Do you have a mathematical formula/solution method/theorem of your own creation?

[Yasha]

I've devised or found quite a number of mathematical formulae, identities and solution methods in the past. Most of them are used to solve particular types of mathematical puzzles. But among them there is a method I found of which I'm fairly proud. I can't show it here because it's going to take up a lot of space (simple calculations but very, very long) but I'm going to show you what it can do.

Take this random sequence for example: 3, -21, 8, 9, -4

I can use my method to find a polynomial function that represents this sequence where x= 1,2,3,4,5

I found the function to be (95/24)x^4 - (637/12)x^3 + (5905/24)x^2 - (5399/12)x + 256.

I can always find a polynomial function that represents a sequence regardless of how random it looks and the terms need not be integers but they have to be rational to ease the calculation.


So, had you ever come across any mathematical formula/theorem/method to solve particular problems by yourself before? If you had you can share them with us here. It can be something you deduced by yourself but later found out that it's actually a well-known formula/theorem or it can be a certain creative way you use to solve a particular problem that is different from the conventional method your teacher and classmates use.

[Link]

I figured out the circumference of a circle in the 5th grade intuitively... but my natural ability with math fell off at some point. I just do what I'm shown to do.

[Yasha]

I figured out the circumference of a circle in the 5th grade intuitively... but my natural ability with math fell off at some point. I just do what I'm shown to do.


Yeah, I think schools make many of us lose creativity and originality.

[myle]

Take this random sequence for example: 3, -21, 8, 9, -4

I can use my method to find a polynomial function that represents this sequence where x= 1,2,3,4,5

I found the function to be (95/24)x^4 - (637/12)x^3 + (5905/24)x^2 - (5399/12)x + 256.

I can always find a polynomial function that represents a sequence regardless of how random it looks and the terms need not be integers but they have to be rational to ease the calculation.


So, had you ever come across any mathematical formula/theorem/methods to solve particular problems by yourself before? If you had you can share them with us here.

This reminds me of the interpolation polynomial. One of the simplest method to do that is the Newton. You can also use Lagrange or just solve a system.

[sel]

I found out the newton-raphson intuitively when messing around once (for solving a polynomial equation)

[infinite]

I once did some integration by myself, without ever heard of it before. The only thing that was my teacher was: Do the oppositte of dy/dx.

One thing I did found myself was a winning formula for any Matrix-game that's 2 x 2.
You could solve the formula without computer :amuse, bad part... best strategy made the game even...

[Zaru]

I worked quite long on this one but I think it's without fallacies and absolutely plausible:

Uchiha = shit

[QuoNina]

That's a 5 equations, 5 unknowns problem (linear equations). If your method is a formulated approach, then that does remind me of the calculation techniques we learned using matrices and the determinant to calculate solutions for linear equations. Matrix is a convenient way to group coefficients and solve or perform operations on linear equations. (I am just telling this in case you never encounter matrix operations.) Linear equations are solved very systematically (although the numbers might be ugly) using matrix operations.

Hm, I don't remember I've really come up with something worth mentioning. But I remember, when I was in middle school, my math teacher asked me several times to go up to the board to explain to the class the "alternative" ways to solve problems in our class exercises. It's mostly just different approaches to problems instread of creations of something. And my teacher's true purpose was to give me chances to practice public speaking skills. She found me quite fascinating for being very eloquent in private yet couldn't explain a simple thing smoothly in front of a crowd. So she kept on putting me on the spot. I was her lab rat basically. ^o^

[martryn]

No. Not any that I remember. Tell you the truth, I can't really think of anything I'd need that doesn't already have a pretty good solution.

[impersonal]

No. Not any that I remember. Tell you the truth, I can't really think of anything I'd need that doesn't already have a pretty good solution.

Well, so far it's the same for everyone in this thread ;)

For some reason, it's much easier to find things a second time, even when you have never heard of them. Perhaps it's because we live in the paradigm (read: the mindset) which follows from these solutions.

