3. e and Calculus
300 years ago Euler showed the importance of the number e through analysing certain functions with the Calculus. I shall attempt to follow in his footsteps (although I have no idea exactly what he did). I shall also give a totally fictional possibility as to how the number became to be known as 'e'. Some people say that it was simply named after Euler himself, but nobody knows for sure. I have a different theory!
Before I begin I’d like to point out that the standard formula for finding the integral of a power of x:
fails when the power is -1. The problem of how to find the integral of the reciprocal of x troubled mathematicians for some time and was only resolved once the number e was put in its rightful place as one of the fundamental constants of the universe. Possibly, the resolution of this problem was what put the number e onto this pedestal. This article attempts to recreate the possible train of thought that might have been taken to reach this conclusion and, amazingly, it does not set out to solve this problem at all but rather seems to stumble upon it quite by accident!
Ooh… look familiar? Remember in the last article on e when I expanded this expression?
Now let’s call this special value of a the next letter in the alphabet that we haven’t used yet…hmmm, now let's see....That’d be 'e' !!!!
So we now know that:
That is interesting; we have found a function that has its y value equal to the gradient. But still this number is just a curiosity, not actually all that useful…
Let’s play around with it and see what crops up…
For some arbitrary constant .
Suddenly the number gains a very special place in maths – particularly with respect to Calculus.
There is a problem with this formula though: logs cannot take a negative argument, so this is only valid when Ax is positive. For negative values we have to consider the area under the x-axis curve for negative x values. Because of the symmetry of the function 1/x, it is plain to see that the area under a negative region is the same as that under the corresponding positive x region. Therefore:
The negative sign before the second integral is because of the negative area under the x-axis represented by the first integral.
And so, all because of this strange number, we suddenly have a result that people had been looking for for years. Once this mystery was unlocked, the doors were opened to numerous discoveries in all sorts of areas of maths and the sciences. It really is a rather extraordinary number!