I think I’ll resurrect this thread into the brave new world of the new forum design. There are just too many good links out there. Here’s Feynman’s textbook reviewing experience: http://www.textbookleague.org/103feyn.htm
I have nothing to do with TTL (although what they say is probably true)-that’s just the first link I could find. Speaking of Feynman there’s also this, also from Surely You’re Joking, Mr Feynman!, but it’s just an excerpt: http://www.ee.ryerson.ca:8080/~elf/abacus/feynman.html
And speaking of calculating machines…http://downlode.org/Etext/power.html
I thought of the first link because I was recently thinking about bad, sloppy textbooks. Particularly (and this is probably something most people have never really thought of) how it’s pretty stupid to define i=sqrt(-1) instead of i being one number, along with -i, such that i^2=-1. There probably seems to be no difference until you consider why sqrt(25)=5 and not -5 (the convention of always choosing the non-negative root to avoid multivalued functions is common in the US although maybe different elsewhere). Well, by convention sqrt(25)=5 because 5 is positive and -5 is negative. So what else can a student possibly think when he sees sqrt(-1)=i instead of -i? Obviously he’ll think it’s because i is positive and -i is negative…if he thinks at all, which excludes me when I first learned it.
Not only does this not make sense because the complex numbers can’t be (totally) ordered, it reinforces the common existing misconception that -x is a negative number because of the - sign. It also leads to having to tell students that sqrt(1*1)=sqrt(1)sqrt(1) and sqrt(-11)=sqrt(-1)sqrt(1), but goddammit sqrt(-1-1) does not equal sqrt(-1)*sqrt(-1)!!! I think the tiny inconvenience of having a multivalued function makes up for all this confusion, and multivalued functions are introduced later with complex numbers anyway (http://en.wikipedia.org/wiki/Imaginary_unit#i_raised_to_the_power_of_i ).
Don’t even get me started on how almost all textbooks strongly imply that i was just invented to expand the classes of equations we can solve instead of just stumbled upon when looking for the cubic formula. Inventing i to solve x^2=-1 would be like inventing blayp to solve cow-x=7 as far as 16th century mathematicians were concerned.
Rant over lol. Enjoy the links.
edit: Oh! I forgot to mention a good book. Linear Algebra by Shilov. Real cheap, I mentioned it on the first page too. It’s for a SECOND course, and yes I’m yelling to strongly discourage anyone from using this as a first text. You will hate it. I love it, although there are a lot of typos that sometimes make the math wrong, which sucks but they possibly come from the translator and can be spotted and fixed if you’re careful, which you should be. Would probably be good for physicists and engineers too, since Shilov doesn’t try to be overly slick to show off what a hotshot he is at the possible expense of student understanding (I’ve seen Rudin accused of this by another mathematician lol)