*Navigation Panel:* (These buttons explained below)

**Answers and Explanations**

## The Number *e* as a Limit

So far we've talked about "continually compounding" interest: interest
is assumed to be earned continually. In many bank accounts, however,
interest only starts becoming eligible to earn further interest on
specific dates (such as the end of each month, or the end of each
year) when it is officially added in to your balance. The calculation
of compound interest is different in this case, and this different
calculation gives a nice mathematical definition of the number
*e*.
Suppose the simple interest rate is 100% for a certain period. Under
simple interest alone, the money will double during that period. Under
continually compounding interest, it will be multiplied by *e*. But
what happens if interest is compounded at *n* equally spaced dates
during that period (such as at the and of each month), rather than
continually?

In the interval before the first compounding date, you are just
earning simple interest on the original balance. Let's call the
original balance *B*. The interest rate for this interval is 1/*n*,
since the total interest rate for all *n* intervals is 1 (100% is
just another way of writing 1). So, on the first compounding date, one
gets paid interest in the amount of (1/*n*)*B*. The balance
after this payment is therefore *B *+ (1/*n*)*B* which
equals (1 + (1/*n*)) *B*. In other words:

(balance at end of first interval)
= (1 + (1/*n*)) times (balance at start of first interval)

= (1 + (1/*n*)) *B*.

During the second interval (between the first and second compounding
dates), one is again earning interest at a rate of 1/*n*, but
this time on this new balance (1 + (1/*n*))*B*.
Therefore,
(balance at end of second interval)
= (1 + (1/*n*)) times (balance at start of second interval)

= (1 + (1/*n*)) times (1 + (1/*n*)) *B*

= .

If you do this same calculation again, you find that the balance at
the end of the third interval is
and so
on. The balance at the end of the *n*th and final interval is
.
Therefore: if the interest is compounded *n* times during the
period, the final balance is times the
original balance.

Compounding at more and more frequent intervals means that one is
approximating more and more closely the idea of *continual
compounding* which we talked about earlier, in which the final
balance is *e* times the original balance. What this means is
that

*The number **e* is the limit of the quantity
as *n* goes to infinity.

This gives a nice mathematical definition of the number *e*.

[Go On]

This page last updated: September 1, 1997

Original Web Site Creator / Mathematical Content Developer:
Philip Spencer

Current Network Coordinator and Contact Person:
Any Wilk - mathnet@math.toronto.edu

*Navigation Panel:* Go backward to The General Situation

Go up to Does the Number e Have Special Meaning?

Go forward to The Number *e* in Calculus

Switch to text-only version (no graphics)

Access printed version in PostScript format (requires PostScript printer)

Go to University of Toronto Mathematics Network
Home Page