tag:blogger.com,1999:blog-5544661968326910027.post2864109851301899210..comments2024-02-05T02:24:14.386-08:00Comments on Three J's Learning: 23 isn't prime (a short bedtime story)JGR314http://www.blogger.com/profile/11702319994021721608noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-5544661968326910027.post-7883676378335720722016-06-03T09:54:46.564-07:002016-06-03T09:54:46.564-07:00I have another math lie to add to this list, inspi...I have another math lie to add to this list, inspired by <a href="https://tjzager.wordpress.com/2016/05/13/straight-but-wiggled" rel="nofollow">TJ Zager straight-but-wiggled</a>. My comment:<br /><br />I'm now adding the "math teacher" definition of straight-line to my list of unintentional half-truths that we tell students. This one is a little complex, so probably most students will never learn the variety of full, technical definitions of a straight line. Here's one version in a <a href="http://www.math.illinois.edu/~dwest/regs/linemetric.html" rel="nofollow">metric space</a>. I'm not totally sure, but have the feeling that all of the (visually) wiggly lines in this post could actually be straight (under suitably defined metric for the plane).<br /><br />Vector spaces have another definition. Fun exercise: will the two notions be the same in a normed vector space?<br /><br />A crazy exercise (or fun game for the teacher to play at home) would be to use the students' (apparent) misconception and explore the following:<br />- what properties do we want a [straight line] to have? For example, shortest distance between two points.<br />- what if we assumed that these [lines are straight] and these other [lines] are not? This means we are magically giving the property we just agreed to these examples and saying these other ones don't have that property.<br /><br />What else would be true in our mathematical system? Do we get any results that can't work together (a contradiction)? If not, what other cool things can we discover about the new world we have just created? What's the same/different compared with our "usual" world?<br /><br />It just happens that this particular exploration playing with the definition of "straight line" is really rich and includes: non-euclidean geometry, differential geometry, general relativity.JGR314https://www.blogger.com/profile/11702319994021721608noreply@blogger.comtag:blogger.com,1999:blog-5544661968326910027.post-34505904043924909232015-01-29T06:53:39.117-08:002015-01-29T06:53:39.117-08:00A related discussion focusing on the old quip abou...A related discussion focusing on the old quip about how angle-side-side doesn't prove congruence, <a href="http://curiouscheetah.com/BlogMath/fibs-ssa/" rel="nofollow">Fibs</a>, and an extensive follow-up analysis of <a href="http://curiouscheetah.com/BlogMath/ssa-constraints/" rel="nofollow">S-S-A</a>.JGR314https://www.blogger.com/profile/11702319994021721608noreply@blogger.comtag:blogger.com,1999:blog-5544661968326910027.post-19082660321835582902015-01-29T00:27:52.698-08:002015-01-29T00:27:52.698-08:00I found the Chris Lusto post that was talking abou...I found the Chris Lusto post that was talking about <a href="https://linesoftangency.wordpress.com/2012/02/22/pretty-little-lies/" rel="nofollow">pretty little lies.</a><br /><br />Just to be clear: Chris is awesome. I don't mean to criticize him specifically and some of the examples he cites are tricky. Perhaps it is a subtle case-by-case call on which approach will lead to greater student understanding.JGR314https://www.blogger.com/profile/11702319994021721608noreply@blogger.comtag:blogger.com,1999:blog-5544661968326910027.post-60689748769063317462015-01-28T23:46:02.983-08:002015-01-28T23:46:02.983-08:00Divide by zero:
(1) Studying limits, we recognize ...<b>Divide by zero:</b><br />(1) Studying limits, we recognize that there is a sense in which we can understand division by zero, with 0/0 cases being the most exciting, but 1/0 and -1/0 type limits having an important place, too. A related example is the Dirac delta-function.<br />(2) It also suggests you can always divide by things that are non-zero. This isn't true for rings with zero divisors, for example matrix rings or integers mod 6.<br /><br /><b>1-1+1-1+1... </b> for this, there are two ways that I know to extend the convergence definition of summability:<br />(1) <a href="http://en.wikipedia.org/wiki/Ces%C3%A0ro_summation" rel="nofollow">Cesaro Summation</a><br />(2) <a href="http://en.wikipedia.org/wiki/Analytic_continuation" rel="nofollow">Analytic continuation</a><br /><br />These ideas have been popularly presented in <a href="https://www.youtube.com/watch?v=w-I6XTVZXww" rel="nofollow">Numphile's -1/12 video</a>. Frankly, I don't love their presentation because it takes something many people already find uncomfortable, convergent infinite series, and then seems to violate their intuition. That leaves them with the unsettling seeming confirmation that they never really understood what was going on.<br /><br />Admittedly, Numberphile also made this about my specific example: <a href="https://www.youtube.com/watch?v=PCu_BNNI5x4" rel="nofollow">Grandi's series video.</a> In that video, they do a better job of being more helpfully explicit. I will resist the temptation to make the observation that one presenter is a physicist and the other is a mathematician.JGR314https://www.blogger.com/profile/11702319994021721608noreply@blogger.comtag:blogger.com,1999:blog-5544661968326910027.post-85918738163748269502015-01-28T20:28:23.451-08:002015-01-28T20:28:23.451-08:00What is the white lie behind divide by zero and 1+...What is the white lie behind divide by zero and 1+1-1+1-1...?Heathhttps://www.blogger.com/profile/16219966444108277892noreply@blogger.com