Important Numbers
The Number Pi
The original definition of the number pi, or ,
is that it is the ratio of the circumference of a circle (the distance around) to its diameter
(the distance across the circle).
Here is the number to 1000 decimal places according
to the symbolic math app Maple:
Thanks to J.M. Borwein (of Simon Fraser University) and his brother P.B. Borwein and D.V.Chudnovsky and his
brother G.V.Chudnovsky, the value of is now known to
several billion decimal places.
Click here
to link to a site that gives several algorithms
that can be used to calculate to many decimal places.
The number e
Consider the function y = 2^{x}, which is graphed to the right.
This type of function is known as an exponential function because the variable x is in
the exponent. Notice that the higher up the curve one goes, the steeper the curve becomes.
Other exponential functions with other bases such as y = 3^{x} and
y = 4^{x} have similar shapes.
Question: Is there an exponential function with the property that the slope (steepness) of the function
exactly equals the height of the function at every point?
Answer: Yes. It is the exponential function y = e^{x}, where the base
e is approximately 2.718.
Here is the number e to 1000 decimal places according to the symbolic math app Maple:
This graph of the exponential function y = e^{x} shows that at the point where the
height is 1 the slope is 1; at the point where the height is 5 the slope is 5; at the point
where the height is 10 the slope is 10; etc.
In the field of calculus, which is concerned with slopes and the rates of change of things,
a function with the property that the slope equals the height will obviously play a fundamental role.
The exponential function y = e^{x} can be written in the form of an
infinite power series:
The 3 dots indicate that the series goes on forever. The exclamation mark ! denotes the
factorial function. If the series is truncated
(cut off) after a few terms then we get a reasonable approximation for the function.
For example if we let x=1 and keep just 5 terms of the series then this becomes:
which already gives e to within ½ %.
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