Mathematical Equation of the Week (MEoW)

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?

Note by Sharky Kesa
4 years, 10 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

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This number amazed me:

$1741725= 1^7+7^7+4^7+1^7+7^7+2^7+5^7$

- 4 years, 10 months ago

Check out Narcissistic Number. There're even more than that!

- 4 years, 10 months ago

Wow they are much!! Thanks for sharing that :)

- 4 years, 10 months ago

- 4 years, 10 months ago

Why is it fascinating?

Staff - 4 years, 10 months ago

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

- 4 years, 10 months ago

@U Z Hahahaha. I was asking why Ramanujan's formula for pi is interesting to you. Isn't the one than comes from arctan more simple and beautiful?

Staff - 4 years, 10 months ago

Many great mathematicians could'nt give its accurate value , and Ramanujan gave an interesting formula.

Which one is that which comes from arctan that is more simple and beautiful?

- 4 years, 10 months ago

@U Z $\pi = 4(1- \frac{1}{3} + \frac{1}{5}-\frac{1}{7}+\cdots )$

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)

Staff - 4 years, 10 months ago

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:

$π = 3 + \frac{4}{2 × 3 × 4} - \frac{4}{4 × 5 × 6} + \frac{4}{6 × 7 × 8} - \frac{4}{8 × 9 × 10} + …$

$π = 2 × \frac{2}{1} × \frac{2}{3} × \frac{4}{3} × \frac{4}{5} × \frac{6}{5} × \frac{6}{7} × \frac{8}{7} × \frac{8}{9} × …$

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

- 4 years, 10 months ago

@U Z Hm, I remember wishing 'Happy Pi Day' to my beloved ones

Staff - 4 years, 10 months ago

Prove it! :P Does anyone has a proof of it?

- 4 years, 10 months ago

Prove what? The faster convergence rate? Or the arctan formula for pi? Or Ramanujan's formula?

I have a proof for my formula

Staff - 4 years, 10 months ago

Ramanujan's formula!

- 4 years, 10 months ago

I have not

Staff - 4 years, 10 months ago

For $n = 1,2,3,4,5,6$ is true that:

$14^n+16^n+45^n+54^n+73^n+83^n = 3^n+ 5^n + 28^n + 34^n+ 65^n + 66^n + 84^n$

- 4 years, 10 months ago

Can you tell me why is this happening?

- 4 years, 10 months ago

Probably magic. You can check here to see more than that.

- 4 years, 10 months ago

Nice! Thanks for sharing!

- 4 years, 10 months ago

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

- 4 years, 10 months ago

$\frac{1}{7}=0.142857...$

- 4 years, 10 months ago

Cyclic number.

- 4 years, 10 months ago

$\displaystyle e^{\pi \sqrt(163)} \approx 262537412640768743.99999999999925007 \approx 640320^3+744-0.00000000000075$ Coincidence

What do you think?

Credits : Tumblr (& The Incredibles), Ramanujan, Wikipedia.

- 4 years, 10 months ago

- 4 years, 10 months ago

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.

- 4 years, 10 months ago

What does that even mean?

Staff - 4 years, 10 months ago

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'.

- 4 years, 10 months ago

$\frac{\pi}{2} = \frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdot\frac{8}{7}\cdot\ldots\\ \frac{\pi}{4} = 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\ldots\\ \frac{\pi^2}{6} = 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\ldots\\ \frac{\pi^3}{32} = 1-\frac{1}{3^3}+\frac{1}{5^3}-\frac{1}{7^3}+\frac{1}{9^3}+\ldots\\ \frac{\pi^4}{90} = 1+\frac{1}{2^4}+\frac{1}{3^4}+\frac{1}{4^4}+\frac{1}{5^4}+\ldots\\ \frac{2}{\pi} = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2+\sqrt{2}}}{2}\cdot\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdot\ldots\\ \pi = \cfrac{4}{1+\cfrac{1^2}{3+\cfrac{2^2}{5+\cfrac{3^2}{7+\cfrac{4^2}{9+\ldots}}}}}\\$

Staff - 4 years, 10 months ago

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!!!

- 4 years, 10 months ago

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.

- 4 years, 10 months ago

I loved this one as well.

$\dfrac {d}{dx} (e^x) = e^x$

- 4 years, 10 months ago

$a^2 + b^2 = c^2$ (Pythagoras Theorem)

Edit-These positive integers $a,b,c$ form the sides of a right triangle necessarily.

- 4 years, 10 months ago

$\sin{(2°\times{10}^{-n})}-\sin{(2°\times{10}^{-(n+1)})}\approx\pi\times{10}^{-(n+2)}$

Try to derive it and you'll see the magic.

- 4 years, 10 months ago

Oh, nice. LOL

Staff - 4 years, 10 months ago

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.

- 4 years, 10 months ago

For the first time, there is not a single downvote(see!) in such an open discussion.

- 4 years, 10 months ago

Yeah there is. Check out Samuraiwarm's comment below. 1 downvote.

