It is the limit approached by #(1+1/n)^n# as #n# increases without bound.

It is the limit approached by #(1+n)^1/n# as #n# approaches #0# from the right.

It is he number that the sum:

#1+1/1+1/2+1/(3*2)+1/(4*3*2) + 1/(5*4*3*2) + . . . # approaches as the number of terms increases without bound.

It is the base of the function with y intercept #1#, whose tangent line at #(x, f(x))# has slope #f(x)#. This function turns out to be the exponential function #f(x) = e^x#.

It is the base for the growth function whose rate of growth at time #t# is equal to the amount present at time #t#.

It is the value of #a# for which the area under the graph of #y=1/x# and above the #x#-axis from #1# to #x# equals #1#.

If we define #lnx# for #x>+1# (as we often do in Calculus 1) as the area from #1# to #x# under the graph of #y=1/x#, then #e# is the number whose #ln# is #1#.