Other Should I Become a Mathematician?

[dkotschessaa]

I got a good question...

a. Why did Stewart's calculus textbook take off so successfully? and is anyone out there bold enough to toss some detailed minuses, and detailed pluses to the text?
[and like Thomas and Finney the earlier books were better...]

I'd like to know the answer to that too, though I think I can contribute somewhat to the discussion. I think the strongest point of Stewart's book may also be what makes it so disliked by many students - namely, that is extremely concise.

Most students complain that it is "hard to read" and that "it doesn't have enough examples." I think the philosophy of the text is to keep students away from the "plug and chug" method of heading straight for the homework problems, looking for examples that are similar, and re-arranging the necessary formulas.

The explanations are actually very good but have to be read very carefully and "unpacked." Sometimes there are VERY IMPORTANT details that are relegated to a small, fine print marginal sentences. When I took notes out of this book I would often re-write what was contained in a single paragraph to something (for my understanding) that would fill a whole page of notes.

I looked at earlier editions of the book and it seems to have gotten thinner and thinner as the new editions came out. Stewart is putting supplements online, but most of the students I studied with weren't even aware of this (even though it is advertised in the book.)

-Dave K

 

[A. Bahat]

That problem is much more fun in reverse! (Finding a rational parametrization of the unit circle, that is.) Give this one a try:
parametrize the curve y²=x³+ax+b

 

[RJinkies]

a. Why did Stewart's calculus textbook take off so successfully?

dkotschessaa - I'd like to know the answer to that too, though I think I can contribute somewhat to the discussion. I think the strongest point of Stewart's book may also be what makes it so disliked by many students - namely, that is extremely concise.

I hear people like and dislike Larson/Edwards aka Larson/Hosteller/Edwards] which goes from being a junk book to crystal-clear at times depending on who's opinion and what edition.

On LE/LHR:
[like a typical intro calc book - it's not rigorous enough, has too much brute force, too many applications, not enough mathematics, not enough creativity.]
[I have many of the same criticisms of this book as I do of the Stewart, although I do think this book does a slightly better job in the very beginning]
[This book does provide the concepts and theory critical to an understanding of calculus. Unfortunately, it is in a wordy, technical, abstract, and thoroughly annoying format...This book gives you plenty of abstract proofs that look like bull@!#t, but falls far short of my engineering book in encouraging an understanding of calculus. The truth is, this book gives you hundreds of formulas to memorize, instead of a relative few like my engineering book that can cover every problem.]

LHR/LE went through 10 editions, and suffered a *lot* in the 4ed from 1993 with the horrible idea of using computers and graphic calculators and other stuff. Thomas-Finney at least in the 80s just plopped in all the freaky 3D graphs and didnt need you to play with software or odd CAI stuff] But i wonder if the later editions got better and dropped those fads... and the ratings went up. It's interesting since people think the minuses of stewart's book apply here too.

-----
I think the secret to Stewart getting cult status by some is due to his influence of Polya with trying to show students how to actually solve the problems. [Something i forgot was hidden in my notes lol]

Here are some of the gems people said about Stewart [and other texts]

a. [i recommend the Second edition of stewart. it went downhill after that. - Mathwonk]

b. [A few people I know have trudged all the way through Apostle's tome, and found they had to skip over entire sections reverting to stewarts book to tell them what the hell is going on intuitively.]

c. [Presentation of Applications Confuse Students - 2 out of 5]
[This book was used at my undergraduate and graduate institutions; I am currently forced to teach out of it. I don't understand why it's considered such a great book. I have seen many students confused by it, and I find it mediocre as a reference text.]
[It is my belief that calculus should be presented in a simple and pure way so that students can master the fundamentals, and then (simple) applications be presented later. Instead, this book introduces fairly complex and "ugly" applications right from the start.]
[The net effect is that students using this book often fail to master the fundamentals of the subject, and find calculus overwhelming and confusing.]
[The book's covering of advanced topics is better than the earlier chapters, but there are far better calculus books out there, and I would not under any circumstances recommend this one.]

