That Number e

One of the more interesting results from limits as x goes to infinity, is the limit of (1+1/x)^x as x goes to infinity.  This limit, introduced by Leonhard Euler, came out of the consideration of exponential functions. (It shows up in finance when interest is compounded.)

As x goes to infinity, 1/x goes to zero.  One might then be tempted to think that (1+1/x)^x goes to 1.  It does not.  The exponent is also going to infinity and so as the expression 1+1/x goes to 1, the expression (1+1/x)^x  stays between 2 and 3.  Here, using the variable n instead of x, is a table of values of (1+1/n)^n as n gets large.

Mathematicians have given this limit a name, defining the constant e to be this limit.

See this Wikipedia webpage on this interesting constant!  The constant e is much more important in mathematics than pi!

Here is e to 100 decimal digits, compliments of WolframAlpha.

2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427

What are the strangest constants in mathematics?  Surely the top three are e, pi and i.  At the end of the first part of this course, we will ask the question, What is e^{pi i}? If we combine these three strange constants we get a very simple answer... but that is later.