tag:blogger.com,1999:blog-3748625510348961342.comments2016-12-31T10:34:22.028-08:00solidangl.esBill Shillitohttp://www.blogger.com/profile/17774101901445053590noreply@blogger.comBlogger31125tag:blogger.com,1999:blog-3748625510348961342.post-26644066691633581212016-12-31T10:34:22.028-08:002016-12-31T10:34:22.028-08:00Excellent article! I'd like to see you extend ...Excellent article! I'd like to see you extend the examples to include division and multiplication as well. It may be difficult to deal with division as something that's simple to understand as shorthand for something. 6 / 2 is short hand for 6 * 1/2 but students certainly are not used to that view. And the notion of left to right is important to include as a 3rd thing to remember since subtraction and division are not commutative.<br /><br />Bob Orchardhttp://www.blogger.com/profile/12615009252327101120noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-91934163931358797822016-12-26T03:02:04.392-08:002016-12-26T03:02:04.392-08:00I hate to say this but what is the ACTUAL function...I hate to say this but what is the ACTUAL function to solve the problem you propose because the problem in today's world is no one has a FUNCTION to solve problems efficiently.Amuro Kazamahttp://www.blogger.com/profile/17142044174072293847noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-48773914876427939792016-06-29T21:41:36.795-07:002016-06-29T21:41:36.795-07:00Why stopping at the + x exp relation, why not keep...Why stopping at the + x exp relation, why not keep going the pattern with the relations due the Conway chained arrow notation? With these simple relations, computation of the V of parallel wires was obtained, it was something nice! Maybe something deeper may happen if the system is generalized.Daniel de França MTd2http://www.blogger.com/profile/01281817409696805377noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-50442861286712909062016-06-29T21:41:19.622-07:002016-06-29T21:41:19.622-07:00This comment has been removed by the author.Daniel de França MTd2http://www.blogger.com/profile/01281817409696805377noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-9382962772830697412016-02-02T18:26:11.335-08:002016-02-02T18:26:11.335-08:00I just ran it to this post via another website. I...I just ran it to this post via another website. I am glad to see an article about alternatives to PEMDAS. For several years, I stopped using PEMDAS in my sixth grade classroom and their misconceptions like adding before all subtraction, went away completely. Well, until one of the kids came to class having his older brother help him with his homework. Suddenly he was making so many mistakes. I asked him what his brother did to "help" him and he answered "PEMDAS." I recently got all the other teachers at my school onboard with dropping PEMDAS and our scores on order of operations went up tremendously. I didn't know anyone else used GEMA. Again, like you said, it really isn't necessary to use any mnemonic at all, but it was the only way to get some to drop PEMDAS. Lovely article!LD Helferhttp://www.blogger.com/profile/15362613994636151867noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-13261392684388020082015-11-14T12:02:54.975-08:002015-11-14T12:02:54.975-08:00As the author wisely noted at the end of his artic...As the author wisely noted at the end of his article, it never hurts to have too many parentheses! <br />/Bravo, --problem eliminated!Lonewolf™http://www.blogger.com/profile/17592367417704960634noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-31773975960037273592015-08-16T09:00:27.189-07:002015-08-16T09:00:27.189-07:00Love it!Love it!Paul Ohttp://www.blogger.com/profile/06294340052733992008noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-18584089043896129122015-05-20T14:16:29.116-07:002015-05-20T14:16:29.116-07:00A computer scientist might argue "2" is ...A computer scientist might argue "2" is the "natural" logarithm base (anything binary). <br /><br />PS: damn this commenting system is weird.Victor Hernandeshttp://www.blogger.com/profile/00064218464586434649noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-27256876967819952012015-05-20T14:14:47.808-07:002015-05-20T14:14:47.808-07:00This comment has been removed by the author.Victor Hernandeshttp://www.blogger.com/profile/00064218464586434649noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-66242648144597530652015-04-29T18:21:13.317-07:002015-04-29T18:21:13.317-07:00Interesting idea, but I'm not at all a fan of ...Interesting idea, but I'm not at all a fan of that "visual cancellation rule" as phrased. There's no real mathematical reason behind it - it's all smoke and mirrors, which is pedagogically dangerous. You're relying on the "b^\b^" resembling a fraction that "cancels" to be 1, after which a multiplication sign magically appears next to the exponent. Math should be logical, not magical. Cancelling things is ubiquitous, sure, but you have to know *why* it works.