EggMath: Embryo Calculus
The Number e
Among all exponential
functions b^{t}, the one with
base e has many special properties.
For any exponential function
b^{t}, the
slope of the tangent
line is proportional to b^{t}. The
constant of proportionality here is 1 only when
the base is the special number e. This can
be taken as the defining property of e.
As we shall see, this number e is not a
whole number. In fact it's not even a fraction
(such as 22/7 or 87/32). If you try to write it
as a decimal, you find that the decimal expansion
never ends and never repeats; instead it's an
example of an irrational number. Its value
is approximately:
e = 2.7182818284590...
How to Compute
e
Computing the precise value of the number
e is a surprisingly mysterious task. We
will describe two different ways to do this, even
though the full explanation of why the methods
work is unfortunately a bit beyond our reach.
(For that, you need to know some calculus).
The first method is as follows:
Use the sliding bar to change n in the
formula (1+1/n)^{n}.
Notice how, as n increases, the answers
seem to be getting closer and closer to a fixed
number. This limiting value, is the magic
number e. In mathematical language, this
is expressed by saying:
The number e is the
limit as n tends to
infinity of the expression
(1+1/n)^{n}.

The second method is as follows:
 step 1: compute 1+1
 step 2: compute 1+1+1/2
 step 3: compute 1+1+1/2+1/(2x3)
 step 4: compute 1+1+1/2+1/(2x3)+1/(2x3x4)
 step 5: compute
1+1+1/2+1/(2x3)+1/(2x3x4)+1/(2x3x4x5)
As you can see, it starts to get a bit
cumbersome to write each new term that gets added,
even though the pattern should be clear. We can
make it a bit easier by introducing a useful piece
of notation. Instead of 2x3, we write
3!; instead of 2x3x4 we write
4!, etc. In this notation, n!
(pronouced `n factorial') stands for the product
of all the integers from 1 up through
n. We can thus write step 6 as:
 step 6: compute
1+1+1/2!+1/3!+1/4!+1/5!+1/6!
You can probably now guess the pattern:
 step n: compute
1+1+1/2!+1/3!+...+1/(n1)!+1/n!
(The dots mean `continue in this way until you
get to'.)
We can write this sum (from step n) more
compactly using the greek letter Sigma, as in the
applet. Using the slider to change n, and
hence the number of terms in the sum.
Mysteriously, as you add more and more terms to
this series, you find that the answer gets closer
and closer to  you guessed it  e. We
thus get our second recipe for e:
The number e is the limit as
n tends to infinity of the
sum
1+1+1/2!+1/3!+...+1/(n1)!+1/n!

Although our two methods both give e as
a limiting value, the second one converges much
more rapidly, as you can see with the applets.
The Function e^{t}
In fact, both these limiting methods can be
used more generally to compute the exponential
function e^{t}. That is, it turns
out that the value of e^{t} can be computed as
the limit (as n tends to infinity) of:
(1 + t/n)^{n}
or as the limit (as n tends to infinity)
of:
1 + t + t^{2}/2! + t^{3}/3! + ... +
t^{n1}/(n1)! +
t^{n}/n!.
Either one of these formulas tells us how to
compute exponential functions even when t
is not a whole number. Again, the second method,
with the summed factorials, converges much faster.
For this reason it is the one used in practice by
calculators and computers.
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