Can only get to *e* in a limiting process:
(*e* doesn't have a geometry like *π* does).

Fun with *e*:

**Money** with *e*:

future value FV (accumulated value) of *continuosly compounded interest* r of principal P in t years:

FV=P*e ^{rt}*

(continuous growth. inflation etc.)

"

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

i.e. slope is itself. "Most important property"

NB. derivative of general exponential function *b*^{x} is *b*^{x} ln b

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

Hanging/sagging/drooping chain/rope/cable is a catenary curve: *a cosh(x/a)*

Steiner's problem: where's the max value of x^{1/x}? i.e. x^{th} root of x

*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**
A *derangement* is a permuatation that is a complete rearrangement,
i.e. all items move.

Example: bca and cab are the derangements of abc.

Example: badc bcda bdac cadb cdab cdba dabc dcab dcba are the derangements of abcd.

3:2 4:9 5:44 6:265 7:1854 8:14833 9:133496

Probability of a permutation being a derangement
is 1/*e* (in the limit as n-->∞)
≈ 0.367879441

≈ 36.78% of permutations are derangements

#permutations = n!

#derangements !n = floor(n!/*e* + 1/2)

n!/!n = *e*

**Factorials**

Roll an n-sided die *n* times. Probability never is a particular side (e.g. a 1) is 1/*e*.

If have N equally likely outcomes. Do N experiments. Probability that a certain outcome
never happens is 1/*e*.

Gambling game: N slots, one the winner. N tries(plays). Chance of losing all (ie. never winning) is 1/*e*.

demo

The average number of random real numbers between 0 and 1 needed to sum
to more than 1 is *e*.

I.e. need an average of *e* numbers uniformly distributed in [0,1] to sum >1.

demo

**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) | %primes |
---|---|---|---|---|

10^{6}
| 13.81 | 72,382 | 78,498 | 7.8498 |

10^{9}
| 20.72 | 48,254,942 | 50,847,534 | 5.0847534 |

10^{12}
| 27.63 | 36,191,206,825 | 37,607,912,018 | 3.7607912018 |

10^{15}
| 34.54 | 28,952,965,460,216 | 29,844,570,422,669 | 2.9844570422669 |

**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

Google's IPO goal was to raise $2,718,281,828 (*e* billion dollars)

Entropy S = k ln W

I think that I shall never see a number as lovely as *e*.

**Logarithmic spirals.**

*x(t)=a e^{bt}cos(t)*