Basically, post the weirdest, most beautiful or just the coolest mathematical equation you have ever come across. Here is an example:
\[e^{ix} = \cos{x} + i \sin{x}\]
Only one entry per person is allowed and the winner is determined by the number of up votes. Any down votes will be added on as up votes. Good luck, have fun and do math.
P.S.
Who understands the abbreviation? Why is it significant?
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> This is a quote
2 \times 3
2^{34}
a_{i-1}
\frac{2}{3}
\sqrt{2}
\sum_{i=1}^3
\sin \theta
\boxed{123}
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Top NewestThis number amazed me:
\[1741725= 1^7+7^7+4^7+1^7+7^7+2^7+5^7\]
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Check out Narcissistic Number. There're even more than that!
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Wow they are much!! Thanks for sharing that :)
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Why is it fascinating?
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It seemed to me fascinating because
\(Oa = Ob , OA=OB \)
\(\frac{AB}{ab} =\frac{OA}{Oa}\)
\(\frac{Perimeter of the outer polygon}{Perimeter of the inner polygon} = \frac{n.AB}{n.ab} = \frac{AB}{ab}=\frac{OA}{Oa}\)
\(\frac{Circumference of outer circle}{Circumference of inner circle} = \frac{OA}{Oa} = \frac{Radius of outercircle}{radius of inner circle}\)
Thus
\( \frac{Circumference of outer circle}{radius of outercircle} = \frac{Circumference of inner circle}{radiusof inner circle}\)
You know \(\frac{Circumferenceofanycircle}{Diameter} =\) the constant\( \pi\)
The approximate values were given as \( \pi = \frac{22}{7}\) , more accurate was \(\frac{355}{113}\)
You can see now why it is fascinating@Agnishom Chattopadhyay
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Which one is that which comes from arctan that is more simple and beautiful?
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To get this, you should realise arctan(1)=pi/4 and the. Just substitute arctan(1) with its taylor series.
However, your formula is better because it has faster convergence)
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Pi, written as π, equals 3.1415926… and its digits goes on forever, without any repeating pattern. Numbers with this property are called irrational numbers. Many ancient mathematicians – including the famous Pythagoras – were horrified when they discovered that such bizarre and ‘impure’ numbers exist.
Today many mathematicians believe that Pi has an even more curious property: that it is a normal number. This would mean that the digits from 0 to 9 appear completely at random, as if nature had rolled a 10-sided dice, again and again, to find the next digit. It also means that if you think of any string of digits, like 123456789, it has to appear somewhere in the digits of Pi – but you might have to calculate millions of digits.
We could even convert an entire book, like the works of Shakespeare, into a very long string of digits (a = 01, b = 02, and so on). If Pi is normal, this string must also appear somewhere in its digits. But even if we used all computers on Earth to calculate more and more digits of Pi, we would probably have to look for longer than the age of the universe… The First few Digits of Pi
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628
0348253421170679821480865132823066470938446095505822317253594081284811174502841027019
3852110555964462294895493038196442881097566593344612847564823378678316527120190914564
8566923460348610454326648213393607260249141273724587006606315588174881520920962829254
0917153643678925903600113305305488204665213841469519415116094330572703657595919530921
8611738193261179310511854807446237996274956735188575272489122793818301194912983367336
2440656643086021394946395224737190702179860943702770539217176293176752384674818467669
4051320005681271452635608277857713427577896091736371787214684409012249534301465495853
7105079227968925892354201995611212902196086403441815981362977477130996051870721134999
9998372978049951059731732816096318595024459455346908302642522308253344685035261931188
1710100031378387528865875332083814206171776691473035982534904287554687311595628638823
53
There are many different ways to calculate Pi, some of which use sequences or series of numbers. One example is the following series discovered by Gottfried Wilhelm Leibniz (1646 – 1716). As you add more and more terms, following the same pattern, the result will get closer to Pi:
\(π = \frac{4}{1} – \frac{4}{3} + \frac{4}{5} – \frac{4}{7} + \frac{4}{9} – \frac{4}{11} + …\)
Another sequence, published by Nilakantha Somayaji (1444 – 1544), is even better since it gets closer to Pi with fewer terms:
\(π = 3 + \frac{4}{2 × 3 × 4} – \frac{4}{4 × 5 × 6} + \frac{4}{6 × 7 × 8} – \frac{4}{8 × 9 × 10} + …\)
This formula was published by John Wallis in 1655:
\(π = 2 × \frac{2}{1} × \frac{2}{3} × \frac{4}{3} × \frac{4}{5} × \frac{6}{5} × \frac{6}{7} × \frac{8}{7} × \frac{8}{9} × …\)
Using powerful computers, Pi has been calculated up to 10 trillion digits (that’s a 1 with 13 zeros)! Because Pi is so easy to understand, yet important in many areas of mathematics, it enjoys an unusual popularity in our culture (unusual, at least, for areas of mathematics). There even is a Pi Day on 14 March or 22 July, since \(\frac{22}{7}\) is a close approximation to pi. @Agnishom Chattopadhyay
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Prove it! :P Does anyone has a proof of it?
