What Are the Secrets Behind the Number e and Its Calculations?

[robertjford80]

I'm trying to figure out what the number e is all about.

100 * .1 = 110
110 * .1 = 121
110 * .1 = 133.1

that should be equal to 100e^.4, right?

well, 100e^.4 = 134.99, not 133.1

What am I doing wrong?

 

[Dick]

I have no idea what you are doing, right or wrong. 100*.1=10. Can you explain?

 

[robertjford80]

the formula for calculating continuous compound interest is

A = Pe^rt

Where A = final amount
P = initial amount
r = rate
t = time

if you start with 100 dollars and the rate is 10% after 3 payment periods it should be 134.99 based on the above formula.

well, 100 * 1.1 is 110, 110 * 1.1 = 121, 121 * 1.1 = 133.1, not 134.99

 

[Dick]

the formula for calculating continuous compound interest is

A = Pe^rt

Where A = final amount
P = initial amount
r = rate
t = time

if you start with 100 dollars and the rate is 10% after 3 payment periods it should be 134.99 based on the above formula.

well, 100 * 1.1 is 110, 110 * 1.1 = 121, 121 * 1.1 = 133.1, not 134.99

Your formula is only valid for continuously compounded interest. Not for interest paid at intervals.

 

[robertjford80]

ok, thanks, i thought they were the same but i was wrong.

 

[e^(i Pi)+1=0]

e is what happens when you continuously compound something over an infinitely short interval.
$x\stackrel{lim}{\rightarrow}∞$ (1+$\frac{1}{x}$)^x=e

 

[Ray Vickson]

ok, thanks, i thought they were the same but i was wrong.

They can be made to give the same results at integer values of time t = 1,2,3,..., but you need to adjust the rate. In order to have a continuous interest rate r give a true annual interest of i you need to have
$$e^r = 1 + i, \text{ or } r = \ln(1+i).$$

In your example, to get a true annual interest rate of 10% you need to take a continuous interest rate of 9.531017980%, giving r = 0.0953101798. If you take, instead, a continuous rate of 10% you get a true annual rate of $i = e^{0.1}-1 = 0.105170918,$ or about 10.5171%. This is the origin of the differences you note.

RGV