Theoretical/Rigorous precalculus textbook recommendation(s)
gaya....@gmail.com
I've been out of highschool for a couple of years and want to review
highschool mathematics (at more than an applied/computational level).
I've looked at some precalculus textbooks (i.e., James stewart), but
they are not very theoretical and are very applied, can anyone
recommend very theoretical precalculus textbooks (that cover algebra,
trig, analytic geom, functions, etc.). Preferrably with emphasis on
"why" every step is done, and with the idea of stuff like proofs and
implications (which I really don't know muchabout, but I know it has
something to do with notation like A=>B).
thanks
Isura
This seems like exactly what you are looking for:
http://www.artofproblemsolving.com/Books/AoPS_B_About.php
Good luck!
Shmuel (Seymour J.) Metz
11/29/2005
at 08:25 PM, gaya....@gmail.com said:
>I've looked at some precalculus textbooks (i.e., James stewart), but
>they are not very theoretical and are very applied, can anyone
>recommend very theoretical precalculus textbooks (that cover algebra,
>trig, analytic geom, functions, etc.).
Try to find a Geometry textbook dating from the 1950's or earlier.
As for Algebra, etc., I don't know of anything targeted to a HS
audience. There are undergraduate texts that I could recommend once
you are comfortable with the concept of formal proofs, but it doesn't
sound like you're ready for them yet.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
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gaya....@gmail.com
Shmuel (Seymour J.) Metz wrote:
> In <1133324712.6...@g43g2000cwa.googlegroups.com>, on
> 11/29/2005
> at 08:25 PM, gaya....@gmail.com said:
>
> >I've looked at some precalculus textbooks (i.e., James stewart), but
> >they are not very theoretical and are very applied, can anyone
> >recommend very theoretical precalculus textbooks (that cover algebra,
> >trig, analytic geom, functions, etc.).
>
> Try to find a Geometry textbook dating from the 1950's or earlier.
>
> As for Algebra, etc., I don't know of anything targeted to a HS
> audience. There are undergraduate texts that I could recommend once
> you are comfortable with the concept of formal proofs, but it doesn't
> sound like you're ready for them yet.
>
Hi. I actually want to review HS math, but not at the same level as it
was taught to me, but at a more formal definition, precise level. I'm
more mature now so I was looking for a precalculus (HS review) textbook
that was more precise, had proofs, and was rigorous. I remember HS
math being very computational with ZERO proofs of anything (even the
books we used).
Since I lost all this "experience" with HS math, I think it might be a
good thing because now I can find a book that didn't present things
just to give us experience, but presented things because they were
logical and had proofs.
Most of my younger friends (who still remember HS math, because they
just graduated) know most of the math because of repetition or
"experience" not because they understood WHY. I am just trying to
relearn HS math and learn it for the WHY not just the experience.
Sorry, I don't know how to explain my thoughts. I think I probably
should use a different word instead of "experience" (which I can not
think of). Sorry my english/explanations are not the best. It's kind
of like being taught limits and methods to solve limits, without
knowing the precise definition (which I was shown by my brother to
involve absolute values of distances). Same with highschool geometry -
being shown similar triangles and stuff without knowing the actual
axioms and stuff.
So if anyone knows any precalculus (HS review) textbooks that are
proof/axiom based, please let me know.
Thanks
Stephen J. Herschkorn
If you can find it,
Lehmann, Charles H. College Algebra. New York: John Wiley & Sons, 1962
might be the kind of thing you are looking for for algebra. It
rigorously proves things like why
(-a) (-b) = - a b and the remainder and factor theorems. It
acknowlegdes when the proofs of some theorems are too advanced ofor its
audience, but then you are welcome to study more advanced courses later.
The book is a little old-fashioned in its content. For example, I would
not worry about the last chapter on interest and annuities. Some of its
topics - e.g., variation and much of the theory of equations it covers -
have fallen out of fashion . Others, such as partial fractions and
probability, are usually postponed until later courses. Still, if you
can go through this entire book, you will know and understand much more
"precalculus" than at least 99% of the students who take calculus.
I happen to have this book because my older brother had it. I suspect
there are others from the same era (i.e., the early 60's) which are
similar to this.
