I hesitated to post this, since I think clearer, shorter, better written textbooks are an admirable goal, and I keep a (short) list of good text books I have found over the years. But based on the preview on the website, I'm going to keep referring those who ask me to Gelfand's wonderful basic math texts instead (Algebra, Method of Coordinates, Functions and Their Graphs, Trigonometry -- anyone who wants further pointers to these, private message me).
There are also some howlers in the preview text, such as "after thinking very hard the mathematicians were able to classify all the different number like objects into sets" and then lists the naturals, integers, rationals, reals, and complex numbers. Except that these are nested subsets of each other, not disjoint, and the four normed division algebras are the reals, complexes numbers, quaternions, and octionions, so if you're going to talk about all the number like objects without including those last two, you're off the mark.
I haven't seen the physics sections, but I will say that teaching physics is actually remarkably difficult. I tend to recommend 1960s editions of Halliday and Resnick (not the recent ones!), though I will probably switch to recommending Karl Wiemann's work (http://c21.phas.ubc.ca/).
I absolutely agree that mechanics and the differential and integral calculus should be taught together, though. They don't make any sense without each other.
"after thinking very hard the mathematicians were able to classify all the different number like objects into sets"
Granted I am only going from the quote but I see nothing in there that indicates he considers them disjoint. Given most people have seen the Euler diagram of numbers since grade school i think the default interpretation would be correct. Could be clarified but not a howler.
quaternions, and octionions,
Considering the intended audience I do not see what purpose that would serve beyond intimidating the audience with how clever he is. Counter productive. This is just intro physics - the places where quaternions might be useful are just not there. And in the cases where introducing the terms actually served a purpose one would be better served by just going with the neater geometric algebra framework.
There are some helpful suggestions in some posts but it would be nice if people swapped their easily flabbergasted expert tones for a more helpful one. He has a vision and has begun acting on it. Has a good start with lots of potential. No one gets it just right from the outset.
> Given most people have seen the Euler diagram of numbers since grade school i think the default interpretation would be correct.
This kind of assumption is one reason so many textbooks suck so much. This assumption is not true for me. I don't know why you'd assume it would be true for "most people."
Indeed, stating that these objects result from some classification of all number-like objects seems misleading at best: Most mathematicians would, just as one example, certainly consider p-adic numbers as number like objects.
It would be better to point out what really matters in the hierarchy N, Z, Q, R, C:
Everybody will accept that N is an interesting object for counting.
Z is the quotient group of N under addition, more elementary: In Z you can compute arbitrary differences of elements of N, and it is the smallest "reasonable" such object.
Similarly, Q is the quotient field of Z, i.e. "the smallest reasonable object containing Z that allows division of non-zero elements".
R is the completion of Q under a natural metric (i.e., "fills the gap on the number line of Q"), and therefore allows one to have a reasonable notion of limits and calculus.
C finally is the algebraic closure of R, so that every non-constant polynomial has a root in C.
Thus, these objects are all constructed from the natural numbers to satisfy certain "niceness" properties.
Of course, it would take quite some space to explain this in a very elementary fashion. So perhaps it is instead better to simply refrain from such a statement in a textbook aiming for shortness above all, or at least make sure that any statement in this direction is substantiated and supported by a suitable reference to a place where your readers can learn more.
A textbook that gains conciseness from vague statements and half-truths would seem to be much worse than a textbook of twice the length that explains its content well.
My choice was to cover N,Z,Q operationally -- as in what you can do with them. Everyone knows about +,-,/ and so I think it makes sense to connect with this previous knowledge of the reader rather than get into the formality of sets and set containment. I stand by my choice.
> normed division algebras are the reals,
> complexes numbers, quaternions, and octionions,
Can you see why I would not want to talk about quaternions and octionions in a chapter which is meant to introduce math to people who have math phobia?
> these are nested subsets of each other, not disjoint,
>
Everybody can see why you wouldn't want to do that, but there's no point in lying or, if you prefer, selectively informing. Complex numbers and quaternions will be equally mysterious to the true novice; there's nothing intrinsically scary about the words.
> I tend to recommend 1960s editions of Halliday and Resnick (not the recent ones!)
I'm curious why. I remember reading the second edition back in 1998. Recently, I got one of the new editions (8th) but the new ones seem too verbose. What went wrong?
In textbook publishing, it's very important to put out a new edition every year or two. Otherwise your sales will disappear, since everyone will buy the book used. For a book like Halliday and Resnick that's been around since the 1960s, it doesn't take a very large number of people selling theirs to satisfy the demand of all current physics students.
So you put out a new edition in which you shuffle all the exercises so that students can't do their homework, and you have someone mess with the text and the formatting to make it look like a real change. You add glossy pictures, because you get a much bigger visual impact from changing the pictures than from actually changing content.
For an old book, this is a problem because often the authors are dead or retired, or think the book is just fine. Now you have to find someone who would like his name added to a classic text who will sign off on the job. Today it's Halliday, Resnick, and Crane. For Arfken's old mathematical methods for physics text, it was Weber.
Thanks for pointing to Gelfand. I'd never heard of him. Intriguing that University of Chicago assigns these texts. (Heads up— there's no contact info in your profile).
There are also some howlers in the preview text, such as "after thinking very hard the mathematicians were able to classify all the different number like objects into sets" and then lists the naturals, integers, rationals, reals, and complex numbers. Except that these are nested subsets of each other, not disjoint, and the four normed division algebras are the reals, complexes numbers, quaternions, and octionions, so if you're going to talk about all the number like objects without including those last two, you're off the mark.
I haven't seen the physics sections, but I will say that teaching physics is actually remarkably difficult. I tend to recommend 1960s editions of Halliday and Resnick (not the recent ones!), though I will probably switch to recommending Karl Wiemann's work (http://c21.phas.ubc.ca/).
I absolutely agree that mechanics and the differential and integral calculus should be taught together, though. They don't make any sense without each other.