If e=(1+1/n)^n = 2.71828, does that mean e = 1 raised to the power of infinity is also 2.71828?
OT: Euler's number (2.71828)
Started by ●November 13, 2014
Reply by ●November 13, 20142014-11-13
On 11/13/2014 8:44 PM, Bill Bowden wrote:> If e=(1+1/n)^n = 2.71828, does that mean e = 1 raised to the power of > infinity is also 2.71828? >Keep hitting those buttons, and report back when you get there. ;) (Hint: the whole point of limits is that lim(x->x0) {f(x)} isn't necessarily equal to f(x0).) Cheers Phil Hobbs
Reply by ●November 13, 20142014-11-13
On 2014-11-14, Bill Bowden <bperryb@bowdenshobbycircuits.info> wrote:> If e=(1+1/n)^n = 2.71828, does that mean e = 1 raised to the power of infinity is also 2.71828?ah, you mean lim(n=>infinity) (1+1/n)^n> does that mean e = 1 raised to the power of infinity is also 2.71828no, for e it has to be 1+1/infinity ^ infinity :) although 1.000001 ^ 1000000 _will_ get you 2.71828 -- umop apisdn
Reply by ●November 14, 20142014-11-14
On Thu, 13 Nov 2014 17:44:33 -0800, Bill Bowden wrote:> If e=(1+1/n)^n = 2.71828, does that mean e = 1 raised to the power of > infinity is also 2.71828?1^infinity = 1. The hypothesis e = (1 + 1/n)^n is untrue for any finite n. The _limit_ of (1 + 1/n)^n, as n goes to infinity, is e -- but that's a different story than just e = (1 + 1/n)^n. -- www.wescottdesign.com
Reply by ●November 15, 20142014-11-15
On Thursday, November 13, 2014 5:44:40 PM UTC-8, Bill Bowden wrote:> If e=(1+1/n)^n = 2.71828, does that mean e = 1 raised to the power of infinity is also 2.71828?Sorry, arithmetic with 'infinity' isn't that easy. Consider (1+1/n) ^(2*n) which, in the limit of large 'n', becomes e^2 Also, 2.718281828459045235... is Napier's constant; Euler's constant, 0.57721566 is (I think) gamma = lim(n-> infinity) { alog(n) - Sum(M=1...n){ 1/M } } and there are "Euler numbers" E_n tabulated in my trusty old Abramowitz and Stegun. Wikipedia calls 'e' Euler's number. Either there's some national name differences, or perhaps Wikipedia is in error?
Reply by ●November 19, 20142014-11-19
"Tim Wescott" <tim@seemywebsite.com> wrote in message news:V6ydnVCCRutV1vvJnZ2dnUU7-R-dnZ2d@giganews.com...> On Thu, 13 Nov 2014 17:44:33 -0800, Bill Bowden wrote: > >> If e=(1+1/n)^n = 2.71828, does that mean e = 1 raised to the power of >> infinity is also 2.71828? > > 1^infinity = 1. > > The hypothesis e = (1 + 1/n)^n is untrue for any finite n. > > The _limit_ of (1 + 1/n)^n, as n goes to infinity, is e -- but that's a > different story than just e = (1 + 1/n)^n. > > -- > www.wescottdesign.comIn a RC circuit of 1 volt, 1 ohm, 1 farad, and 1 second, the voltage across the resistor will be 1/e at the end of 1 time constant (R*C). This is confusing since it seems there should be some exact value of voltage. I have trouble with irrational numbers, but I guess it's the same problem as working out the square root of 2? . -Bill --- news://freenews.netfront.net/ - complaints: news@netfront.net ---
Reply by ●November 22, 20142014-11-22
On Wed, 19 Nov 2014 17:16:59 -0800, Bill Bowden wrote:> "Tim Wescott" <tim@seemywebsite.com> wrote in message > news:V6ydnVCCRutV1vvJnZ2dnUU7-R-dnZ2d@giganews.com... >> On Thu, 13 Nov 2014 17:44:33 -0800, Bill Bowden wrote: >> >>> If e=(1+1/n)^n = 2.71828, does that mean e = 1 raised to the power of >>> infinity is also 2.71828? >> >> 1^infinity = 1. >> >> The hypothesis e = (1 + 1/n)^n is untrue for any finite n. >> >> The _limit_ of (1 + 1/n)^n, as n goes to infinity, is e -- but that's a >> different story than just e = (1 + 1/n)^n. >> >> -- >> www.wescottdesign.com > > In a RC circuit of 1 volt, 1 ohm, 1 farad, and 1 second, the voltage > across the resistor will be 1/e