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Answers and Explanations
Does the number e have any real
physical meaning, or is it just a mathematical convenience?
Yes, the number e does have physical meaning. It occurs
naturally in any situation where a quantity increases at a rate
proportional to its value, such as a bank account producing interest,
or a population increasing as its members reproduce.
Obviously, the quantity will increase more if the increase is based on
the total current quantity (including previous increases), than if it
is only based on the original quantity (with previous increases not
counted). How much more? The number e answers this question.
To put it another way, the number e is related to the how much more
money you will earn under compound interest than you would under
Therefore, we will address the following topics:
- Simple and Compound Interest:
a discussion of the meaning of simple and compound interest, and related
concepts in more general situations such as population growth.
- A Physical Meaning for the Number e:
how the concepts of simple and compound interest lead
to a physical interpretation of the number e in one
particular case (the case in which your money would double under
- The General Situation:
how the number e enters in to the picture when we aren't in this
one particular case.
- The Number e as a Limit:
what happens when interest is compounded only at specific times
instead of continually, and how this yields a mathematical
definition for the number e as a certain limit.
- The Number e in Calculus:
why the number e is an especially important and natural one in calculus.
You may select any of the items above, or follow
the [Go On] links to see
them discussed in order.
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