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What are some surprising equations/identities that you have seen, which you would not have expected?
This could be complex numbers, trigonometric identities, combinatorial results, algebraic results, etc.
I'd request to avoid 'standard' / well-known results like $ e^{i \pi} + 1 = 0$.
Please write a single identity (or group of identities) in each answer.
I found this list of Funny identities, in which there is some overlap.
$$ \begin{align}\frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{239} \\\,\\\,\\ \frac{\pi}{4} = 5 \arctan \frac{1}{7} + 2 \arctan \frac{3}{79}\end{align}$$
Both can be shown easily using polar form, complex multiplication.
Some zeta-identies have been much surprising to me.
Let's denote the value $\zeta(s)-1$ as $\zeta_1(s)$ then $$ \small \begin{array} {} 1 \zeta_1(2) &+&1 \zeta_1(3)&+&1 \zeta_1(4)&+&1 \zeta_1(5)&+& ... &=&1\\ 1 \zeta_1(2) &+&2 \zeta_1(3)&+&3 \zeta_1(4)&+&4 \zeta_1(5)&+& ... &=&\zeta(2)\\ & &1 \zeta_1(3)&+&3 \zeta_1(4)&+&6 \zeta_1(5)&+& ... &=&\zeta(3)\\ & & & &1 \zeta_1(4)&+&4 \zeta_1(5)&+& ... &=&\zeta(4)\\ & & & & & &1 \zeta_1(5)&+& ... &=&\zeta(5)\\ ... & & & & & & & &... &= & ... \end{array} $$ There are very similar stunning alternating-series relations:
$$ \small \begin{array} {} 1 \zeta_1(2) &-&1 \zeta_1(3)&+&1 \zeta_1(4)&-&1 \zeta_1(5)&+& ... &=&1/2\\ & &2 \zeta_1(3)&-&3 \zeta_1(4)&+&4 \zeta_1(5)&-& ... &=&1/4\\ & & & &3 \zeta_1(4)&-&6 \zeta_1(5)&+& ... &=&1/8\\ & & & & & &4 \zeta_1(5)&-& ... &=&1/16\\ ... & & & & & & & &... &= & ... \end{array} $$
$$\int_0^1\frac{x^4(1-x)^4}{1+x^2}\mathrm{d}x=\frac{22}{7}-\pi$$
It's interesting how something so bizarre on the left hand side yields the tiniest of errors in one of the most famous approximations of $\pi$.
This one is one of my favorite:
1. $$ e^{\pi i} + 1 = 0$$
This simple equation links five fundamental mathematical constants:
Moreover, the three basic arithmetic operations occur exactly once each: addition, multiplication and exponentiation; and these are magically wound into one single relation$(=).$
The beauty lies in the fact that an irrational number, raised to the power of an imaginary number multiplied with another irrational number, exactly becomes zero when added to $1.$
As quoted by Benjamin Peirce, a noted American 19th-century philosopher,mathematician, and professor at Harvard University, "it is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth."
This identity is a special case of Euler's Formula: $$e^{ix}=\cos(x)+ i\sin(x)$$ It's almost mystical that these values are even related to one another.
2. The solution to this equation: $$1+\frac{1}{\phi}=\phi$$ Which is The golden ratio:$$\phi=\frac{1+\sqrt5}{2}=1.6180339887 . . .$$Which can turn into recurrence equation: $$\phi^{n+1}=\phi^n+\phi^{n-1}$$ Beautiful how it is also related to Fibonacci numbers: $$1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , ...$$ Where if you divide any consecutive Fibonacci numbers, in the infinite horizon will converge to, again, the golden ratio: $$\lim_{n\to\infty}\frac{F(n+1)}{F(n)}=\phi$$
3. Tupper's Self Referential Formula
When plotted with k=960939379918958884971672962127852754715004339660129306651505519271702802395266424689642842174350718121267153782770623355993237280874144307891325963941337723487857735749823926629715517173716995165232890538221612403238855866184013235585136048828693337902491454229288667081096184496091705183454067827731551705405381627380967602565625016981482083418783163849115590225610003652351370343874461848378737238198224849863465033159410054974700593138339226497249461751545728366702369745461014655997933798537483143786841806593422227898388722980000748404719
$0 \le x \le 106$ and $k \le y \le k + 17$, the resulting graph looks like this:
4. Ramanujan's golden ratio equation:
$$\int_{-\infty}^\infty \! e^{-x^2}dx = \sqrt{\pi}$$
6. Cauchy's Integral Formula: $${f^{\left( n \right)}}\left( a \right) = \frac{{n!}}{{2\pi i}}\oint_\gamma {\frac{{f\left( z \right)}}{{{{\left( {z - a} \right)}^{n + 1}}}}dz}$$ The derivative of a analytic function given as a closed path integral in the complex plane.
