$\pi$ and e (the base of the natural logarithm) are two of the most important and widely studied mathematical constants. Although they seem very different from each other, they are actually closely related through various mathematical formulas.

One of the most famous relationships between $\pi$ and e is given by Euler’s identity:

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

This formula relates e, $\pi$, the imaginary unit i, and the number 1 in a beautiful and unexpected way. It is considered one of the most elegant and profound equations in all of mathematics.

Another connection between $\pi$ and e comes from calculus. The derivative of sine is cosine, and the derivative of cosine is negative sine. Therefore, if you take the derivative of sine repeatedly, you will get a sequence of functions that alternate between sine and cosine. Similarly, if you take the derivative of e^x repeatedly, you will get a sequence of functions that are all equal to e^x. These two sequences of functions can be combined using complex numbers to give the formula:

$e^{i\frac {\pi}{4}} = \frac {1+i}{\sqrt 2}$

This formula relates e, $\pi$, and the complex number $\frac {1+i}{\sqrt 2}$ in a way that has many important applications in mathematics and physics.

There are many other connections between $\pi$ and e that have been discovered over the years, including formulas involving integrals, infinite series, and continued fractions. These connections highlight the deep and surprising interrelationships that can exist between apparently unrelated mathematical concepts.