The elusive exponential: what is e, and what does it mean?
Most people first meet e in calculus or logs class in high school. And most think of e along with π. And in a sense, this is right, as both are transcendental and irrational.
Transcendental numbers are non-algebraic numbers: they are not real, and they are not solutions to polynomials with integer coefficients. Most real numbers we deal with are algebraic numbers, thus, not transcendental. For instance, the equation x²+1=0 gives x=√-1, a complex number that is also an algebraic number.
Now let's look at irrationality.
Most of us are familiar with the idea that π’s digits go on forever. This is the same with e. As such, both the two numbers are irrational — they cannot be written as a ratio of two integers. In order to understand e, let’s start by representing e using continued fractions, a fraction of the form
Now we can use an algorithm to express ‘e’ as a continued fraction.
e=2.71828… (Key point here being the 3 points representing it goes on infinitely.)
⌊e⌋ = ⌊2.71828…⌋ = 2 (Taking the floor function — the smallest integer less than the number)
e–2 = 0.71828… (subtracting the value)
Taking 1/0.71828…
⌊1.392214…⌋ = 1 (And now we repeat the process)
1.392214…–1=0.392214
(Repeating once again)
And so on. We have generated the infinite continued fraction representation of e.
If we cut this infinite fraction off at an arbitrary point to use integers less than 1000, we get this fraction below
which is accurate to 4 decimal points.
Now that we have represented ‘e’ with an infinite, continued fraction, we can define what e actually is. Many of you might associate e with compound interest, as compound interest led to e’s discovery.
Compound interest is different from simple interest. In simple interest, the amount of interest paid is calculated as a percentage of the original amount, and this interest amount is not changed as time goes on. Compound interest, however, is interest that is calculated on an initial amount, then re-calculated with the initial amount plus the previous interest added.
The equation for compound interest is as follows.
Say we invest $1 in a bank and that money gets compounded with 100% interest rate n times per 1 year. Although unrealistic, this simplifies calculations.
For starters, let n=1 so the money gets compounded once in the year.
Now, let n=12 so the money gets compounded every month for one year.
Now, let n=365 so the money gets compounded every day for one year.
Does 2.7145… look familiar? If we keep compounding the money more and more times —as n approaches ∞ —the expression seems to be getting closer to e and in fact, mathematically we can say the limit of the above expression equals e.
What we’ve just uncovered is the meaning and the origin of the number e, and the applications of this are nearly unlimited. Hopefully, the next time you come across this transcendental number, you’ll be able to see a little more behind it than you would have before.