I was quite happy with a few "discoveries" I made in philosophy, until I read authors who said the same thing and went beyond, decades or centuries before me.

[sel]

I was quite happy with a few "discoveries" I made in philosophy, until I read authors who said the same thing and went beyond, decades or centuries before me.

I know!

I was so annoyed when hearing my friend telling me about Descartes :mad

[mislead]

I once calculated the chromatic polynomial of a tree by accident. I was so proud of myself till I realized it was something utterly trivial.

:(

I do have a tendency to have good intuition for analysis, and rather frequently think "I knew this was true." when seeing theorems on lectures.

[martryn]

I've had a couple of scientific ideas, especially concerning the nature of light and of the nature of photons in particular, but I'm pretty sure that most of my ideas, probably all of them, have already been addressed.

As for formula used and what have you, I'd come up with some stuff during exams in college but they were all merging a few ideas here and there to come up with something, and what physics student didn't sit in an open book exam and not come up with some crazy equation that happened to give the right answer only to discover when they got their exam back, graded, that the equation they used was simply taking two or three other ones and merging them without realizing that that was what they were doing at the time.

It's especially easy to do something like that when matrices are involved, I find.

[Red]

No one fucking laugh or I neg you till yesterday comes again.

I was 13 and still schooling in Africa and they gave us a simplified rundown of the fundamental particles. At that stage they only taught us Neutron,Proton and Electron.

The next class I asked what gives them they're different mass the teacher shot me down by asking how a little rat like me who's barely passing his class expects to understand the subtle mechanics of fundamental particles i.e He didn't know :awesome.

It was until I read how different frequencies gives light it's different colours and how Light is the same thing as x-rays the only difference is the frequencies. Then it clicked What if the same is applied to matter wouldn't it mean that matter is made from the same substance only thing giving them their different properties how they vibrate at their base levels? I felt really smart until I heard of the higgins field (http://en.wikipedia.org/wiki/Higgs_field). Then I felt really dumb .

[Bleach]

2+2>3<2+2

Umm what @_@?

[fxu]

2+2>3<2+2

Umm what @_@?


Sir, you're a genius.

You've just discovered the cure for AIDS .. it all adds up, it all makes sense.

5 is great than 3, yet is less than 5. Amazing yet surprising, how could such theory solve a million lives.

[TH4N4T0S]

I had intuitively discovered how to solve 3x3 matrices using Cramer's rule several years ago. I've discovered a couple of theorems in geometry, but the day I introduced them, I was saddened to learn that they already existed. So annoying. I'm still proud, though. I also had a luck in proving how formulas came to be. I forcefully wanted to prove them because of the mere fact that teaching formulas by "telling them to students" is a very ineffective way of teaching. I wanted to show how important it was to know how formulas were formed. Very seldom did I hear right answers from batchmates when I asked them how the quadratic formula was formulated. It was simply through completing the square . . . I wish education to evolve.

[Xion]

No one fucking laugh or I neg you till yesterday comes again.

I was 13 and still schooling in Africa and they gave us a simplified rundown of the fundamental particles. At that stage they only taught us Neutron,Proton and Electron.

The next class I asked what gives them they're different mass the teacher shot me down by asking how a little rat like me who's barely passing his class expects to understand the subtle mechanics of fundamental particles i.e He didn't know :awesome.

It was until I read how different frequencies gives light it's different colours and how Light is the same thing as x-rays the only difference is the frequencies. Then it clicked What if the same is applied to matter wouldn't it mean that matter is made from the same substance only thing giving them their different properties how they vibrate at their base levels? I felt really smart until I heard of the higgins field (http://en.wikipedia.org/wiki/Higgs_field). Then I felt really dumb .

Damn, your teacher sounded like an asshole. :oh

How did you actually put up with that crap?

