tag:blogger.com,1999:blog-3889388775673754833.post1036730645508907560..comments2022-11-27T10:19:27.872-05:00Comments on Stealing Commas: Basic Math - Division Ruleschris the cynichttp://www.blogger.com/profile/06872875475212333027noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-3889388775673754833.post-62830660709106459722011-12-27T15:45:44.387-05:002011-12-27T15:45:44.387-05:00And these are infinitely important rules to know, ...And these are infinitely important rules to know, so that when you are driving in your car and trying to find the prime factors for the number on you 100K+ odometer, you can figure out in advance whether a given number is worth attempting as a factor.<br /><br />Yes, this is how I pass the time on long trips. For convenience, I never bother starting with a number which cannot be divided by 6 and I rarely start with a number that cannot be divided by 36. The fun part is trying to finish the factoring before the number changes again. (Okay, I did factor out 111,111 despite being obviously divisible only by 3 with the tricks I knew. I was pleasantly surprised to find it factored neatly within my mental math capacity.)Mousehttps://www.blogger.com/profile/08075516515255227054noreply@blogger.comtag:blogger.com,1999:blog-3889388775673754833.post-58863172756177062582011-12-26T10:23:53.876-05:002011-12-26T10:23:53.876-05:00It's fun when you change number bases: some of...It's fun when you change number bases: some of the rules stayed kindasorta the same, and others you can forget. (I was a CS major with some boring classes. And I got 'New Math' when I was younger: base 7 and base 12 are both familiar.)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3889388775673754833.post-28482041024443440712011-12-18T08:45:06.118-05:002011-12-18T08:45:06.118-05:00You're right, it's been years since I'...You're right, it's been years since I've been in a math class and I forgot the exception.<br /><br />Pulling out a number theory book, I see that zero is explicitly excluded. a (not equal to zero) divides b if b=ac, a, b, and c all integers.<br /><br />I don't like that definition for a few reasons. Mostly it just seems inelegant. There's also the fact that intuitively it doesn't make a lot of sense to me to say, "a divides b because because there exists and integer c, which does not remotely divide b, such that ac = b." If c can't divide b then what the hell is it doing in the equation?<br /><br />I don't think that zero being indeterminate really makes a difference. If we say that zero divides zero that still doesn't help us determine the the result. 0=0*c where c is an integer, yes, but there are an infinite number of things c can be, so it's still indeterminate.<br /><br />Which brings me to how I would define divisibility if I really thought that it was for some reason important to exclude zero (and obviously someone did.)<br /><br />I'd say, the b integer b is divisible by an integer a iif there exists a unique integer c, such that b=ac. Pretty sure that gets you the same exact results, but it doesn't leave you saying, "Zero obviously satisfies this rule, why'd we make an exception?"<br /><br />Infinity isn't an integer, so I think the only time this would come up is when a and b are both zero.<br /><br />It has been a while though.chris the cynichttps://www.blogger.com/profile/06872875475212333027noreply@blogger.comtag:blogger.com,1999:blog-3889388775673754833.post-22880548340591911302011-12-18T04:24:29.602-05:002011-12-18T04:24:29.602-05:00Indeterminate forms aren't usually treated as ...Indeterminate forms aren't usually treated as definitive for number theory.<br /><br />0^0 = 1 in number theory by convention, even though it's an indeterminate form when encountered as a limit.<br /><br />On the other hand, I'm pretty sure the standard convention in number theory is that n|0 for all n *except* n=0Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3889388775673754833.post-29567149501750196172011-12-17T23:35:09.175-05:002011-12-17T23:35:09.175-05:00You consider 0 to be divisible by 0? You might wan...You consider 0 to be divisible by 0? You might want to review your indeterminate forms.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3889388775673754833.post-40270489348382989102011-12-17T19:17:32.607-05:002011-12-17T19:17:32.607-05:00Oh wow! A rule for 13? My 13yo learned the seven...Oh wow! A rule for 13? My 13yo learned the sevens rule in his number theory class this summer, but they didn't do the rule for 13.<br /><br />I was really excited to learn the rule for 7 this summer, because *my* high school Advanced Math teacher said, "There's a rule for seven, but it's much more complicated than just doing long division," and wouldn't tell us what it was.cjmrhttp://cjmr.livejournal.comnoreply@blogger.comtag:blogger.com,1999:blog-3889388775673754833.post-55352368536310846532011-12-17T18:43:51.958-05:002011-12-17T18:43:51.958-05:00x^2 = (x+c)(x-c) + c^2 (because x^2 = x^2 + xc - x...x^2 = (x+c)(x-c) + c^2 (because x^2 = x^2 + xc - xc - c^2 + c^2).<br /><br />This is useful for squaring numbers in your head, where x is the number you want to square and c is a number you pick (any number will work) to make the math easy.<br /><br />So, let's square 82. a good number to pick for c is "2" or "8" or "18" (if you have 18 squared memorised, otherwise it's a poor choice).<br /><br />82^2 = (82+2)*(82-2) + 2^2 (this step is just for completion, in my head i usually skip to the next step)<br />82^2 = 84*80 + 4<br />8*8 = 64, 8*4 = 32. therefore. 84*80 = 6400+ 320 = 6720. add 2^2 and you get 82^2 = 6724.<br /><br />the reason I said 18 was a good number to pick:<br /><br />82^2 = (82+18)(82-18) + 18^2<br />100*64 = 6400. 18^2 = 324. 6400 + 324 = 6724.<br /><br />Once you get the hang of that, try 3 digit numbers. You can do them in multiple steps:<br /><br />321^2 = 300*342 + 21^2<br />300*342 = 102600 there will probably be a carry, so start saying "one hundred three thousand" out loud while you work out the rest. it'll make it look like you solved it faster. hang on to the 600 on your fingers if you have to. if you didn't know there would be a carry, just start saying "one hundred" and hold on to 2600 on your fingers.<br /><br />meanwhile. 21^2 = 20*22 + 1 = 441. now finish giving the answer "forty one". 321^2 = 103041.<br /><br />--PAnonymousnoreply@blogger.com