Steen F.H., Ballou D.H. Analytic Geometry, 3rd ed. Lexington, Mass.:
Xerox College Publishing, 1974
is also a wonderful little book devoted entirely to the topic of its
title. The first ten chapters (three whereof are devoted to conic
sections) provide practically everything you will need to know, and
more, before you study calculus. I admit it is overkill for most
students, but it may what you seek.
--
Stephen J. Herschkorn sjher...@netscape.net
Math Tutor on the Internet and in Central New Jersey and Manhattan
Shmuel (Seymour J.) Metz
11/30/2005
at 12:50 PM, gaya....@gmail.com said:
>Hi. I actually want to review HS math, but not at the same level as
>it was taught to me, but at a more formal definition, precise level.
The problem is that the public high schools are not interested in
teaching real Mathematics and are not buying such books, so nobody is
writing them. They used to teach Geometry in a semi-rigorous fashion,
which is why I suggested finding an old Geometry book. Once you are
comfortable with proofs, you might try a good undergraduate text in
Real Analysis, e.g., Mathematical Analysis by Apostol.
My experience is limited to the USA; it may be that HS texts published
elsewhere are more rigorous.
gaya....@gmail.com
>
> As for Algebra, etc., I don't know of anything targeted to a HS
> audience. There are undergraduate texts that I could recommend once
> you are comfortable with the concept of formal proofs, but it doesn't
> sound like you're ready for them yet.
>
Thanks. I live near a large univ library, so I could use those
recommendations.
Shmuel (Seymour J.) Metz
12/01/2005
at 01:48 PM, gaya....@gmail.com said:
>Thanks. I live near a large univ library, so I could use those
>recommendations.
These are old, but good
Halmos, "Finite Dimensional Vector Spaces"
Apostol, "Mathematical Analysis"
If your precalculus curriculum includes set theory, try
Halmos, Naive Set Theory
Does anybody have a suggestion for an accessible text on synthetic
Geometry?
Dave L. Renfro
Something similar recently came up in an ap-calculus list
thread that I made a post in this morning (but which might
not show up until Monday because the list is moderated), at
http://mathforum.org/kb/thread.jspa?threadID=1308735
This got me looking though some books I have,
and I found a couple that are worth mentioning.
John M. H. Olmsted, "Prelude to Calculus and Linear Algebra",
New York Appleton-Century-Crofts, 1968, xx + 332 pages.
John L. Kelley, "Algebra: A Modern Introduction",
D. Van Nostrand Company, 1965, viii + 335 pages.
Both of these texts do a great job of covering precalculus
mathematics at a level that's quite a bit above anything
you'll find in the usual texts. Kelley dives into a bit more
linear algebra than is probably appropriate for an "honors"
precalculus level text, so I'd recommend Olmsted (which
also has a broader coverage of topics than Kelley) for
the original poster in this thread.
Dave L. Renfro
Dave L. Renfro
>> I've been out of highschool for a couple of years and
>> want to review highschool mathematics (at more than an
>> applied/computational level).
>>
>> I've looked at some precalculus textbooks (i.e., James
>> stewart), but they are not very theoretical and are very
>> applied, can anyone recommend very theoretical precalculus
>> textbooks (that cover algebra, trig, analytic geom,
>> functions, etc.). Preferrably with emphasis on "why"
>> every step is done, and with the idea of stuff like proofs
>> and implications (which I really don't know muchabout,
>> but I know it has something to do with notation like A=>B).
Dave L. Renfro wrote:
> Something similar recently came up in an ap-calculus list
> thread that I made a post in this morning (but which might
> not show up until Monday because the list is moderated), at
>
> http://mathforum.org/kb/thread.jspa?threadID=1308735
>
> This got me looking though some books I have,
> and I found a couple that are worth mentioning.
>
> John M. H. Olmsted, "Prelude to Calculus and Linear Algebra",
> New York Appleton-Century-Crofts, 1968, xx + 332 pages.
>
> John L. Kelley, "Algebra: A Modern Introduction",
> D. Van Nostrand Company, 1965, viii + 335 pages.
>
> Both of these texts do a great job of covering precalculus
> mathematics at a level that's quite a bit above anything
> you'll find in the usual texts. Kelley dives into a bit more
> linear algebra than is probably appropriate for an "honors"
> precalculus level text, so I'd recommend Olmsted (which
> also has a broader coverage of topics than Kelley) for
> the original poster in this thread.