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I have a proof for my formula
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For \[n = 1,2,3,4,5,6\] is true that:
\[14^n+16^n+45^n+54^n+73^n+83^n = 3^n+ 5^n + 28^n + 34^n+ 65^n + 66^n + 84^n\]
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Can you tell me why is this happening?
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Probably magic. You can check here to see more than that.
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A number which rearrange itself upto multiplication by 6
142857 \(\times\) 1 = 142857
142857 \(\times\) 2 = 285714
142857 \(\times\) 3 = 428571
142857 \(\times\) 4 = 571428
142857 \(\times\) 5 = 714285
142857 \(\times\) 6 = 857142
Here is the interesting part
142857 \(\times\) 7 = 999999
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\(\frac{1}{7}=0.142857...\)
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Cyclic number.
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\(\displaystyle e^{\pi \sqrt(163)} \approx 262537412640768743.99999999999925007 \approx 640320^3+744-0.00000000000075\)
Coincidence
What do you think?
Credits : Tumblr (& The Incredibles), Ramanujan, Wikipedia.
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What's so special about this equation?
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This equation's speciality is that it uses irrational numbers to get very, very, very, very close to an integer. Check this one out: \(\displaystyle e^\pi -\pi = 19.999099979189 \approx 20\)
Is that a coincidence or what?!
P.S: You might wanna check out 163 and Ramanujan's constant.
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What does that even mean?
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That means that with irrational numbers, we can get as close as we want, at least for practicality's sake, to an integer. That just blows my mind!
P.S: You might want to check out Almost Integer for more such 'coincidences'.
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From http://math.stackexchange.com/questions/8814/funny-identities?rq=1
\[ \frac{\pi}{2} = \frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdot\frac{8}{7}\cdot\ldots\\ \frac{\pi}{4} = 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\ldots\\ \frac{\pi^2}{6} = 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\ldots\\ \frac{\pi^3}{32} = 1-\frac{1}{3^3}+\frac{1}{5^3}-\frac{1}{7^3}+\frac{1}{9^3}+\ldots\\ \frac{\pi^4}{90} = 1+\frac{1}{2^4}+\frac{1}{3^4}+\frac{1}{4^4}+\frac{1}{5^4}+\ldots\\ \frac{2}{\pi} = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2+\sqrt{2}}}{2}\cdot\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdot\ldots\\ \pi = \cfrac{4}{1+\cfrac{1^2}{3+\cfrac{2^2}{5+\cfrac{3^2}{7+\cfrac{4^2}{9+\ldots}}}}}\\ \]
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Fermat's last theorem.. \(a^{n}+b^{n}=c^{n}\) for all a,b,c,n being integers gives real integral solutions only if n<=2 . The proof of the solution took 7 years of hard solitary work!!!
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Honestly, the equation that comes into my mind was
\[x^2-x-1=0\]
which is related to the golden ratio and the Fibonacci Sequence.
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I loved this one as well.
\[\dfrac {d}{dx} (e^x) = e^x\]
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\(a^2 + b^2 = c^2\) (Pythagoras Theorem)
Edit-These positive integers \(a,b,c\) form the sides of a right triangle necessarily.