<br /><br />That being said, I would be interested in seeing a linear form of the radical symbol and my reflected version of it.Bill Shillitohttp://www.blogger.com/profile/17774101901445053590noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-25252635717616844032015-04-29T17:42:35.800-07:002015-04-29T17:42:35.800-07:00Great idea. I had been searching for a notation t...Great idea. I had been searching for a notation that could be "typed" in linear text form. To that end, I had played around with combinations of the caret ^ and forward and reverse diagonals / \<br /><br />Given the linear notation for power:<br /><br />Power: b ^ e = p<br /><br />For root, I suggest the symbol ^/ (caret, slash):<br /><br /> p ^/ e = (b^e) ^/ e = b<br /><br />My root notation represents an "abbreviation" of the identity: e-th root of p equals p to the power 1/e,<br />that is: (b^e) ^/e represents (b^e) ^ (1/e) = b ^ (e * 1/e) = b ^ 1 = b<br /><br />But to get this advantage, I must reverse the order of base and power in the traditional notation.<br /><br />So, in similar vein, your log symbol could be represented by the symbol: ^\ (caret, back-slash):<br /><br />log: logb(p) = b ^\ p = b^\ (b^e) = e<br /><br />This is suggestive of a "visual cancellation rule" <br />b^\ (b^ e) "equals" ( b^ \ b^) e "equals" 1 * e = e.Joseph Austinhttp://www.blogger.com/profile/11397759210330425019noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-40938453703237601762015-04-28T16:50:42.594-07:002015-04-28T16:50:42.594-07:00My only problem with the new notation stems from t...My only problem with the new notation stems from the laziness of the writer. Since 2 is the most natural root and e is the most natural base for logarithms, we don't bother writing the 2 when we take the square root or writing the e when we take logarithm base e (whether you denote this as log or ln is a matter of preference). It wouldn't make sense to compromise on which should be considered natural for both notations since the e^th root doesn't naturally come up and log base 2 doesn't come up much (except maybe in computer applications). I could see how there would be confusion as to why leaving the number off when the sign is in one direction implies something different than leaving the number off when the sign is in the other direction.<br /><br />That said, I like this idea and think the advantages that you explained outweigh this disadvantage.Robert Williamshttp://www.blogger.com/profile/08525366375056584669noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-38437970010485462222015-04-27T21:19:44.643-07:002015-04-27T21:19:44.643-07:00The commentary here which compares implicit multip...The commentary here which compares implicit multiplication with function notation brings to mind another possible adjustment of notation. In particular, as Eric Schecter pointed out long ago (http://www.math.vanderbilt.edu/~schectex/commerrs/), the fact that these two things appear so similar leads to even more insidious errors.<br /><br />For instance, 2(x+y)=2*x+2*y is just fine. But sin(x+y)=sin(x)+sin(y) is not. It's no surprise that this mistake is common among math students though, on account of how the parentheses are, as you noted, overloaded.<br /><br />I would propose that the obviously solution is simply to quit enclosing function arguments with parentheses. Just stop overloading them, and the problem should go away.<br /><br />But how? We could switch to square brackets. In fact, we already use them for this in probability and statistics for some reason. For instance, with expectation: E[X+Y]. Oddly enough, though, expected value is the one commonly used named function that IS linear. E[X+Y]=E[X]+E[Y] is true. I have no idea why we've decided not to use parentheses in this case. Anyway, it sets a precedent that we should follow with all functions.<br /><br />Right now, a student could be quite nonplussed when his teacher goes bananas over his trying to do things like sin(x+y)=sin(x)+sin(y), but if it were sin[x+y], perhaps he would be more cautious, thinking, "OMG SQUARE BRACKETS I BETTER BE CAREFUL IT MAY NOT BE A LINEAR FUNCTION!"