at the end of 1 time constant (R*C). > This is confusing since it seems there should be some exact value of > voltage. I have trouble with irrational numbers, but I guess it's the > same problem as working out the square root of 2?If you have an RC circuit with exactly one ohm, exactly one farad, that starts with exactly one volt on the cap and runs for exactly one second until you can make an exact voltage measurement, then you have somehow exited the real universe and are inhabiting the Land of the Platonic Ideal. In that case, you will probably have no problem with instinctively and intuiting the value of 'e', and everything about it. Put more realistically: you can't achieve an exact one-ohm resistor, or an exact one-farad cap. Even if you could, you couldn't start with exactly one volt on the cap, you couldn't time off exactly one second, and at the end of your exact one-second interval, you couldn't exactly measure the voltage on the cap. If you just buy off the shelf parts and do the experiment, ending up with a voltage that's within 10mV of 0.368V will be doing well. -- www.wescottdesign.com
Reply by ●November 22, 20142014-11-22
In article <QOidnYvBe-DNfu3JnZ2dnUU7-cWdnZ2d@giganews.com>, tim@seemywebsite.com says...> > On Wed, 19 Nov 2014 17:16:59 -0800, Bill Bowden wrote: > > > "Tim Wescott" <tim@seemywebsite.com> wrote in message > > news:V6ydnVCCRutV1vvJnZ2dnUU7-R-dnZ2d@giganews.com... > >> On Thu, 13 Nov 2014 17:44:33 -0800, Bill Bowden wrote: > >> > >>> If e=(1+1/n)^n = 2.71828, does that mean e = 1 raised to the power of > >>> infinity is also 2.71828? > >> > >> 1^infinity = 1. > >> > >> The hypothesis e = (1 + 1/n)^n is untrue for any finite n. > >> > >> The _limit_ of (1 + 1/n)^n, as n goes to infinity, is e -- but that's a > >> different story than just e = (1 + 1/n)^n. > >> > >> -- > >> www.wescottdesign.com > > > > In a RC circuit of 1 volt, 1 ohm, 1 farad, and 1 second, the voltage > > across the resistor will be 1/e at the end of 1 time constant (R*C). > > This is confusing since it seems there should be some exact value of > > voltage. I have trouble with irrational numbers, but I guess it's the > > same problem as working out the square root of 2? > > If you have an RC circuit with exactly one ohm, exactly one farad, that > starts with exactly one volt on the cap and runs for exactly one second > until you can make an exact voltage measurement, then you have somehow > exited the real universe and are inhabiting the Land of the Platonic > Ideal. In that case, you will probably have no problem with > instinctively and intuiting the value of 'e', and everything about it. > > Put more realistically: you can't achieve an exact one-ohm resistor, or > an exact one-farad cap. Even if you could, you couldn't start with > exactly one volt on the cap, you couldn't time off exactly one second, > and at the end of your exact one-second interval, you couldn't exactly > measure the voltage on the cap. > > If you just buy off the shelf parts and do the experiment, ending up with > a voltage that's within 10mV of 0.368V will be doing well.That is, if you have a Volt meter that reads exactly! Jamie
Reply by ●November 22, 20142014-11-22
