances just refuse to be rational.
Do you know all those repeating decimals you learned about in middle school or high school such as 0.3333...,0.1111...,0.1666...,0.142857142857.... Well they all have fraction representations that can be easily figured out.
For 0.1666...
16-1=15: This is your numerator
The 6 repeats so you put a 9. The 1 doesn't repeat so you put a 0 to the left: So your denominator is 90
Now reduce and you have 1/6.
For 0.123333...
123-12=111: This is your numerator
The 3 repeats so you put a 9 and this time two zeros to the left since you have two part that don't repeat: So your denominator is 900
Now reduce and you have 37/300.
For 0.232323...
You make the repeating part your numerator since there's no non repeating part to subtract.
You make the denominator two 9's since your repeating part has two parts to it.
The fraction is 23/99
There's an easier way to explain it using algebra and it explains why these guidelines work but the above way is easier to do in your head if you have all the guidelines down.
Here's another fun bit of trivia. 7 X 7=49. Except this isn't always true. Sometimes 7 X 7=50. This is thanks to significant figures. If you're measuring something, you have to use significant figures, because your answer can't be more accurate than your variables.
Add any two even numbers together and you will get an even number. Add any two odd numbers together and you will get an even number. If you want to get an odd number you have to pick one even and one odd number to add together. Let m and n be in the set of Natural numbers, then the proof is as follow:
Say I have two even numbers 2m and 2n. 2m+2n = 2(m+n). Say I have two odd numbers 2m+1 and 2n+1. (2m+1) + (2n+1) = 2m+2n+2 = 2(m+n+1). Say I have one odd and one even number 2m and 2n+1. (2m) + (2n+1) = 2(m+n)+1.
In the The Twelve Days of Christmas if one adds up the total number of gifts received by the last day, one has received 364 gifts. So one gift on each day that isn't Christmas.
There's a joke computer file system called PiFS that claims its data free (except for the metadata file) and you can store and retrieve any file. The play on this is that any irrational number (a number that does not end or have repeating digits) contains any finite sequence of numbers. All you have to do is find out where in pi this sequence is.
People have noted that pi, e, and other irrational numbers are illegal numbers because they have to contain said illegal number.
The number 12345679 has some...interesting behaviors when multiplied by integers. If you multiply it by a multiple of three, and then pad with zeros up to nine digits, then the result will be the same three digits repeated three times (ie: 037037037, 185185185). If your multiplier is not a multiple of three, the resulting nine-digit number will have no repeating digits. Specifically, it will contain all the digits from 0-9 except one, and the missing one won't be a multiple of 3. (Ie: 049382716, 987654320). The above works for the multipliers from 0 through 81. For larger numbers you have to slice the number up into nine-digit chunks, stack them up, and add them together. This results in the above pattern continuing. (This works because 12345679 * 81 = 999999999. I have no idea why the rest of it happens though.)
Chaos Theory gets into some very strange places. A few examples:
Ok, so you have a function that resembles a hump. ANY FUNCTION resembling a hump. You make a Bifurcation diagram of the function. It will always look similar in many different ways to another function that has that general shape. NO MATTER WHAT FUNCTION YOU PICK
Math and logic as we know it all comes about because of Euclid who lived in 300 BCE Greece. With only five Axioms, ideas he believed and were self evident, and five Postulates, ideas which could not be proven and must be held true on faith, did he set out and start what would become mathematics. His book The Elements has lasted for over 2000 years of changing history.
“A point is that which has no part” – Euclid, The Elements.
Euclid's fifth Postulate, which is the first parallel postulate:
If a straight line falls on two straight lines in such a manner that the interior angles on the same side are together less than two right angles, then the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
For 2000 years people tried to prove this could be proven from his other assumptions. They failed. Removing it brings about two other geometries: Spherical where there are no parallel lines, and Hyperbolic where there are infinite parallel lines.
In Spherical and Hyperbolic geometries, the only similar triangles, two or more triangles having the same angular measurements, are congruent triangles, two or more triangles that share the same angular measurements as well as lengths of each side.