Peter Kaminski

I have "e" on my financial calculater (the venerable HP 12c).

It's a big factor in determining the value of an annuity based on compound interest. Hence the appropriateness for a financial transaction like an IPO.

In calculating interest: i=prt (interest is the principal times the rate times the number of terms)

If an annuity compounds monthly and the annual interest rate is nominally 12 percent then the interest (i) is the principal times the annual rate divided by the number of payments) times the number of payments.

E^x (e to the x) is a way of determining the "effective" interest rate that applies when the principal is continuously compounded.

Here's a myth to start: It's also the source of the letter "e" in front of e*Trade...

Not to mention eBay

The author of the above comment is mistaken (or at the very least confused his terminology).

If:

i = interest accrued over t terms (i.e. the effective interest)

p = principal amount

r = rate of interest per term (as a multiplier -- i.e. 1.07 means 7% interest)

t = number of terms

Then:

i = p * (r^t)

If you like, you could call the number (r^t) the "effective interest" and give it the letter "e", but it's not at all the same as the number e.

Just thought I'd also note that the Googleplex also has a building titled e (right next door to pi).

Hi, Jacob!

Check the "Ask Dr. Math: Compounding Interest and 'e' " and "The Number e as a Limit" links above. You can use 'e' to calculate the result of continuous compounding (compounding with zero interval) because (1+(1/n))^n approaches 'e' as n goes to infinity.

Since I am a graduate of the University of Cambridge (BA, MA (Cantab.) Pure & Applied Mathematics) I don't need lessons from "Dr. Math" :)

The point I have obviously failed to make above is that there is no reason to assume an infinite number of compounded interest terms.

In finance you need to know the "effective interest" for a fixed (i.e. finite) period. It is completely meaningless to ascribe meaning to a number arrived at after waiting an infinite amount of time.

If you read Dan Miller's comment again you'll see that he was talking about valuing an annuity and not working out a geometric progression.

Forgive me for being a pedant here.

I suppose I can see why investment bankers would want to know a quick way to do approximate calculations using a calculator.

I just hope that my tax bill isn't worked out using the exponential approximation rather than the proper way I noted above.

I hate approximations :)

I'd like to take a turn at being pedantic. "I'm an incurable nitpick," as somebody once said.

The original article is incorrect on one point and misleading on another. The number e can certainly be expressed as the ratio of two numbers, but at least one of these numbers must not be an -integer-. An irrational number is one that cannot be expressed as the ratio of two -integers-. Every number is equal to itself over 1.

"Trancendental" means that the number is not the root of a polynomial with integer (equivalently, rational) coefficients. (A number that is is called "algebraic.") For example, the square roots of 2 and 3 are irrational, but they are not transcendental, because they are solutions to the equations x^2 - 2 = 0 and x^2 - 3 = 0, respectively. I'm not saying the author of the original post is incorrect, but what he says seems too unclear to be of much help in understanding the concept correctly.

I don't know that irrational or even transcendental qualify as "fancy" properties. In set-theoretic terms, there are many "more" transcendental numbers than rational ones; the rational numbers are countable, as are the algebraic ones, in fact, but real numbers are not. Most real numbers are not algebraic.

Is the sum of a rational and irrational number always irrational? What is the proof?(if true)