Fun with *e*:

Exponential function is the only function whose derivative is itself,
i.e. (*e*^{x})' = *e*^{x}

So the slope of the tangent line at any point on *e*^{x} curve is *e*^{x}.

Here are some example tangent lines at various points on *e*^{x}.

Exponential function is the only function whose integral is itself.

So the area under the curve of *e*^{x} from -∞ to x is *e*^{x}.

*e ^{x}* and its reciprocal 1/

Average of *e ^{x}* and

Half the difference of

*e*^{x2} 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:

**Derangements**

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)

**Factorials**

**Primes**

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

(which equals e^{N}/N)

N | ln N | N/ln N | π(N) |
---|---|---|---|

10^{6}
| 13.81 | 72,382 | 78,498 |

10^{9}
| 20.72 | 48,254,942 | 50,847,534 |

10^{12}
| 27.63 | 36,191,206,825 | 37,607,912,018 |

10^{15}
| 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