My Formula:

Let X be a connected complex manifold of complex dimension n. X is then an orientable smooth manifold of dimension 2n, so its cohomology groups lie in degrees zero through 2n. Assume that X is a Kähler manifold, so that there is a decomposition on its cohomology with complex coefficients:

H^k(X, \mathbf{C}) = \bigoplus_{p+q=k} H^{p,q}(X),

where Hp,q(X) is the subgroup of cohomology classes which are represented by harmonic forms of type (p, q). That is, these are the cohomology classes represented by differential forms which, in some choice of local coordinates z_1, \ldots, z_n, can be written as a harmonic function times dz_{i_1} \wedge \cdots \wedge dz_{i_p} \wedge d\bar z_{j_1} \wedge \cdots \wedge d\bar z_{j_q}. (See Hodge theory for more details.) Taking wedge products of these harmonic representatives corresponds to the cup product in cohomology, so the cup product is compatible with the Hodge decomposition:

\cup : H^{p,q}(X) \times H^{p',q'}(X) \rightarrow H^{p+p',q+q'}(X).

Since X is a complex manifold, X has a fundamental class.

Let Z be a complex submanifold of X of dimension k, and let i : Z → X be the inclusion map. Choose a differential form α of type (p, q). We can integrate α over Z:

\int_Z i^*\alpha.\!\,

To evaluate this integral, choose a point Z and call it 0. Around 0, we can choose local coordinates z_1,\ldots,z_n on X such that Z is just z_{k+1} = \cdots = z_n = 0. If p > k, then α must contain some dzi where zi pulls back to zero on Z. The same is true if q > k. Consequently, this integral is zero if (p, q) ≠ (k, k).

More abstractly, the integral can be written as the cap product of the homology class of Z and the cohomology class represented by α. By Poincaré duality, the homology class of Z is dual to a cohomology class which we will call [Z], and the cap product can be computed by taking the cup product of [Z] and α and capping with the fundamental class of X. Because [Z] is a cohomology class, it has a Hodge decomposition. By the computation we did above, if we cup this class with any class of type (p, q) ≠ (k, k), then we get zero. Because H^{2n}(X, \mathbf{C}) = H^{n,n}(X), we conclude that [Z] must lie in H^{k,k}(X, \mathbf{C}). Loosely speaking, the Hodge conjecture asks:

Which cohomology classes in Hk,k(X) come from complex subvarieties Z?

I made that up myself. :nod

[Bleach]

Sir, you're a genius.

You've just discovered the cure for AIDS .. it all adds up, it all makes sense.

5 is great than 3, yet is less than 5. Amazing yet surprising, how could such theory solve a million lives.

I know man. This stuff is copyrighted btw so dun steal it!

[GrimaH]

I was practicing the multiplication tables when I was 7, then discovered that multiplication actually went beyond 12 x 12.

[Haruko]

Sir, you're a genius.

You've just discovered the cure for AIDS .. it all adds up, it all makes sense.

5 is great than 3, yet is less than 5. Amazing yet surprising, how could such theory solve a million lives.

I believe 2+2 is 4.

[WelcomeToRapture]

You believe wrong.
Our mathematical system is wrong.

[AbnormallyNormal]

fuck the system 2 + 2 make 3

[Commander Red]

I once made 2 functions for appreaching Pi. However, it's so trivial, it's not special at all.

[WelcomeToRapture]

I once figured out that sin90+theta = sin 90- theta.

Damm Im smart.

[Fiera]

I once had three apples and ate two of them. I had one apple left. :wtf Srsly though no!

[myle]

Who ate it?
Seriously, I think we are off topic.

[yuri]

no i dont.

[Shaidar Haran]

I once had three apples and ate two of them. I had one apple left. :wtf Srsly though no!Wouldn't you technically still have three apples? Or do you just have an amazingly fast digestive system?

[AbnormallyNormal]

there really is no such thing as "two apples" if you think about, since no two apples are EXACTLY the same.