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\[\sin{(2°\times{10}^{-n})}-\sin{(2°\times{10}^{-(n+1)})}\approx\pi\times{10}^{-(n+2)}\]
Try to derive it and you'll see the magic.
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Oh, nice. LOL
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Inclusion-Exclusion: Either you like something, or you don't.
If you like this, then upvote it, or else, if you don't like it, then downvote it.
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For the first time, there is not a single downvote(see!) in such an open discussion.
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Yeah there is. Check out Samuraiwarm's comment below. 1 downvote.
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-- You haven't read the note carefully. Despite me bolding the text, your despicable eyes have failed to read it --
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See this
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It is similar to saying " 11 + 2 = 1 " (where the arithmetic is performed on the clock)
It is similar to saying "1 + 1 = 0 ". (Where the arithmetic is performed in the field of order 2)
Neither of these would be "acceptable", and are only technically right by omission. I value clarity of expression much more than showing off.
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@Sharky Kesa I just trolled your meow...
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EDIT OF SATVIK GOLECHHA'S COMMENT:
@Sharky Kesa I just trolled your (meow)
I just trolled your what?
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Read this slowly:
MATHISNOWHERE
What does it say?
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read like "matt, his now - hurry"
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I neither like it nor dislike it :3
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Expected from you. You can still upvote it.
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it is...........................
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1 = 1
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what is so great in that?@math man
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idk lol
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\[\zeta(-1) = 1+2+3+\dots = -\frac{1}{12}\]
where \(\zeta(s)\) is the Riemann zeta function.
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Please state the system of arithmetic in which this statement is true.
Otherwise, it is similar to saying " 1 + 1 is equal to 10 "
\( \vdots \)
(where I am working in binary)
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wait waa?
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The Riemann Zeta function also links with quantum mechanics and is said to encode a 'formula' for the distribution of primes. Whether it does is the million dollar question!
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Taylor Series -
\(f(x) = \sum _{ n\quad =\quad }^{ \infty }{ \frac { { f }^{ (n) }(0) }{ n! } { x }^{ n } } \)
I have one more(if it is allowed):
https://qph.is.quoracdn.net/main-qimg-7beca9cf6c835bdfeabac3541a778973?converttowebp=true
And one more:
http://en.wikipedia.org/wiki/Tupper%27sself-referentialformula
*The last 2 are just for sharing
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1+(e^(i*pi))=0
five fundamental numbers in one equation... almost all you guys are already familiar with this
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\({ G }_{ \mu v }=8\pi G\left( { T }_{ \mu v }+{ \rho }_{ \Lambda }{ g}_{ \mu v } \right) \)
Einstein field equation
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\(V-E+F=2\)
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And what is that about?
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Euler's formulae
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This is not for competition. However this is what I remember since I was of teen age.\(\color{blue}{152207 * 73= 11111111.(Eight~ '1's)\\~ It~ is ~ clear~ that ~if ~we~use~N*73, N=1,2,...9,~~\\we~ get~ EIGHT~ Ns.~\\152207~is ~divisible ~by ~11. ~(11|152207)} \)
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\[\Im(i^{i})=0\]
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substitute x=pi. then eqution becomes e^(i pi)+1=0. this relation connects 5 most important numbers of mathematics, which makes this most beautiful relation. :)
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Haha...The abbreviation is Meow :)
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You all have really good entries. I don't know I only have one equation that may seem not as good as yours. Still have a look. e^{i(pi)}= -1 This equation is derived from De Moivre's Theorem(Sorry Sharky Kesa) But I loved it more than the original theorem since it is of the form An irrational number raised to the product of an imaginary and irrational number which gives us a real number. P S I don't know how to write 'pi' in Greek.
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The sequence 1-1+1-1+1-1... an Oscillatory series equals 1/2 even though all are integers .
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Interesting! But can you explain why that happens?
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Refer to this note https://brilliant.org/discussions/thread/interesting-sums/ . Hope it helps ! :)
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