7. Ramanujan's Infinite series for calculation of $\pi$. It converges faster
$$ \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}$$
8. Batman Curve
The batman curve is a piecewise curve in the shape of the logo of the Batman superhero originally posted on reddit.com on Jul. 28, 2011. It can written as two functions, one for the upper part and the other for the lower part, as:
9. The Schrodinger Equation:
$$H\Psi(x,t) = i\hbar\frac{\partial}{\partial t}\Psi(x,t)$$
Another one, which occured to me when I began to learn about double-sums in the context of divergent summation. I really had to chew on this, that the sum of the vertical sums can be different from the sum of the horizontal sums... And just different by the exact value of 1. So this had some appeal as another example of Where is the missing 1 in the equation? (From an older essay of mine):
Ramanujan stated this radical in his lost notebook:
$$\sqrt{5+\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5-\dots}}}}}}} = \frac{2+\sqrt 5 +\sqrt{15-6\sqrt 5}}{2}$$
I still don't have any idea on this one.
Another surprising equation $$\bbox[7pt,border:3px #FF69B4 solid]{\color{red}{\large 213 \times 122 = 25986}}$$ Now read above expression in reverse order $$\bbox[7pt,border:3px #FF69B4 solid]{\color{red}{\large 68952 = 221 \times 312}}$$
Personally I find this very interesting: $$ \lim_{n\to\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}=\frac{1}{2}. $$
I've found this to be rather surprising:
$\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{2^n}=1$
$\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{2^n\ln(2^n)}=1$
As it essentially yields the identity:
$$\sum\limits_{n=1}^{\infty}\frac{1}{2^n}=\sum\limits_{n=1}^{\infty}\frac{1}{2^n\ln(2^n)}$$
It is surprising because obviously:
$$\forall{n\in\mathbb{N}}:\frac{1}{2^n}\neq\frac{1}{2^n\ln(2^n)}$$
In fact, the above inequity holds for every value of $n$, except for $n=\log_2e$.
Still, when summing up each of these infinite sequences, the result is $1$ in both cases.
$$(x_1^2+x_2^2+x_3^2+\dots+x_{16}^2)(y_1^2+y_2^2+y_3^2+\dots+y_{16}^2) = z_1^2+z_2^2+z_3^2+\dots+z_{16}^2$$
where the $z_i$ are rational functions of the $x_i, y_i$. One would have thought that $n$ square identities are only for $n = 1,2,4,8$, but non-bilinear ones in fact are for all $n = 2^m$.
It is still strange for me $$ i^i = e^{\pi(2k-\frac{1}{2})}. $$ And so, one could say $i^i\in\mathbb R$.
Note that $i^i$ is a sequence of real numbers and actually $i^i\not\in\mathbb R$, but still $i^i\subset\mathbb R$.
A rather simple one $$2^4 = 4^2$$
You can use this one to "proof" (as a prank) that $x^y = y^x$
$$(1+2+3+\cdots+n)!=1!3!5!\cdots(2n-1)!$$for $n=0,1,2,3,4$.
$$\sum_{n=1}^\infty(n\,\operatorname{arccot}n-1)=\frac12+\frac{17\pi}{24}-\ln\sqrt{e^{2\pi}-1}+\frac1{4\pi}\operatorname{Li}_2\left(e^{-2\pi}\right),$$ where $\operatorname{Li}_2$ is the dilogarithm.
If $x^n + y^n + z^n=0$ and $xyz \ne 0$, then $$\frac{(x^n-y^n)^2}{(xy)^n} + \frac{(y^n-z^n)^2}{(yz)^n} + \frac{(z^n-x^n)^2}{(zx)^n} = -9.$$
Using the mystical ennead to calculate the decimal expressions for fractions of 7. Start with this figure:
Then follow the connected path, giving the sequence 1 4 2 8 5 7. Then you write this sequence starting on each digit, in order, giving
. 1 4 2 8 5 7 = 1/7
. 2 8 5 7 1 4 = 2/7
. 4 2 8 5 7 1 = 3/7
. 5 7 1 4 2 8 = 4/7
. 7 1 4 2 8 5 = 5/7
. 8 5 7 1 4 2 = 6/7
One of the most worth-mentioning identities may probably be Carl Friedrich Gauss's compution of $\cos(\frac{2\pi}{17})$:
\begin{align} \cos(\frac{2\pi}{17})&= \frac1{16}[-1+\sqrt{17} + \sqrt{34-2\sqrt{17}}+2\sqrt{17+3\sqrt{17}-\sqrt{34-2\sqrt{17}}-2\sqrt{34+2\sqrt{17}}}]\\ \end{align}
which has a significant role in regular heptadecagon construction.
