tag:blogger.com,1999:blog-7718462793516968883.post3507544714189573785..comments2021-01-05T07:04:40.286-05:00Comments on MadMath: Google Divide by ZeroDeltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-7718462793516968883.post-12250956363203861822012-09-11T01:36:45.060-04:002012-09-11T01:36:45.060-04:00Yeah, that graphical calculator interface is new s...Yeah, that graphical calculator interface is new since I wrote that post. Slightly disappointing that it returns infinity (although that's correct in the extended reals).Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-12720125655063293092012-09-10T21:28:46.471-04:002012-09-10T21:28:46.471-04:00So, you can sort of force google calculator to att...So, you can sort of force google calculator to attempt to divide by zero: enter a non-division-by-zero expression, and then use the calculator buttons to enter, for example 8 / 0. It returns infinity though...<br />http://imgur.com/r3cmdAnonymoushttps://www.blogger.com/profile/00455373209587558415noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-29184438403653132092012-04-02T00:15:18.722-04:002012-04-02T00:15:18.722-04:00^ Well, you got me to look more closely at your ar...^ Well, you got me to look more closely at your argument above (3/10), and it's pretty good. But I think an opponent would say that summation is only equal to e^x for all x granted a prior definition of 0^0 = 1.<br /><br />Intriguingly, this lead to me tracking down a contradiction on this point in an old calculus book of mine (Stein/Barcellos 5E, 1992). They do say of that summation, "this equation clearly holds when x = 0, since it then reduces to the equation e^0=1" (p. 631), yet elsewhere they state "0^0 is not defined" (p. S-31). Fascinating.<br /><br />Anyway, I'm personally convinced that we should define 0^0=1 for purposes like these and others (binomial theorem). Thanks a bunch for the input.Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-26997292376720995142012-03-27T18:02:44.195-04:002012-03-27T18:02:44.195-04:00It doesn't matter what the sequence 0^b conver...It doesn't matter what the sequence 0^b converges to. There is no law that says all functions have to be continuous. x/x converges to 1 as x tends to 0, but that doesn't mean 0/0 is 1.<br /><br />(Please forgive the extreme lateness of this reply)Glowing Face Manhttps://www.blogger.com/profile/07717328290680086281noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-40565152291435989052012-03-11T12:53:57.281-04:002012-03-11T12:53:57.281-04:00^ Yes, but then they can come back with what the s...^ Yes, but then they can come back with what the sequence 0^b (b->0) converges to, etc.Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-89152880981494855072012-03-10T23:05:19.339-05:002012-03-10T23:05:19.339-05:00Next time, catch them off guard. Without any ment...Next time, catch them off guard. Without any mention of 0^0, show them the series:<br />sum_{i=0}^{infty} x^i/i!. Ask what it converges to.<br />"Why, e^x, of course!"<br />Ask them, isn't that rather controversial?<br />"Nonsense, there couldn't be anything more fundamental in mathematics!"<br />I see... and 0!, it is 1, yes?<br />"Of course..."<br />Alright then. We are agreed, 0^0=1.Glowing Face Manhttps://www.blogger.com/profile/07717328290680086281noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-14800877799037253732012-03-09T14:21:21.109-05:002012-03-09T14:21:21.109-05:00^ Now, my understanding is that there's legiti...^ Now, my understanding is that there's legitimate debate about whether one defines 0^0 = 1 or not. As someone one said, the function a^b has a discontinuity at (0,0) no matter which way you go with it it. (Either looking at a^0 =1 mostly or 0^b = 0 mostly.)<br /><br />I agree with Euler, Donald Knuth, etc., that it's best to define 0^0 = 1. But I tried arguing that we do that in our in-house custom algebra books where I teach, and it was in fact declined.Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-69383935444477327922012-03-09T13:08:09.063-05:002012-03-09T13:08:09.063-05:00The plot thickens: enter 0^0 and Google correctly...The plot thickens: enter 0^0 and Google correctly outputs 1 (this is something which Wolfram Alpha, unbelievably and inexcusably, gets wrong, saying 'indeterminate' even while saying that sum_{i=0}^{infty}x^i/i!=e^x, which, for x=0, is inconsistent with 0^0 being "indeterminate")<br /><br />I don't think it's a joke nor an oversight. When you enter 6/0, you haven't entered a meaningful mathematical question, so there is no reason to respond mathematically.Glowing Face Manhttps://www.blogger.com/profile/07717328290680086281noreply@blogger.com