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:
(continuous growth. inflation etc.)
"e is an interesting number" "It sure is pertty"
Exponential function is the only function whose derivative is itself,
i.e. (ex)' = ex
i.e. slope is itself. "Most important property"
NB. derivative of general exponential function bx is bx ln b
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.
Hanging/sagging/drooping chain/rope/cable is a catenary curve: a cosh(x/a)
Steiner's problem: where's the max value of x1/x? i.e. xth root of x
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:
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
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.
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.
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)||%primes|
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.