How does continuous compound interest work?
Well, suppose the magical "Bank of Bernoulli" offers you 100 percent interest per annum. Or, in decimal terms, a return of 1.0.
Great. You put a dollar in the bank and the bank gives you two dollars at the end of the year, one dollar representing the principal, the other the yield. Only one simple computation has to be made, so we can say of this situation that n = 1.
It gets better. Bank of Bernoulli [so named after the great 17th century mathematician Jacob Bernoulli] decides to compound semi-annually. You put in one dollar, and on June 30 you are credited with 100 percent interest for half a year, or $0.50. In the second half of the year, you earn the same 1/2 of the underlying annual rate, for the redefined principal amount, this time on $1.50. So you earn $0.75 between June 30 and December 31. This means you have $2.25 waiting for you at the end of the year.
You can keep this up, presuming a calculation at the end of every quarter (n=4) or month (n-12) and so forth.
You will find that at each stage
A(n)=(1+1/n)n
Where A is the final amount in the bank. So, for example, the amount when n = 2, with the mid-year break we mentioned above, is 1 + 1/2, or 1.5, to the power of 2. That gives us 2.25.
If there is a daily compounding, so that when n is 365, then the amount available in the account at the end of the year is found by adding 1 to 1/365, then raising the result to the power of 365. In that case, A, the final amount at the end of the year, is $2.71. The amounts continue to grow as the input n grows, but they grow by diminishing amounts, so that over time they are levelling off, to a limit.
Can we calculate the continuous compound interest, as n approaches infinity? Bernoulli came from a banking family, so his interest in the mathematics of it apparently began very young.
It turns out that the proper formula for yield is Pert = A. This is easy to remember for English speaking folks who know the word "pert"! It means that the amount of principal, multiplied by e [Bernoulli didn't call it that when he first worked this out in the 1680s, but we will make do with archaisms], then raised to a power defined by the time the money spends earning interest and the underlying interest rate, gives us the final amount.
The underlying interest rate is expressed as a fraction, so that for the superscript "t" above we don't use, say, "4%", we use 0.04.
Or, for our Bank of Bernoulli example, we say that r is 1 and t is 1, so the exponent that results from multiplying them ... is one.
The initial investment, P, is one dollar. Since everything else in this formula is one, the amount, A, must be given us by e. And if you follow this through, you find that the number e, the limit of this process approaching infinity, is irrational, i.e. there is an infinity of digits beginning 2.71828. So (with a quite modest rounding up) you will get $2.72 at the end of the year.
That is where, so to speak, Achilles catches the tortoise. Yes, Bernoulli's discussion arose in the early days of calculus and the refutation of Zeno that calculus allows. Bernoulli knew Leibniz and was in a sense 'in on' the early days of calculus as a branch of mathematics.
So: why is the number named e, and called Euler's number? Why isn't it called b?
More to come.
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