d. [Stewart does not sugarcoat or resort to gimmicks or superficiality in order to make the material learnable]

e. [I am teaching honors calc this fall and cannot find a good book. I do not mean Spivak or Apostol, those are too hard for my "honors" course. There only seem to be really weak books for non honors, or really hard books for super honors courses, Any good plain old intermediate honors books out there? I don't want to be difficult but I also dislike heavy books, and space wasted on technology, or bundled CD's. I want clear explanations, some rigor, and a logical sequence of ideas, intelligently written. I have considered the old Courant, but it looks a bit unattractive on the page for todays kids. I once liked Stewart, and Thomas Finney, but subsequent editions have been dumbed down. - Mathwonk]

f. [stewart is a joke compared to spivak. i.e. stewart (2nd edition) is a good non honors book. spivak is an excellent super honors book,(not just regular honors). - Mathwonk]

-------Two promising books, which few know about, but i like a fair deal, is something, I'd like others to chime in about...

Leithhold [circa 1968]
and
a. 43 The Calculus 7 aka TC7 (Hardcover) - Louis Leithold - Harpercollins 1995 - 1216 pages
[extremely approachable text - well liked for Third World Engineering types]
[dates before 1968]b. 44 Calculus and Analytic Geometry - Second Edition - Sherman K. Stein - McGraw Hill 1973
[aka Calculus in the First Three Dimensions - 1967 First Edition]
[Sherman Stein - PhD Columbia 1953]
[Taught at the University of California, Davis - retired 1993]
[a gem to have]
[brilliant method]
[This book is literally the best basic calculus text you can possibly get.]
[Reading this book really gave me an true understanding of basic calculus.]
[Stein offers several suggestions on how to solve certain problems. It is a shame - this book does not attract the amount of attention it deserves.]
[I did get stuck a couple times]
[when I need to refresh some calculus and geometry techniques, this book is really the best to sharpen my intuition and understanding of calculus.]
[I wish all math books were like this]
[If math is not your strongest skill and you need to learn some higher calculus this book will be your excellent companion helping you to gain the insight and intuition you need.]
[may disappoint the reader who is looking for rigor]
[perfect book to gain insight in calculus]
[unique calculus book with a physical bent - tech book guy - los angeles]
[This book starts out with integration and presents the main ideas in a very concrete fashion. Although many books would have the student think otherwise - the techniques of calculus were developed to solve concrete physical problems and model natural phenomena. This book does a good job of helping the student realize this.]
[Another thing I like about this book is that it actually assumes the reader knows pre-calculus mathematics rather than trying to review everything. The inclusion of pre-requisite material is what usually drives calculus books into phone book size.]

[First Edition] - 1967 [or 1968?] [could be called Calculus in the First Three Dimensions - 613 pages]
[Second Edition] - 1973 - the one i like
[Third Edition] - 1982
[Fourth Edition] - 1987
[Fifth Edition] - 1992

------
Stein was the first calculus book in my house! It was tossed at me about 1974-1975 when i was in elementary school. Some stuff was easy but in places, i did feel stonewalled. But it was probably due to youth and not enough algebra, or just finding it at the time, clear clear clear uh oh impossible [hide the book for a whole year]

Other books I'd like an opinion on was

d. Calculus - Harley Flanders
[first edition - 70s/80s?]
[second edition 1985 WH Freeman]
All i know about him is he got a BA in 1946 at Chicago
50s - Faculty Berkeley
50s fellowship at Caltech
60s - Purdue
Tel Aviv University 1970-1977
Ann Arbor Michigan 1985-1997

people like his first textbook Differential Forms from 1964 and Dover has it out now, but it's not an elementary textbook.

[I did a mistake a few days ago thinking it was Flanders, but the book was Edwards]

e. Advanced Calculus: A Differential Forms Approach - Harold M. Edwards - 1969/now Birkhauser 1994
[I'm just going to come out and say it: this book is the best treatment of multivariable calculus that I've seen. Unlike the usual multivariable textbook, this book gives lucid, clear, and elegant explanations and proofs for nearly all principles introduced, i.e. the method of Lagrange Multipliers. The author never keeps you guessing; he starts low and builds up quickly and brilliantly.]
[An inviting, unusual, high-level introduction to vector calculus, based solidly on differential forms. Superb exposition: informal but sophisticated, down-to-earth but general, geometrically and physically intuitive but mathematically rigorous, entertaining but serious. Remarkably diverse applications, physical and mathematical.]
[In fact this book looks decidedly 19th century in places. This is the opposite to a book by Lang, Dieudonne or Rudin. To be fair the author has gone to great lengths to motivate the mathematics and for this reason it may well be very popular with engineers and physicists.]