There are many fantastic equations that I've seen, but the one that definitely sticks out as top in my mind is the Atiyah-Singer Index theorem (it can be written in many ways; this is the way that I first learned it).
$$\operatorname{Ind}(D) = (-1)^n \int_M \frac{\operatorname{ch}(E)-\operatorname{ch}(F)}{\operatorname{e}(TM)} \operatorname{Td}^{hol}(TM \otimes \mathbb C)$$
Here $M$ is a $2n$-dimensional smooth compact manifold, with $E$, $F$ vector bundles over $M$, $D:\Omega^0(E) \rightarrow \Omega^0(F)$ is an elliptic differential operator, $\operatorname{Ind}(D) = \dim \ker D - \dim \operatorname{coker} D$, $\operatorname{ch}$ denotes the Chern class, $\operatorname{e}$ is the Euler class, and $\operatorname{Td}^{hol}$ is the holomorphic Todd class. Note that in this formulation the choice of the divisor depends naturally on the choice of $D$, a fact which is obscured in the notation.
It's probably the only equation I've ever seen in math which forced me to think about two entire fields in a different way. The mere fact that something like this connecting analysis and topology at such a deep level could be true is really incredible.
$$\cos \left(20\right) \cos \left(40\right) \cos \left(80\right) = \frac{1}{8}$$ for angles in degrees. This identity is interesting for its historical association with the teenage Richard Feynman. From Genius by James Gleick:
"He and his friends traded mathematical tidbits like baseball cards. If a boy named Morrie Jacobs told him that the cosine of 20 degrees multiplied by the cosine of 40 degrees multiplied by the cosine of 80 degrees equaled exactly one-eighth, he would remember that curiosity for the rest of his life, and he would remember that he was standing in Morrie's father's leather shop when he heard it."
$$(1+i)(1+2i)(1+3i) = (1-i)(1-2i)(1-3i)$$
This one is no less-
Let $d$ be the distance between Incenter($r$) and Circumcenter ($R$) Then-
$$R^2-d^2=2Rr$$ and this one $$\frac{1}{R-d}+ \frac{1}{R+d}=\frac{1}{r}$$
from $1$ years ago
$$\tan x=\cfrac{x}{1-\cfrac{x^2}{3-\cfrac{x^2}{5-\cfrac{x^2}{7-...}}}}$$
The series $$\sum_{n=1}^{\infty} \frac{n^{13}}{e^{2\pi n} - 1} = \frac{1}{24}$$ is not entirely obvious. (At this time WolframAlpha is unable to find its closed form.)
I was really amazed by discovering that the squared arcsine function has a pretty nice Taylor series at the origin: $$\arcsin^2(z)=\frac{1}{2}\sum_{n\geq 0}\frac{(2z)^{2n}}{n^2\binom{2n}{n}}$$ Even more amazed by the variety of techniques one may employ to prove such identity: combinatorial convolutions, hypergeometric transformations, (poly)logarithmic integrals, the Lagrange inversion theorem, the residue theorem, Legendre polynomials, Euler's Beta function, creative telescoping... They're all pretty interesting.
There are many involving infinite sums of number theoretic functions: $$\sum_{i = 1}^\infty \frac{\phi(i)}{i^k} = \frac{\zeta(k - 1)}{\zeta(k)}$$ $$\sum_{i = 1}^\infty \frac{\tau(i)}{i^k} = \zeta(k)^2$$ $$\sum_{i = 1}^\infty \frac{\sigma(i)}{i^k} = \zeta(k - 1)\zeta(k)$$ $$\sum_{i = 1}^\infty \frac{\mu(i)}{i^k} = \frac{1}{\zeta(k)}$$
And for some reason, the Riemann zeta function pops up in each. (I have no idea if these are well-known or not, I just thought they were very surprising, because I learned about these functions in a very discrete, number theory context, and the zeta function in, well, not that.)
$$
2^{67}-1 = 193,707,721 × 761,838,257,287
$$
This identity was found by Cole in the early 20th century. He later said that the result had taken him "three years of Sundays" to find.
