tag:blogger.com,1999:blog-3748625510348961342.post5648553532173813458..comments2020-09-15T07:44:25.397-07:00Comments on solidangl.es: The Implications of Being ImplicitBill Shillitohttp://www.blogger.com/profile/17774101901445053590noreply@blogger.comBlogger2125tag: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!Lonewolfâ„˘https://www.blogger.com/profile/17592367417704960634noreply@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)Â˛.<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...Quintopiahttps://www.blogger.com/profile/11935053984682797775noreply@blogger.com