I just got a couple of books yesterday (in the mail from
amazon.com) that might be almost perfect for the original
poster:
Israel M. Gelfand and Alexander Shen, "Algebra", Birkhauser,
1993, viii + 149 pages.
Israel M. Gelfand and Alexander Shen, "Trigonometry",
Birkhauser, 1993, viii + 229 pages.
After looking through these books, I'm surprised that
no one in sci.math was sufficiently aware of them to
recommend them (including myself). Are these books *that*
little known, or do others not think as highly of them
as I do? There's a 4 page review of "Algebra" by Richard
Askey in The American Mathematical Monthly (102, 1995,
78-81), by the way.
Dave L. Renfro
Dave L. Renfro
> Israel M. Gelfand and Alexander Shen, "Trigonometry",
> Birkhauser, 1993, viii + 229 pages.
Oops, this book was (first) published in 1999, not 1993.
Dave L. Renfro
gaya....@gmail.com
Dave L. Renfro wrote:
> gaya....@gmail.com wrote:
>
> >> I've been out of highschool for a couple of years and
> >> want to review highschool mathematics (at more than an
> >> applied/computational level).
> >>
They send you book to have you review them?
Are these books at the level that used to be taught pre 1950s ?
It's interesting taht the trig book has far more pages than the algebra
book.
I'll check the library, thanks.
>
> Dave L. Renfro
gaya....@gmail.com
Shmuel (Seymour J.) Metz wrote:
> 12/01/2005
> at 01:48 PM, gaya....@gmail.com said:
>
> >Thanks. I live near a large univ library, so I could use those
> >recommendations.
>
> These are old, but good
>
> Halmos, "Finite Dimensional Vector Spaces"
>
> Apostol, "Mathematical Analysis"
>
> If your precalculus curriculum includes set theory, try
>
> Halmos, Naive Set Theory
>
> Does anybody have a suggestion for an accessible text on synthetic
> Geometry?
>
Thanks!
Dave L. Renfro
>> I just got a couple of books yesterday (in the mail from
>> amazon.com) that might be almost perfect for the original
>> poster:
>>
>> Israel M. Gelfand and Alexander Shen, "Algebra", Birkhauser,
>> 1993, viii + 149 pages.
>>
>> Israel M. Gelfand and Alexander Shen, "Trigonometry",
>> Birkhauser, 1999, viii + 229 pages.
gaya....@gmail.com wrote:
> They send you book to have you review them?
No, I ordered and paid for them on-line about a week ago.
Dave L. Renfro wrote:
>> After looking through these books, I'm surprised that
>> no one in sci.math was sufficiently aware of them to
>> recommend them (including myself). Are these books *that*
>> little known, or do others not think as highly of them
>> as I do? There's a 4 page review of "Algebra" by Richard
>> Askey in The American Mathematical Monthly (102, 1995,
>> 78-81), by the way.
gaya....@gmail.com wrote:
> Are these books at the level that used to be taught
> pre 1950s ? It's interesting taht the trig book has
> far more pages than the algebra book.
> I'll check the library, thanks.
They might not be at your university library, but it
occurs to me now that they might be at a reasonably
good public library. They aren't at the University
of Iowa library, which is why I ordered them, but I
would have ordered them anyway after seeing them.
I have them with me now and looked at them in
light of your comments, and all I can say is that
there are plenty of things in the algebra text that
you rarely see discussed in current books: why equality
of polynomials implies equality of coefficients, how
to factor a^4 + b^4 as two quadratics with real number
coefficients, harmonic means, is it possible to have
an infinite arithmetic progression with exactly two
(not necessarily consecutive) integer terms (no),
certain symmetrical expressions involving roots
of monic polynomials being equal to the coefficients,
the "rationalized numerator" version of the quadratic
formula and its use, symmetric equations (i.e. polynomials
of x + 1/x), max/min matters (quadratic and other),
and the arithmetic and geometric mean inequality.
It's not really their content as much as their style.
Their level is not particularly advanced, and in some
ways they're more elementary than a lot of past or present
precalculus texts. However, they read like they were
written by someone who knows what mathematics is really
about (duh, Gelfand) and who wants to convey this to the
reader, so their focus is on "why", not "how", although
it's surprising (except maybe to Herman Rubin) how much
of the "how" gets taken care of with the "why" in these
books.
Dave L. Renfro