[chickmashine]

Hmm old thread xD Just found this by accident but I thought it's interesting - Well I have an alternative way of solution for quadric functions

f(x) = x²+px+q ll x = y - p/2
0 = (y-p/2)² + p*(y-p/2) + q
0 = y² - py + p²/4 + py - p²/2 +q
0 = y² -p²/4 +q ll switching sides
y² = (p/2)² - q ll square root
y = ( (p/2)² - q )^0,5

and now x = y - p/2 so x = -p/2 +- ( (p/2)² -q )^0,5
__________________________________________________ ______________________

Then a while ago I solved the cubic function but that's too long to post it so if you're interested then just ask me ^^
__________________________________________________ ______________________

Then I've solved several sums like this kinda thing:

The sum of 1+2+3+4+5+6 + ... + 12 = f(12) for this one you just line em up differently->

1 + 2 + 3 + 4 + 5 + 6
+12 + 11 + 10 + 9 + 8 + 7 now if you look at this then the upper and the lower number when added always equal 13. and that exactly six times. This is logically ( x + 1 ) * x/2

This solution can be used to sum up any number you want so for example: 1+2+3+...1000= ( 1000 +1 ) *1000/2 = 500500

Eventhough that's trivial you need it for this kind of solution:

when you sum up this: 1² + 2² +3² + 4² + 5² ... 12² = f(12) The first thing you have to know is that

x² -(x-1)² = x² - x² +2x-1 = 2x-1 This is the distance between every x² number but you have to take 0² = 0 as a given

so for example:

(You have to add the solution beforehand because you're only calculating the distance to the former quadric number.)

1² = 1
2² = 2*2 -1 + 1 = 1 + 3 = 4
3² = 2*3-1 + 4 = 1 + 3 + 5 = 9
4² = 2*4 - 1 +9 = 1 + 3 + 5 + 7 = 16

So now if you want to add up several quadric number you don't have to add them up like this: 1² + 2² + 3² + 4² but instead you can write:

1+1+1+1+3+3+3+5+5+7 or you can write it like this:

7
5 + 5
3 + 3 + 3
1 + 1 + 1 + 1 so in a triangle and this is actually equal to 1²+2²+3³+4² but obviously you cant make an equation just out of this so... I turn it around twice!

1
1 + 3
1 + 3 + 5
1 + 3 + 5 + 7

1
3 + 1
5 + 3 + 1
7 + 5 + 3 + 1

now if we add these number triangles we get this:


9
9 + 9
9 + 9 + 9
9 + 9 + 9 + 9 and this divided by 3 eaquals the first triangle ^^

So let's difine this triangle: it is in fact: (2 * x + 1) * (the sum of 4 = (x + 1)*x/2 and then the triangle / 3

so it is: (2*x+1)*x*(x+1) /6

yeah that's the kind of stuff I do, but no real math like you do :P It's really cool by the way!

[shirish]

I once calculated the chromatic polynomial of a tree by accident. I was so proud of myself till I realized it was something utterly trivial.

:(

I do have a tendency to have good intuition for analysis, and rather frequently think "I knew this was true." when seeing theorems on lectures.
Now that's cool. I think though that a lot of people have some kind of an ad hoc idea about the chromatic poly if they are doing data structures as it is. Well, maybe not the same thing but something similar to graph coloring anyways.
But if you formulated the formal version by yourself, kudos to you mate.

[Connie]

Hm...

The Purpose of Mathematics = x

Find x.

[shirish]

^
x = Life

[impersonal]

Hm...

The Purpose of Mathematics = x

Find x.

There are several solutions, actually. In fact, there's almost as many solutions as there are :
a) Mathematicians
b) People financing mathematics (states, universities...)
c) People benefitting from mathematics (other scientists...)

So, that's a shitload of purposes, which include such things as "good grades", "prestige", "technological improvements", "self-esteem", etc, etc.



(In other terms, I don't think there's a point in looking for "The" purpose of mathematics; maths don't have a purpose; people who do maths or use maths have one.)