<br /><br />There is one place where his caution would still not help him, however. He would still be susceptible to the Freshman's Dream. Exponents are not currently written using function notation, even though they perform a decidedly non-linear operation. Because of this even if we committed to using square brackets for function arguments, we'd still write (x+y)².<br /><br />Come to think of it, isn't exponentiation the only operator in high school math books that is applied on the right? The only one that is right-associative? I'm starting to doubt the sanity and utility of exponent notation. But that's a completely different argument...Quintopiahttp://www.blogger.com/profile/11935053984682797775noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-78115844630824381482015-04-27T18:59:55.327-07:002015-04-27T18:59:55.327-07:00"literally pointing to the part we're loo..."literally pointing to the part we're looking for in the corresponding exponential expression"<br /><br />Yes, yes, and again yes! This is exactly what I meant. Glad to see that others are understanding why I chose this notation.<br /><br />If you feel this should be a thing, please share it with whoever you think may be interested!Bill Shillitohttp://www.blogger.com/profile/17774101901445053590noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-17144656494600205202015-04-27T13:43:00.100-07:002015-04-27T13:43:00.100-07:00am completely on board with this proposal. I never... am completely on board with this proposal. I never liked log or ln. I always have to remind myself what they mean when I see them.<br /><br />In light of Shillito's proposal, I think I understand why: <br /><br />We have this intuitive difference between procedures that are operations and procedures that are functions. Or maybe not even a merely intuitive difference? PEMDAS, after all, is a thing even if the E should really be expanded to include roots and logs.<br /><br />Yes, we could denote everything with functions. add(3,mul(6,9)) = 57. But we don't, do we? We all know how inefficient that would be, and how much it would obscure the algebraic manipulations we do all the time.<br /><br />No. We recognize EMDAS, specifically, as operations and assign them unique notational forms which help us to perform the manipulations we need to do.<br /><br />And note that these operations come in pairs. Plus has minus. Both operations, denoted with symbols that catenate their arguments. Multiplication has division, which likewise have nicely symbolic forms. Exponentiation uses an intuitive placement of arguments to convey its sense, but still, a semantically meaningful way of writing the arguments.<br /><br />Exponentiation is a little odd, though, in that its two arguments do not have equal relationships to one another. Their roles are fundamentally different. Thus, exponentiation has two inverses: taking roots, and finding logarithms.<br /><br />Why, then, do roots get an operator representation, while logs are written with what amounts to functional notation? There's not a lot of representational daylight between log_2(x) and sin(x).<br /><br />So yeah. I'm all for an operator representation for logs, especially one that so nicely mirrors the one for roots and at the same time helps people remember what each one does by literally pointing to the part we're looking for in the corresponding exponential expression.<br />Jason Blackhttp://www.blogger.com/profile/08181267035103592296noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-78788400248309947642014-11-26T03:47:09.385-08:002014-11-26T03:47:09.385-08:00El perro labrador retriever es una de las razas má...El perro labrador retriever es una de las razas más apreciadas por las personas. Es un can excesivamente noble <a href="http://www.perrolabrador.net/" rel="nofollow">cuidados perro labrador retriever</a>, trabajador y cumplidor.albina N murohttp://www.blogger.com/profile/08139646674252673476noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-42786999487009350872014-11-26T03:29:07.360-08:002014-11-26T03:29:07.360-08:00El dolor en la espalda es una de las dolor en la e...El dolor en la espalda es una de las <a href="http://eldolordeespalda.es/" rel="nofollow">dolor en la espalda</a> molestias más frecuentes que sufren las personas. Existen estrategias como el pilates que permite tratar y prevenir el dolor de espalda.albina N murohttp://www.blogger.com/profile/08139646674252673476noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-92105048965264751492014-11-08T20:11:46.066-08:002014-11-08T20:11:46.066-08:00The