Of all the things in mathematics, the last thing we expect is to witness two absolutely irrational constants appear alongside each other in way more than one single situation.
In this short post we’ll try to uncover what makes e and π somewhat related to each other. Within the next few hundred words, we’ll update the way we view these constants and get acquainted with a simple flowchart that directly connects the two of these.
Firstly, to set the stage let’s remind ourselves that traditionally π is witnessed in everything circly! On a two dimensional plane should you plot all the points that are equidistant from a central one, you end up with an infinite number of concentric circles.
Interestingly, for all these concentric circles when the circumference gets divided by the distance from the centre the answer turns out to be the twice of π. This is true for every circle. And we all accept that.
Now save some of your money in an awesome bank that offers you 100% interest per year that gets compounded every instant! After an year you’ll end up with the money you saved multiplied by e. Let the money sit for twice the unit time and it gets multiplied by e squared.
The circle for π and compound interest for e are the most common rendezvous points between a student and these constants. Yet hidden deep within these examples is a common ground. Both of them involves an infinitesimal change occuring at every instant.
If we would have compounded at 100% per annum for a year we would have simply doubled our principal. Had we turned only once while going from one equidistant point of a central point to another located at right angles, we would have ended up with a perimeter that is twice the distance from centre at those points.
In the first case e popped up when we started to compound at every instant. The circle formed and π popped up when we started to turn by an infinitesimal amount at every instant.
There’s no doubt that Euler’s formula connects π and e the most elegantly. We’ll therefore take the Euler highway to connect the two. We won’t derive the formula neither dive into any details but just point out the intervening machinery separating π and e.
We saw how π is inbuilt into the definition of the circle. When we represent a circle in our usual cartesian coordinates we like to do it by designating the x and y coordinates of the curve of the circle to sine and cosine of x respectively. Let’s pause here for a second.
Now e initially came up as the constant that the exponential function evaluates with x as 1. And what is the exponential function? It’s just an infinite series that is the expansion of a expression which is a trade of between how fast you want to grow and how long you want to grow. All very compound interest-y.
Just that this exponential function is found when there exists a limit towards infinity with regard to the iteration of our growth and zero with regards to the amount of growth per iteration. You expand this out and you get a polynomial that yields ever diminishing higher order terms. Evaluate the same at 1 and the series gives you e. An infinite series gives you a constant with infinite digits? Well that seems to fit.
Interestingly enough, both sine and cosine of any variable could also be expressed in terms of a series similar to e such that when you combine sine and cosine in a particular ratio (that of the imaginary number i) you surprisingly end up with the exponential function! The involvement of i here is purely justified as it has the beautiful property of alternating between +1 and -1 as it gets raised to higher and higher powers.
And that’s it! e is related to sine and cosine and so is π and that connects the two. e gets connected to trigonometric ratios when we think analytically and π gets connected to trigonometry when we think geometrically. And in the end we end up with an algebraic expression with an awfully named imaginary number starring in a cameo.