[Mider T]

lol no, I fail at math.

[impersonal]

I do have a tendency to have good intuition for analysis, and rather frequently think "I knew this was true." when seeing theorems on lectures.

I'm the same. I used to have one-on-one oral interrogations in maths (pretty common in the french educational system, during the 2-3 years after the baccalaureat; it's called "classes préparatoires" or "prépa"), and I frequently got away with a correct grade, whithout having worked at all, because I could intuitively see the solutions of the problems.

This didn't work out so well in written exams in which the examiner isn't there to help you once he sees that you're "on the right path". Remember, kids - correct intuitions don't make up for hard work :(.

[shirish]

(In other terms, I don't think there's a point in looking for "The" purpose of mathematics; maths don't have a purpose; people who do maths or use maths have one.)

It has the purpose of being the language for accurate description of physical phenomenon and their interaction

[1mmortal 1tachi]

I suspect this man has:

http://www.youtube.com/watch?v=OqY_q7riL-w

:zaru

[shirish]

^ I was able to do that in twice as much time as that guy in my head
;__;

[impersonal]

It has the purpose of being the language for accurate description of physical phenomenon and their interaction

Utter bullshit. I do computer science, so I know for a fact that maths have other purposes than physics. (And the video is also sufficient proof that you are completely wrong).

[shirish]

Utter bullshit. I do computer science, so I know for a fact that maths have other purposes than physics. (And the video is also sufficient proof that you are completely wrong).
Oh and computation isn't a physical phenomenon?

:lmao

And how does the video prove anything against me? Think again buddy.

[zoro_santoryu]

Check this one outsmile-big Its very simple

X^2 - X^2 = X^2 - X^2

LHS is factorised into X(X - X)
RHS is factorised using different off 2 sqaures = (X - X)(X + X)

Thus X(X - X) = (X - X)(X + X)
Thus X = X + X (X - X) cancels
X = 2X

Therefore 1 = 2

[AbnormallyNormal]

Check this one outsmile-big Its very simple

X^2 - X^2 = X^2 - X^2

LHS is factorised into X(X - X)
RHS is factorised using different off 2 sqaures = (X - X)(X + X)

Thus X(X - X) = (X - X)(X + X)
Thus X = X + X (X - X) cancels
X = 2X

Therefore 1 = 2

X - X = 0. you cant divide by zero

[1mmortal 1tachi]

^ I was able to do that in twice as much time as that guy in my head
;__;


Sweet. I can't do it at all.

:dupe

I think the rest of that video is on the TED talks site, if anyone wants to see it. He does a bunch of audience participation calculations like that. The last one is actually the most complex and is pretty impressive.

:zaru

[The Space Cowboy]

Mathematical Truths can neither be created nor destroyed. They may only be changed from one Theorem to another.

[cacophony]

Sir, you're a genius.

You've just discovered the cure for AIDS .. it all adds up, it all makes sense.

5 is great than 3, yet is less than 5. Amazing yet surprising, how could such theory solve a million lives.

i could have sworn 2 + 2 was 4

[adee]

Mathematical Truths can neither be created nor destroyed. They may only be changed from one Theorem to another.
Wat?

Math is a language of our own creation, sure we can change around mathematical facts if we find it better translates Physics for us.

[avraell]

Yeah, I think schools make many of us lose creativity and originality.

I really think so, I find myself surprised when I *think* about things now rather than absorbing accepted view points and deciding which I like best, quite depressing.

[Kokain]

Math is a language of our own creation, sure we can change around mathematical facts if we find it better translates Physics for us.

Math isn't a language like English is a language, nor is it as easy to change. The best you can do is alter the basic axioms of a certain branch of math, but the "facts" derived from those axioms are dependent on logic and thus not subject to arbitrary alterations, though it should be mentioned that the relationship between mathematics, logic and the nature of mathematical truths is its own sizeable can of worms.

Mathematical facts aren't reliant on physics either.