first program I was paid to write was a simple...The first program I was paid to write was a simple calculator, and I implemented parenthesis as two unary do-nothing operators. It worked perfectly, and didn't add any complexity to the implementation. I don't think it's unreasonable to think of parenthesis that way, but it is a bit weird that it doesn't look for matching.Shad Sterlinghttp://www.blogger.com/profile/03239616723077338092noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-86316962816040050712014-09-16T15:05:43.056-07:002014-09-16T15:05:43.056-07:00I wouldn't call it a "huge mathematical b...I wouldn't call it a "huge mathematical blunder". The contradiction reached was intentional - the point was to show that an apparent "fact" about numbers may only actually work in particular situations. In the case of \(\sqrt{a\cdot b}=\sqrt{a}\cdot\sqrt{b}\), this is only guaranteed to work as long as \(a\) and \(b\) are positive real numbers — if we try to work with negative reals or complex numbers, the equation no longer holds, as you pointed out. (Do keep in mind, though, that when working with positive reals, the symbol \(\sqrt{x}\) only refers to the positive root of \(x\)!)Bill Shillitohttp://www.blogger.com/profile/17774101901445053590noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-20860610595245536482014-09-16T13:41:05.500-07:002014-09-16T13:41:05.500-07:00My reply has not shown up? Bill Shillito, you mad...My reply has not shown up? Bill Shillito, you made a huge mathematical blunder in your argument concerning i. Even root radical equations can have extraneous roots, the sqrt(1) is actually + or - 1 with +1 being the extraneous root. shutup_and_just_drink_beerhttp://www.blogger.com/profile/18408390216090324892noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-14544575846404342014-09-15T21:01:10.597-07:002014-09-15T21:01:10.597-07:00I love your responses to my posts. :DI love your responses to my posts. :DBill Shillitohttp://www.blogger.com/profile/17774101901445053590noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-39716626859345865142014-09-15T20:52:06.545-07:002014-09-15T20:52:06.545-07:00Those who claim that Infinity is Not A Number need...Those who claim that Infinity is Not A Number need to read their standards more carefully. According to IEEE-754, single precision (positive/negative) Infinity is the bitstring (0/1)1111111100000000000000000000000, while single precision Not A Number is any bitstring which differs from this one in any of the last 23 bits. Yes, that's right, IEEE-754 defines Not A Number in terms of what differentiates it from Infinity, thus definitively proving that Infinity cannot be Not A Number.Quintopiahttp://www.blogger.com/profile/11935053984682797775noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-61596443769175675162014-08-18T01:36:54.822-07:002014-08-18T01:36:54.822-07:00Great post, I too find this frustrating, and have ...Great post, I too find this frustrating, and have first hand experience of people teaching it wrong! I wrote this recently in a similar vein: <br />https://cavmaths.wordpress.com/2014/06/29/aaargh-ruddy-bidmas/Stephen Cavadinohttp://www.blogger.com/profile/08497166692282461180noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-46010906711303065452014-08-05T06:11:44.964-07:002014-08-05T06:11:44.964-07:00I have a (small) collection of situations where mn...I have a (small) collection of situations where mnemonics are appropriate. PEMDAS is not a part of that collection. Where would you use mnemonics, and how?MariaDhttp://www.blogger.com/profile/00769513929584082597noreply@blogger.comtag:blogger.com,1999:blog-3748625510348961342.post-13092404979573390832014-07-31T23:57:00.496-07:002014-07-31T23:57:00.496-07:00Interesting article. However, I suspect that I tea...Interesting article. However, I suspect that I teach younger pupils than you do and I would add that the crossover period from single or serial arithmetic tasks into multiple operators is fraught with problems. Having a simple hook that students can hang onto is a useful tool; it gives them a flag that they can wave at relevant problems, and each other. I would heartily agree that it should not be taught as a rigid system or a blind process. Understanding how something works and the limitations of a method are important skills and I would always teach them too.Singing Hedgehoghttp://www.blogger.com/profile/04594317170233264009noreply@blogger.com