I thought it was Flanders was the Differential Forms text that people wet their pants about, but it was Edwards...

----

The main thing is that i thought Flanders for the 80s had a book that though weird in places [he obsessed about getting students to draw crappy diagrams a lot] it seemed like a book that almost pushed Thomas and Finney out of place. [then again WH Freeman used to be like Addison-Wesley, almost anything they printed was awesome]
Seemed like a solid unusual book...

--------

another one the MAA liked, but it's probably more for the later books than the initial books was

f. Jerrold E. Marsden and Alan Weinstein - Calculus I, II, III - New York, NY: Springer-Verlag, 1985. Second Edition - three book set

I think Marsden might hath popped the book out in the late late 70s. and for a while it was used when they both taught at Berkeley.

------

g. 72 Allendoerfer, C.B. and Oakley, C.O. Principles of Mathematics - McGraw-Hill 1963
[probably Second Edition 1963]
Mathwonk used it and liked it...but there's a lot of proofs...

----
h. Thomas and Finney...

[The prose is clear and tight. The figures are fantastic. Great examples. Great discussion of the mean value theorem. The discussion of limits is rigorous but not overly so. The 4th edition went overboard on rigor as that was vogue in the early 1970's. Subsequent editions became heavy phone book size calculus texts. Of all the editions of this text this is the one to get. Although there are some other good older calc texts out there this is the cream of the crop.]

[First Edition] 1951
[Second Edition] 1956 - 731 pages [maybe 1953] - Parke III recommendation
[Third Edition] 1961
[Fourth Edition] 1968 - [this went overboard on New Math Rigor]
[Third Alternative Edition] 1972 - 1025 pages
[i can't figure out the alt editions, anyone know??]
[Fifth Edition] 1979
[Sixth Edition] 1984 - a bit of a let down
[Seventh Edition] 1988 - a nice edition

i hear the 8th and 9th are okay - 1991 and 1995
but the 10th-12 editions from 2001 to 2010 arent as good now

seemed like an okay book of the 50s and 60s
and it was up and down in places in the 70s 80s...

but if anyone wants to add any forgotten books 1955-1980 that Thomas-Finney or Apostol or Apostol.. didnt steamroller into obscurity, do tell.

Leithhold, Stein and Flanders were i think three that stood out.. and seem more fun to browse than Stewart. But i think my guess is that Polya's influence is what got Stewart his 10 million house in Toronto lol...Some say

i. Calculus - RT Smith and RB Minton
is better than Stewart...

but i think there were problems with the first edition with proof reading and the binding, and McGraw-Hill offered replacement texts. Nothing like mistakes or falling apart books to ruin a promising beginning...
Not sure when the first edition came out or if it's useable
but the second edition was 2002 McGraw-Hill...

and the third edition is 2007 with like 7000 problems and 1000 examples...

[sounds like a schuams outline with handholding, how can you go wrong!]

anyhoo, some say Smith and Minton beats Stewart, Larson and Anton, so it's worth a look...

------

dk - students complain that it is "hard to read" and that "it doesn't have enough examples." I think the philosophy of the text is to keep students away from the "plug and chug" method of heading straight for the homework problems, looking for examples that are similar, and re-arranging the necessary formulas.

dk - The explanations are actually very good but have to be read very carefully and "unpacked." Sometimes there are VERY IMPORTANT details that are relegated to a small, fine print marginal sentences. When I took notes out of this book I would often re-write what was contained in a single paragraph to something (for my understanding) that would fill a whole page of notes.

dk - I looked at earlier editions of the book and it seems to have gotten thinner and thinner as the new editions came out. Stewart is putting supplements online, but most of the students I studied with weren't even aware of this (even though it is advertised in the book.)

A lot of the older textbooks were short on examples till the 1970s. Hard to read can sometimes show a huge flaw, or sophisication, so it's a hard one to judge. Kittel's solid state physics texts are infamous for being hard to read, but if you're extremely slow and careful you see the method to his madness...

