tag:blogger.com,1999:blog-7718462793516968883.post9080444144600775655..comments2023-03-02T12:12:05.847-05:00Comments on MadMath: Proof of Approximating Radicals to the Closest IntegerDeltahttp://www.blogger.com/profile/00705402326320853684noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-7718462793516968883.post-79989570502970915812013-08-01T02:27:38.071-04:002013-08-01T02:27:38.071-04:00Good question and yes, in fact it's worse, thi...Good question and yes, in fact it's worse, this one fails for lots of integers: like any floor of the average of sequential cubes. Example: Let x = 4. It's closest to the cube 1^3 = 1 (4-1=3 but 8-4=4). But the cube root of 4 ~ 1.59, which is actually closer to 2.<br /><br />Algebraically comparing the cutoffs like I did above: cube-of-average (n+1/2)^3 = n^3 + 3/2*n^2 + 3/4*n + 1/8, while average-of-cubes (n^3+(n+1)^3)/2 = n^3 + 3/2*n^2 + 3/2*n + 1/2. So in this case there's a difference in the n term, indicating the gap between cutoffs gets wider and wider, trending arbitrarily large as n increases. <br /><br />Which is maybe an even better illustration that while you can get a correct <i>interval</i> (bracketing range) with this technique, saying that you're closer to one end or the other by looking at the power is not generally correct.Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-46529547426068760342013-07-29T22:06:49.139-04:002013-07-29T22:06:49.139-04:00Interesting! I'll admit to using the rule of t...Interesting! I'll admit to using the rule of thumb your textbook suggests, though I didn't learn it anywhere, and since I never got past the first year of my math major, lo these many moons ago (another one lost to the humanities, alas), I lacked the mathematical acumen to notice the cases where it's false. I recently had call to do a lot of estimating of cube-roots (don't ask), so I'm curious if there's a similar problem with estimating cubes. Seems as though there'd have to be...LWSCHURTZhttps://www.blogger.com/profile/06635573516962732975noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-69488637622286633382013-07-21T12:38:37.387-04:002013-07-21T12:38:37.387-04:00I totally agree, of course!I totally agree, of course!Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-4082910956041745842013-07-20T13:20:20.538-04:002013-07-20T13:20:20.538-04:00I'd be very uncomfortable too, and not just be...I'd be very uncomfortable too, and not just because of the factual error.<br /><br />For me the bigger concern is that a great by-product of math is that it promotes rigorous thinking -- that is, math should help one become sensitive to assumptions, and help one understand that even reasonable-sounding assumptions can be wrong. But this text is giving an implicit endorsement of a bad assumption.<br /><br />It might be argued that rigorous thinking is too lofty of a goal to ask of remedial students. That might be right (though I think it's a more important skill than estimating square roots). But at the very least, I think a math text should not be endorsing such an assumption!<br />BostonQuadhttps://www.blogger.com/profile/15739476475159681555noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-19149613264861065992013-07-19T17:45:15.199-04:002013-07-19T17:45:15.199-04:00Sounds like a great exercise! Interested in how it...Sounds like a great exercise! Interested in how it will work in-class...Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-32766846976379619422013-07-19T17:26:18.695-04:002013-07-19T17:26:18.695-04:00And that's why I liked the idea of possibly wa...And that's why I liked the idea of possibly walking through it with students. <br /><br />"Is this true always, sometimes, never?" Feels like always, but when we try to show it, we get to sometimes. It seems like a good item for helping students see why proof is necessary.Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-42132314257617977792013-07-19T17:21:50.891-04:002013-07-19T17:21:50.891-04:00Sue, thanks for reading that and commenting. It do...Sue, thanks for reading that and commenting. It does seem like an assumption that's easy to make (even for some book authors) although it's not true for all values of x.Deltahttps://www.blogger.com/profile/00705402326320853684noreply@blogger.comtag:blogger.com,1999:blog-7718462793516968883.post-69191818820051200892013-07-18T18:54:37.404-04:002013-07-18T18:54:37.404-04:00Nice work!
I don't think my textbooks have m...Nice work! <br /><br />I don't think my textbooks have made that claim. But this is a great question to ask students to ponder. I wonder if I might ask it of my pre-calc students at some point.Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.com