eefing amazing:

Fun with e:

Exponential function is the only function whose derivative is itself, i.e. (ex)' = ex
So the slope of the tangent line at any point on ex curve is ex.
Here are some example tangent lines at various points on ex.

Exponential function is the only function whose integral is itself.
So the area under the curve of ex from -∞ to x is ex.

ex and its reciprocal 1/ex = e-x. Symmetric about Y axis.

Average of ex and e-x is the hyperbolic cosine function, cosh.
Half the difference of ex and e-x is the hyperbolic sine function, sinh.

ex2 and e-x2. Not exponential but use e.
e-x2 is the [simplest] Gaussian function. Its infinite integral (i.e. area under its curve) is √π≈1.772.

The standard normal distribution:

Gaussian/normal distribution function:

Probability of a permutation being a derangement (i.e. a complete rearrangement) is 1/e (in the limit as n-->&infinity;) ≈ 0.367879441
#permutations = n!
#derangements !n = floor(n!/e + 1/2)


The Prime Number Theorem: π(N), the number of primes up to N, is approximately N/ln N.
(which equals eN/N)

N ln N N/ln N π(N)
106 13.81 72,382 78,498
109 20.72 48,254,942 50,847,534
1012 27.63 36,191,206,825 37,607,912,018
1015 34.54 28,952,965,460,216 29,844,570,422,669

Euler's constant
γ ≈ 0.57721. Difference between the sum of a harmonic series of n terms and ln n approaches γ as n approaches ∞.

Alternating harmonic series
1-1/2+1/3-1/4+1/5-16+1/7...= ln 2