I think *any* online or CD supplements are a long term death strategy for authors... the worst book with that was
the Quantum Physics book by Gasiorowicz...

[probably the bext textbook combo for QM was
[Cohen-Tannoudji/Gasiorowicz/Griffiths/Liboff/Merzbacher/Sakurai/ Ohanian/Shankar/Feynman/Bransden/Dicke/Schiff]
if you had the right *edition* of Gasio...

Gasio in the First [1974] and Second Edition [1995] was 500 pages
and then the Third Edition was 350 pages
with just 30% of the book taken out, and then plopped online
which i think is almost a criminal thing to do, on top of a ton of mistakes with crappy proofreading...

It was a no nonsense book, but sadly one where like a lot of calculus texts, it's awesome after you took a class but as a text to learn from, it usually seems like knocking your head against a wall. Gasio's book was liked because it was in many ways a replacement for Schiff's 1960s text...

[I truly truly hated this book when I was using it as an undergraduate. It’s thin, explains things with extreme economy of words, and the problems are quite difficult in comparison to the depth of explanation in the chapters. It assumes a very decent mathematical background in linear algebra/Hilbert spaces. That said, now that I understand the material it’s a great reference. I think I would have really liked this book if my math background had been stronger, and it’s still a good source of brushing-up on a few basic topics while taking grad school QM.]

[I have pretty strong math skills and most of the time I have no clue how or why he does things. The text is written very math and equation oriented. There is little to no explanation as to why or what the author is trying to show, he just runs through the equations, section by section. He overly uses terms like "We know that" or "Its clear that" as a means of explanation, and the reader is stuck wondering why something is done.]

[He lays out the concept, manipulates the equation in a few brief steps, and leading to the final equation. Entire sections can be covered in a few sentences.]

[In defense of the text, there is a focus on the physically interesting material, while extraneous mathematical stuff has been skipped. However, the text is too hard for an introduction, but skips too much material to be a comprehensive guide. Perhaps as the second or third quantum mechanics book on your shelf, this book will do, but not as the first. For the mathematically inclined look to Sakurai. For a very readable if non-standard approach see The Feynman Lectures. Or, for a lighter introduction see Griffiths.]

----

[Terrible book. Half it is put online, and it's absolutely gaunt compared to other more comprehensive texts. Completely glosses over many fundamental derivations. Avoid at all costs. - Quaoar]

[It does not teach QM conceptually, instead it just states stuff, giving no reason for why things are done as they are. Overall, this book is terrible - bad for undergrads who will learn little, and horrible for grads who won't learn principles. The sooner this book goes out of print, the better.]

-----

So the best of books can be *ruined* by online gimmicks, or cd rom supplements, or computer crap tie-ins, or graphic calculator or TI-55 button mashing/mathematica problems... I got more respect for a mathbook with APL symbols really...

If a publisher can't cram all the weird stuff into a text or needs to resort to animations, run to your nearest copy of any Sylvanius P thompson from 1914. Heck at least Thomas and Finney in 1988 could cram all the pretty computer pictures into the book without increasing the page count or taking stuff out... and there are still books that could do wonders with black and white or only occasional diagrams.

Again, Courant doesn't put stuff online, or use funny colour photographs or play around with side margins, and people still think it's pretty close to ideal and hard to top, though not a cakewalk...

after a decade all those computer gimmick textbooks, or online gimmick texts end up on the junkpile, not liked anyone cept for the xmas bonfire... Half of the worth of these books are as a reference *later* on... and if we need gimmicks, i'd prefer the 1960s 35-mm film to go with my math or physics text lol

another nightmare were those integrated first year math-first year physics textbooks, all in one... [and the mechanical uni-curse physics textbook seemed like a fad too, i think caltech was the only place that still used it in some classes, where i still think the PSSC films *said* more]

It's strange how any truly creative calculus text, dropped topics that were filler, and included unusual stuff, and sadly the textbooks that get adopted are ones that include the kitchen-sink approach [and not in a good way] out of fear that if something about Newton's method or the new-math delta-episilon stuff is left out, 80% of curriculum writers just don't adopt the textbook.

I don't know if Feynman, PSSC, the Berkeley Physics Series, or Spivak or Courant would fly today with the strangulating feel of a bland curriculum, or the Banesh Hoffman 'tyranny of testing' that pushes out the quirky texts.

