hi, thank you for the detailed comment! i have a writeup with more details on the common definitions (including the natural log one), if you google "better explained math intuition" (on a kindle currently). would love to know what you think. my ultimate goal for e elucidation is to get people to jump from any definiton to the others as they are all linked (i.e. see them as wys to rephrase "write down the equation for perfectly continuous growth").
Well, I have a clever way to explain the relationship between areas under a hyperbola and continuously compounded interest:
dy/dx = 1/x
dx/dy = x
Hopefully that isn't too confusing. If you start with the second relation (continuously compounded interest), a reciprocal gets you to the first (area under a hyperbola) -- the connection between exponentials/logarithms and areas under hyperbolae isn't immediately obvious to most people. Implicit differentiation is kind of scary, but "draw a curve whose slope is its height" requires an art degree.
I like your essay on intuition, but I don't know if your interpretation of the factorial series is really deep enough to explain what is going on. A fully elementary treatment of Taylor's series is a really hard thing to achieve. This, however, could help:
The binomial theorem for integer exponents is easy enough to grasp, though Wikipedia's geometrical explanation is kind of hilarious:
"[...] if one sets a = x and b = Δx, interpreting b as an infinitesimal change in a, then this picture shows the infinitesimal change in the volume of an n-dimensional hypercube, [...]"
I'm afraid infintesimal changes in the volume of n-dimensional hypercubes probably aren't going to make exponential functions any easier to grasp for epsilons...
Thanks for taking a look! My intuition for the dy/dx relations are "amount of growth" and "time needed to grow to the next increment". So for me, e is the amount you have after growing continuously for exactly 1 unit of time (integrate 1/x until it is 1).
I'll have to examine that link while I have time to study (vacationing now) and agree the wiki explanation wouldn't be the most enlightening :).
Comments
hi, thank you for the detailed comment! i have a writeup with more details on the common definitions (including the natural log one), if you google "better explained math intuition" (on a kindle currently). would love to know what you think. my ultimate goal for e elucidation is to get people to jump from any definiton to the others as they are all linked (i.e. see them as wys to rephrase "write down the equation for perfectly continuous growth").
Well, I have a clever way to explain the relationship between areas under a hyperbola and continuously compounded interest:
dy/dx = 1/x
dx/dy = x
Hopefully that isn't too confusing. If you start with the second relation (continuously compounded interest), a reciprocal gets you to the first (area under a hyperbola) -- the connection between exponentials/logarithms and areas under hyperbolae isn't immediately obvious to most people. Implicit differentiation is kind of scary, but "draw a curve whose slope is its height" requires an art degree.
I like your essay on intuition, but I don't know if your interpretation of the factorial series is really deep enough to explain what is going on. A fully elementary treatment of Taylor's series is a really hard thing to achieve. This, however, could help:
http://en.wikipedia.org/wiki/Binomial_theorem#Series_for_e
The binomial theorem for integer exponents is easy enough to grasp, though Wikipedia's geometrical explanation is kind of hilarious:
"[...] if one sets a = x and b = Δx, interpreting b as an infinitesimal change in a, then this picture shows the infinitesimal change in the volume of an n-dimensional hypercube, [...]"
I'm afraid infintesimal changes in the volume of n-dimensional hypercubes probably aren't going to make exponential functions any easier to grasp for epsilons...
Thanks for taking a look! My intuition for the dy/dx relations are "amount of growth" and "time needed to grow to the next increment". So for me, e is the amount you have after growing continuously for exactly 1 unit of time (integrate 1/x until it is 1).
I'll have to examine that link while I have time to study (vacationing now) and agree the wiki explanation wouldn't be the most enlightening :).