- 4 years, 10 months ago

-- You haven't read the note carefully. Despite me bolding the text, your despicable eyes have failed to read it --

- 4 years, 10 months ago

So it was you who did that.

- 4 years, 10 months ago

Yes, an' I edited this comment of yours too :P

- 4 years, 10 months ago

So who was it who did WHAT?!

- 4 years, 10 months ago

Bolded that sentence.

- 4 years, 10 months ago

They might be added but I still have downvoted comments I do not like.

Staff - 4 years, 10 months ago

So it was you who did that -_-

- 4 years, 10 months ago

Yes, I downvoted -1/12 because I can't stand these Kaboobly Doo ideas.

Staff - 4 years, 10 months ago

But that is actually true. In certain maths... That kind of "weird" calculation is commonly seen in physics.

See this

- 4 years, 10 months ago

The issue I have with it, is that it is not made explicit where the calculation is done, which is why we would default to working in the "usual" arithmetic, and say that the sum is infinite.

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.

Staff - 4 years, 10 months ago

What Irony -_- (says master of KD)

- 4 years, 10 months ago

It's not a; kaboobly; idea - there is proof of the outcome. It just seems counter-intuitive because it would seem to equal infinity. However, that is only because you are imagining what happens if you stop it at the 10th terms, 100th term, 1000th term etc.

- 4 years, 7 months ago

@Sharky Kesa I just trolled your meow...

- 4 years, 10 months ago

EDIT OF SATVIK GOLECHHA'S COMMENT:

@Sharky Kesa I just trolled your (meow)

- 4 years, 10 months ago

Depends.

- 4 years, 10 months ago

And, at the end of your edited comment, why did you say 'meow'?

MATHISNOWHERE

What does it say?

- 4 years, 10 months ago

MATH IS NOWHERE OR MATH IS NOW HERE. :D

- 4 years, 10 months ago

Which one do you prefer?

- 4 years, 10 months ago

I prefer "MATH IS NOW HERE"

- 4 years, 10 months ago

Depends.

- 4 years, 10 months ago

Which one do you prefer?

- 4 years, 10 months ago

Depends.

- 4 years, 10 months ago

How so?

- 4 years, 10 months ago

That too depends.

- 4 years, 10 months ago

List why it depends.

- 4 years, 10 months ago

It actually doesn't depend. I was just replying "Depends" for fun.

- 4 years, 10 months ago

That is a very good idea,

Staff - 4 years, 10 months ago

- 4 years, 10 months ago

Which question? You've asked more than one.

- 4 years, 10 months ago

All of them.

- 4 years, 10 months ago

I've got to note that one...

- 4 years, 10 months ago

MAT HIS NOW HER E

read like "matt, his now - hurry"

- 4 years, 10 months ago

I neither like it nor dislike it :3

- 4 years, 10 months ago

Expected from you. You can still upvote it.

- 4 years, 10 months ago

it is...........................

.

.

.

.

.

.

.

.

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1 = 1

- 4 years, 10 months ago

what is so great in that?@math man

- 4 years, 10 months ago

idk lol

- 4 years, 10 months ago

:D

- 4 years, 10 months ago

$\zeta(-1) = 1+2+3+\dots = -\frac{1}{12}$

where $\zeta(s)$ is the Riemann zeta function.

- 4 years, 10 months ago

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)

Staff - 4 years, 10 months ago

wait waa?

- 4 years, 10 months ago

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!

- 4 years, 10 months ago

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

- 4 years, 10 months ago

1+(e^(i*pi))=0

five fundamental numbers in one equation... almost all you guys are already familiar with this

- 4 years, 10 months ago

${ G }_{ \mu v }=8\pi G\left( { T }_{ \mu v }+{ \rho }_{ \Lambda }{ g}_{ \mu v } \right)$

Einstein field equation

- 4 years, 10 months ago

$V-E+F=2$

- 4 years, 10 months ago

- 4 years, 10 months ago

Euler's formulae

- 4 years, 10 months ago

This is not for competition. However this is what I remember since I was of teen age.$\color{#3D99F6}{152207 * 73= 11111111.(Eight~ '1's)\\~ It~ is ~ clear~ that ~if ~we~use~N*73, N=1,2,...9,~~\\we~ get~ EIGHT~ Ns.~\\152207~is ~divisible ~by ~11. ~(11|152207)}$

- 4 years, 7 months ago

$\Im(i^{i})=0$

- 4 years, 10 months ago

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. :)

- 4 years, 7 months ago

Haha...The abbreviation is Meow :)

- 4 years, 10 months ago

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.

- 4 years, 10 months ago

The sequence 1-1+1-1+1-1... an Oscillatory series equals 1/2 even though all are integers .

- 4 years, 10 months ago

Interesting! But can you explain why that happens?

- 4 years, 10 months ago

Refer to this note https://brilliant.org/discussions/thread/interesting-sums/ . Hope it helps ! :)

- 4 years, 10 months ago

Thanks!

- 4 years, 10 months ago