 

[egyptman]

hi.i.love.this.website...>>>but.iam.weak.at.math.:(
im.from.egypt.im.at
3.prepschool.
please
i
wanna
help.:)

 

[ovael]

Mathwonk, could you tell about differences between first and second edition of Allendoerfer's
and Oakley's Principles of Mathematics?

I am currently studying more basic material, and I plan to study that book next but I just don't know
yet which edition to get. I think you recommended the first (1955) in somewhere, but the second edition seems to be almost 100 pages longer, so I was wandering if it contains some useful additions?

I don't really have money to get both editions right now, so I will probably get the first edition if there are no recommendations to do otherwise.

Thanks if you can answer, and also to anyone else who might know. Mathwonk has just been recommending that book a lot.

 

[RJinkies]

Allendeorfer and Oakley - Principles of Mathematics

There's a Third Edition 1969 McGraw-Hill

and some of the books might be with title changes so the diffences could be minor or substantial...
in the day it was likely the same book tweaked for college students...Allendoerfer & Oakley - Principles of Mathematics - McGraw-Hill 1ed 1955 - 540? pages
Allendoerfer & Oakley - Principles of Mathematics - McGraw-Hill 3ed 1969 - 705? pages
Allendoerfer & Oakley - Fundamentals of Freshman Mathematics. McGraw-Hill 1959
Allendoerfer & Oakley - Fundamentals of College Algebra. McGraw-Hill 1967

Allendoerfer - Calculus of Several Variables and Differentiable Manifolds - Macmillan 1974

[he died in 1974]

and i think he was pretty much a fixture at the Uni of Washington, in Seattle from 1951 onwards, being one of the many big cheeses with the New Math...

------'Noting the trend to abstraction in New Math, Morris Kline says "abstraction is not the first stage but the last stage in a mathematical development." ...'

----------

There's been some famous algebra books:
Chrystal i got in the chelsea edition...Peacock - A Treatise on Algebra 1842
Hall and Knight - Elementary Algebra 2ed 1896 - 516 pages
Hall and Knight - Higher Algebra 3ed 1889 MacMillian - 557 pages
George Chrystal - Textbook of Algebra - A&B Black, London 1900/Dover/Chelsea - 1235 pages
Fine - College Algebra - Ginn 1904 - 595 pages
Knebelman and Thomas - Principles of College Algebra - Prentice-Hall 1942 - 380 pages
Ferrar - Higher Algebra - Oxford 1945 - 222 pages
Ferrar - Higher Algebra: A Sequel - Oxford 1948 - 320 pages
Albert - College Algebra - McGraw-Hill 1946 - 278 pages
Welchons and Krickenberger - Algebra - Ginn 1953

you could add after Parke's choices [oddly he mixed up modern algebra with it like Merserve's Fundamental Concepts of Algebra and stuff]...

well you could add

Allendorfer 1955 [and all the other texts he did]
Dolciani 1964

----------------

I'm actually interested in any texts people liked from 1955-1980 actually, since there's a lot of 50s 60s texts that slip through the radar...

-------

one of the stranger ones was Hayden's 1960s book for Allyn and Bacon, talk about being a freaky advanced concepts supplementary text for Honours high school people...

it's got extremely extremely few examples, lots of New Math, and, challenging and scary on most every page. I got one of the two books in the set...

[Algebra Two - Dunstan Hayden and Gay Fischer - Allyn and Bacon 1965 - 454 pages]

IT was ideally meant for three semesters in most cases...

Where I'm assuming they meant 4 quarters for the year, 3 quarters on this text, and one final quarter where they teach the basics of probability and statistics or calculus.

---------

 

[RJinkies]

Here's a really good summary of Allendorger off Amazon:-------
quite a good book on the theory of equations Nov 21 2010
By Bruce D. Wilner - Published on Amazon.com

I used A&O in an experimental high school class in 1976. The book provides thorough, strong, and unique coverage of assorted fun topics in the theory of functions--synthetic division, Descartes's rule of signs, fundamental theorem of algebra, rational root theorem, and such--as well as (as I recall, dredging up thirty-year-old memories as best I can) good stuff on sequences and series. The pedagogy is a bit dated, which is why I withhold the fifth star. But A&O is an enjoyable book that covers lots of stuff one won't find elsewhere. You might also enjoy Hall & Knight, but, like so many British "texts," they don't teach--they just present and assume that you'll follow completely. Even the great Bruce David Wilner gets put off by this approach very occasionally . . .
---------

 

[dustbin]

Mathwonk, could you tell about differences between first and second edition of Allendeorfer's
and Oakleys' Principles of Mathematics?

