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So, we learned that the more times we compound, the more money we make...  What if we could compound continuously?

Let's figure out how the formula for this would work:

We'll invest $1.00 at 100% interest for one year and we'll keep increasing the compounding and see what happens.

       A quick example, so you can follow this:

$1.00 compounded quarterly at
100%
interest for 1 year...

initial amount = $1

split factor = 1.25

number of splits = 4

final amount = 1 * ( 1.25 )^( 4 ) = approximately $2.4414... ... we can leave off the 1

 


TRY IT:

Do the same for compounded monthly.


Let's make a table:

TIMES COMPOUNDED     AMOUNT
annually     $2
semi-annually     $2.25
quarterly   04-exponentials-02.gif $2.4414062...
monthly   04-exponentials-03.gif $2.6130352...
100 times   04-exponentials-04.gif $2.7048138...
1000 times   04-exponentials-05.gif $2.7169239...
10,000   04-exponentials-06.gif $2.7181459...
100,000   04-exponentials-07.gif $2.718268...
1,000,000   04-exponentials-08.gif $2.7182804...

Look at what's happening here.  arrow

Not changing very much anymore, are they?

In fact, they are getting closer and closer to a very special number

e = approximately 2.7182818

It's an irrational number like pi.  It goes on forever and ever and never repeats.

We won't be able to use the split factor for continuous compounding, BUT we WILL be able to use this e guy...  and he came from the split factor!

Continued on the next page