tag:blogger.com,1999:blog-2761684730989137546.post2667739050577711689..comments2024-03-29T09:32:34.853-04:00Comments on Mike Norman Economics: Illustration of an Integralmike normanhttp://www.blogger.com/profile/03296006882513340747noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-2761684730989137546.post-50997864040399010032016-06-22T06:25:51.177-04:002016-06-22T06:25:51.177-04:00Shifting the interval does not help, since rationa...Shifting the interval does not help, since rationals/irrationals are mixed up everywhere on the real line. (Needless to say, the derivative exists nowhere.)<br /><br />This is a classic example from real analysis. There's some good arguments that functions like that are meaningless, but it's hard to exclude them from consideration. You need to do something like get rid of the real line; the problem is - what do you replace it with?<br /><br />In the real world, the described method is pretty much how we calculate integrals numerically. However, we know that these numerical calculations are approximations, and they do not converge to the right answer if we run into degenerate functions like the one I gave above.Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-2761684730989137546.post-44945043729333714032016-06-21T22:46:09.657-04:002016-06-21T22:46:09.657-04:00Well Brian couldnt we just shift the period from 2...Well Brian couldnt we just shift the period from 2 to 3; or 4 to 5 or something... iow dont start at 0 ?Matt Frankohttps://www.blogger.com/profile/11978352335097260145noreply@blogger.comtag:blogger.com,1999:blog-2761684730989137546.post-64332674716431420792016-06-21T19:31:16.247-04:002016-06-21T19:31:16.247-04:00That's what we teach to (non-math major) under...That's what we teach to (non-math major) undergraduates about integrals, and doesn't work.<br /><br />Try integrating: f(x) on the interval [0,1], where:<br /><br />f(x) = { 0, if x is rational (can be written as p/q, where p,q are integers),<br /> { 1, if x is irrational.<br /><br />Note: an any non-zero interval, there is always a rational and irrational number in it. So if we divide f(x) into slices, what is the height of the slice - 0 or 1?<br /><br />Whee, isn't math fun!<br /><br />(The answer is 1; we need to use the notion of Lebesgue intervals to get there.)<br />Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-2761684730989137546.post-89791207053410140942016-06-21T19:11:26.230-04:002016-06-21T19:11:26.230-04:00Why would you want to calculate the area?
This is ...Why would you want to calculate the area?<br />This is a signal of some sort, not a raster. In analog to digital conversion, the quality would depend on the sampling rate and bit size. For example, an 8 bit integer would chop the y-axis into 256 parts.Peter Panhttps://www.blogger.com/profile/09473311771939167712noreply@blogger.com