[Xion]

No. I'm not a pure math major and I have never had a formal mathematics education outside of basic calculus. :(

And to top it off there are not many helpful math resources on the Internet for explaining the advanced shit. :cry

[PDQ]

I was quite happy with a few "discoveries" I made in philosophy, until I read authors who said the same thing and went beyond, decades or centuries before me.
Same here, philosophy or math, there's already a term for it or as my friend calls it "a dead white guy it's named after"

So, had you ever come across any mathematical formula/theorem/method to solve particular problems by yourself before? If you had you can share them with us here. It can be something you deduced by yourself but later found out that it's actually a well-known formula/theorem or it can be a certain creative way you use to solve a particular problem that is different from the conventional method your teacher and classmates use.
I did a lot of messing around with base 2 math long before I knew that different bases other than 10 existed so I derived a bunch of things involving binary arithmetic on my own. Then there was DeMorgan's law and Baye's theorem. There's also the connection between dot products and the law of cosines in trig. And in the end, I always find out someone discovered it. In fact, my math teacher in 11th grade told me anything I think I've come with as original has likely been discovered since the ancient Greeks. On the other hand, things with modern technology you have a better chance of developing because things like networks haven't been around that long, but math has been around forever.

One that I don't know if anyone's formally pointed out was closed end arithmetic expressions for sin/cos of multiples of 3 degrees(or compositions of it's binary components such as 1.5 and (1.5+.75)). Basically I came up with it when messing around with pentagons and I found a proof for the ratio of pentagon side to diagonal being the golden ratio and taking that angle(72) in combination with knowing the ratio of 75 degrees(by taking 60 degrees + 15 degrees using trig identities of adding angles) and taking the difference, I got the value for sin(3 degrees). Of course, I'd imagine there's little use for a closed form expression for the exact value of 3 degrees( or 1.5 or 6 or 9), but it was kinda neat figuring that out.


But yea, even if it's already discovered, that's what I always loved about math, discovering something on your own and knowing for a fact that it is absolutely rooted in something universally true. But that's also what I disliked about probability and statistics.

[Grrblt]

I've independently figured out the pythagorean trigonometric identity: sin2x+cos2x=1

[Kaito Sageko]

This is an "formula" that my friend thought of.
If x=y, solve this:

x²=xy /-y²
x²-y²=xy-y²
(x-y)(x+y)=(x-y)"times"y / : (x-y)
x+y=y
2y=y /:y
2=1
I have a very god case regarding this, but i don't have time to explain it now.

[Masaki]

I cannot follow anything you did there. How did the /-y² disappear after the first line? Where did the x go?

Also, are you guys aware of the "chart method"? My Calc B teacher taught it to us as a shortcut for integration by parts, but my Calc C teacher evidentially didn't know it.

[kingcools]

This is an "formula" that my friend thought of.
If x=y, solve this:

x²=xy /-y²
x²-y²=xy-y²
(x-y)(x+y)=(x-y)"times"y / : (x-y)
x+y=y
2y=y /:y
2=1
I have a very god case regarding this, but i don't have time to explain it now.

if x = y then x-y = 0, u may not divide by zero

[PDQ]

I've independently figured out the pythagorean trigonometric identity: sin2x+cos2x=1

Did you not know the Pythagorean theorem at the time or the definition of sin and cos? Otherwise, isn't that just self-evident?

[Grrblt]

Did you not know the Pythagorean theorem at the time or the definition of sin and cos? Otherwise, isn't that just self-evident?
Only self-evident if you stop and think about it :wink

[Kokain]

Now if money is the 'root of all problems'

The actual quote refers to "the love of money", not "money".

@This thread: when I was around 16 I was pretty interested in n-dimensional geometries, and started off by deriving the properties of various polytopes by dimensional analogy. Nothing too insightful or memorable, since I can't even remember my formulas anymore, or recall being excited about deriving them in the first place. And naturally nothing that I discovered was original, as I later found out.