**1.** $$ e^{\pi i} + 1 = 0$$

This simple equation links five fundamental mathematical constants:

- The number 0, the additive identity.
- The number 1, the multiplicative identity.
- The irrational number π (pi), pivotal in trigonometry and geometry.
- The transcendental constant e, the base of the natural logarithm, widely used in scientific analysis.
- The number i (iota), the imaginary unit of complex numbers, and the square root of -1.

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:
$$e^{ix}=cosx+ i sinx$$
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:

**5**. Gaussian integral:

$$\int_{-\infty}^\infty \! e^{-x^2}dx = \sqrt{\pi}$$

**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:

$$H\Psi(x,t) = i\hbar\frac{\partial}{\partial t}\Psi(x,t)$$