There's also the fact that:
$2^{127} -1 $ is indeed prime, as Mersenne claimed. This was the largest known prime number for 75 years, and the largest ever calculated by hand. Édouard Lucas proved its primality in 1876.
This is the most surprising result that I am the discoverer of.
Consider the diophantine equation $$x(x+1)...(x+n-1) -y^n = k$$
where $x, y, n,$ and $k$ are integers, $x \ge 1$, $y \ge 1$, and $n \ge 3$.
I was led to consider considering this by trying to generalize the Erdos-Selfridge result that the product of consecutive integers could never be a power.
I phrased this as "How close and how often can the product of $n$ consecutive integers be to an $n$-th power?"
Looking at this equation, it seemed reasonable to think that, for fixed $k$ and $n$, there were only a finite number of $x$ and $y$ that satisfied it. This was not too hard to prove.
What greatly surprised me was that I was able to prove that for any fixed $k$, there were only a finite number of $n$, $x$, and $y$ that satisfied it.
The proof went like this:
I first showed that any solution must have $y \le |k|$. This was moderately straightforward, and involved considering the three cases $y < x$, $x \le y \le x+n-1$, and $y \ge x+n$.
The next step really surprised me. I showed that $n < e|k|$, where $e$ is the good old base of natural logarithms.
The proof was amazingly (to me) simple. Since $y \le |k|$ and $2(n/e)^n < n!$,
$\begin{align} 2(n/e)^n &< n!\\ &\le x(x+1)...(x+n-1)\\ &= y^n+k\\ &\le |k|^n+|k|\\ &\le |k|^n+|k|^n\\ &= 2|k|^n\\ \end{align} $
so $n < e |k|$.
I still remember staring at this in disbelief, over forty years later.
I have always been fascinated by Leibniz's formula for $\pi$: $$\dfrac{\pi}{4}=\sum\limits_{n=0}^\infty \dfrac{1}{2n+1}\times(-1)^{n}=1-\dfrac{1}{3}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{9}-\dfrac{1}{11}\dots$$ This can be used to determine the exact value of $\pi$, which is what makes it interesting. $$\displaystyle \boxed{\pi=4\sum\limits_{n=0}^\infty \dfrac{1}{2n+1}\times(-1)^{n}=4-\dfrac{4}{3}+\dfrac{4}{5}-\dfrac{4}{7}+\dfrac{4}{9}-\dfrac{4}{11}\dots}$$
The Fibonacci sequence occurs in the decimal expansion of some "special" fractions.
$\begin{array}{ccccccccc} \frac{1}{89} &= &0.\color{blue}{0} \\ &+ & &\color{blue}{1} \\ &+ & & &\color{blue}{1} \\ &+ & & & &\color{blue}{2} \\ &+ & & & & &\color{blue}{3} \\ &+ & & & & & &\color{blue}{5} \\ &+ & & & & & & &\color{blue}{8} \\ &+ & & & & & & & &\color{blue}{13} \\ &+ & & & & & & & & &\ddots \\ \end{array}\\ \begin{array}{cc}\frac{1}{89} &=0.\color{blue}{011235}\color{red}{9}\ldots \end{array}$
Eventually the pattern is destroyed by carries in the decimal digits of the sum. If you want to go further, simply take a different fraction:
$\begin{array}{lllllllll} \frac{1}{9899} &= &0. &\underbrace{00}_{F_0} &\underbrace{01}_{F_1} &\underbrace{01}_{F_2} &\underbrace{02}_{F_3} &\underbrace{03}_{F_4} &\underbrace{05}_{F_5} &\underbrace{08}_{F_6} &\underbrace{13}_{F_7} &\underbrace{21}_{F_8} &\underbrace{34}_{F_9} &\underbrace{55}_{F_{10}} &\ldots \\ \end{array}$
Again the pattern from this point forward is obscured by carries in the sum (the next two digits are $90$ not $89$).
These patterns are a consequence of the fact that the generating function for the Fibonacci sequence (with $F_0=0, F_1=1$) is
$$f(x)=\frac{x}{1-x-x^2}$$
and taking $x=0.1,0.01,0.001,\ldots$ ensures that the terms in the power series expansion are synonymous with places in the decimal expansion. These values of $x$ are identified with the "special" denominators $89,9899,998999,\ldots$ which are really just entries in the sequence $a_n=10^{2n}-10^n-1$ for $n=1,2,3,\ldots$.