On Sat, 22 Nov 2014 19:33:00 -0500, Maynard A. Philbrook Jr. wrote:> In article <QOidnYvBe-DNfu3JnZ2dnUU7-cWdnZ2d@giganews.com>, > tim@seemywebsite.com says... >> >> On Wed, 19 Nov 2014 17:16:59 -0800, Bill Bowden wrote: >> >> > "Tim Wescott" <tim@seemywebsite.com> wrote in message >> > news:V6ydnVCCRutV1vvJnZ2dnUU7-R-dnZ2d@giganews.com... >> >> On Thu, 13 Nov 2014 17:44:33 -0800, Bill Bowden wrote: >> >> >> >>> If e=(1+1/n)^n = 2.71828, does that mean e = 1 raised to the power >> >>> of infinity is also 2.71828? >> >> >> >> 1^infinity = 1. >> >> >> >> The hypothesis e = (1 + 1/n)^n is untrue for any finite n. >> >> >> >> The _limit_ of (1 + 1/n)^n, as n goes to infinity, is e -- but >> >> that's a different story than just e = (1 + 1/n)^n. >> >> >> >> -- >> >> www.wescottdesign.com >> > >> > In a RC circuit of 1 volt, 1 ohm, 1 farad, and 1 second, the voltage >> > across the resistor will be 1/e at the end of 1 time constant (R*C). >> > This is confusing since it seems there should be some exact value of >> > voltage. I have trouble with irrational numbers, but I guess it's the >> > same problem as working out the square root of 2? >> >> If you have an RC circuit with exactly one ohm, exactly one farad, that >> starts with exactly one volt on the cap and runs for exactly one second >> until you can make an exact voltage measurement, then you have somehow >> exited the real universe and are inhabiting the Land of the Platonic >> Ideal. In that case, you will probably have no problem with >> instinctively and intuiting the value of 'e', and everything about it. >> >> Put more realistically: you can't achieve an exact one-ohm resistor, or >> an exact one-farad cap. Even if you could, you couldn't start with >> exactly one volt on the cap, you couldn't time off exactly one second, >> and at the end of your exact one-second interval, you couldn't exactly >> measure the voltage on the cap. >> >> If you just buy off the shelf parts and do the experiment, ending up >> with a voltage that's within 10mV of 0.368V will be doing well. > > That is, if you have a Volt meter that reads exactly! > > JamiePicky picky. Voltage reading -- how's that. And no cheating with masking tape and a felt pen, either. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●November 27, 20142014-11-27
"Tim Wescott" <tim@seemywebsite.com> wrote in message news:QOidnYvBe-DNfu3JnZ2dnUU7-cWdnZ2d@giganews.com...> On Wed, 19 Nov 2014 17:16:59 -0800, Bill Bowden wrote: > >> In a RC circuit of 1 volt, 1 ohm, 1 farad, and 1 second, the voltage >> across the resistor will be 1/e at the end of 1 time constant (R*C). >> This is confusing since it seems there should be some exact value of >> voltage. I have trouble with irrational numbers, but I guess it's the >> same problem as working out the square root of 2? > > If you have an RC circuit with exactly one ohm, exactly one farad, that > starts with exactly one volt on the cap and runs for exactly one second > until you can make an exact voltage measurement, then you have somehow > exited the real universe and are inhabiting the Land of the Platonic > Ideal. In that case, you will probably have no problem with > instinctively and intuiting the value of 'e', and everything about it. > > Put more realistically: you can't achieve an exact one-ohm resistor, or > an exact one-farad cap. Even if you could, you couldn't start with > exactly one volt on the cap, you couldn't time off exactly one second, > and at the end of your exact one-second interval, you couldn't exactly > measure the voltage on the cap. > > If you just buy off the shelf parts and do the experiment, ending up with > a voltage that's within 10mV of 0.368V will be doing well. > > -- > www.wescottdesign.comI recently read something about Graham's number being the longest known number having an exact value ending in 262464195387. Graham's number is longer than the number of atoms in the earth. Or, as wiki says, the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume. Seems like e should be something shorter than that. http://en.wikipedia.org/wiki/Graham%27s_number -Bill --- news://freenews.netfront.net/ - complaints: news@netfront.net ---