I am currently studying more basic material, and I plan to study that book next but I just don't know
yet which edition to get. I think you recommended the first (1955) in somewhere, but the second edition seems to be almost 100 pages longer, so I was wandering if it contains some useful additions?

I don't really have money to get both editions right now, so I will probably get the first edition if there are no recommendations to do otherwise.

Thanks if you can answer, and also to anyone else who might know. Mathwonk has just been recommending that book a lot.

I do not know if this is still the case, but I read a while back that mathwonk prefers the first edition. I am almost done working through the first edition. At first, I did not like it a whole lot, but as I've worked through it and gotten used to the writing/style, I've come to like it quite a bit, actually. But I cannot give a comparison of the editions :-(

 

[mathwonk]

Yes, in general I always recommend first editions. In this case however I have not seen any other edition since my experience too is simply from having used it in high school in an experimental class in 1959.

As a general rule, the first edition is the authors' own best shot at exactly what they want to say and do, hence it is the best. Usually later editions exist only because the publisher wants to be able to sell more copies and not have to compete with cheaper used copies. So they pressure the author to change it somehow to make suckers, oops, I mean students, buy the newer pricier one.

I myself cannot think of a single book for which I would prefer a later edition. It is tempting to want that extra chapter, but truthfully I seldom even get through all of the shorter version, and if I do, I almost surely do not need the extra stuff in the later version.

Lets put it this way, if you wait until finishing the first edition before worrying about needing the second or third, you will almost never get to that point of having worries.

It is true as mentioned above that some old books are written in a more serious style, i.e. some modern books are written more for students who cannot actually read as well as used to be assumed, so they use smaller words and so on.

However the later editions are not usually much worse than the first one, and then I would be guided by price, although there are a few exceptions as noted below.

Some successful calculus books introduce easier problem sets in later editions to broaden their audience, or actually delete useful material, in favor of including more easier material.

E.g. the 2nd edition of van der waerden's algebra book (the first one available in english) omitted the material on well ordering and restricted to the case of countable fields, so as to include other material on valuations which is less interesting to me personally.

Then the 3rd or 4th edition restored the well ordering and added a chapter on the algebraic riemann roch theorem and topological algebra, neither of great interest to me. To make room for those, it dropped the chapter on elimination theory, which I find quite interesting especially today with the rise of computational methods.

Thus as always, it is helpful to take a look at the books in a library and see which appeals to you.
Since I have not studied this elementary material for some 50 years, it is also quite possible that some more recent books I have not heard of are more palatable and useful. So browse around on the library shelf near this book for others as well.

 

[ovael]

Thanks for answers RJinkies, dustbin and mathwonk.

As I don't have any experience posting forums and didn't yet get the hang of quoting, I will reply
for you here.

RJinkies, that's interesting that Allendoerfer was proponent of New Math. I glanced over his wikipedia page but missed that completely. And I am also interested in old mathbooks, as they seem to be better than what's available nowadays.

By the way if someone is reading this thread and doesn't yet know you can get many books RJinkies mentioned free from archive.org. Like Hall and Knights algebra books.

For example here is Leonhard Eulers Elements of Algebra:

http://archive.org/details/elementsofalgebr00euleDustbin, that's great that you have nearly completed the book. Could you tell a bit more about the experience? Like what kind of math backround you had and do you perhaps now prefer the style it has to other books you have studied previously? Did you think the problems were hard/interesting?Mathwonk, yes i will not worry about those minor differences, I was more curious, as I thought previously that I could get both the first and second edition. We have a system here (Finland) that you can order books from other cities/universities libraries and I was told it's (nearly) free but apparently it isn't.

And sadly we don't have that extensive collection of mathbooks in libraries, not at least elementary books. Local university library has luckily some, and I got my current study materials from there. They are finnish school books from the 1960's, which were first published in the 1940's.
And they are way better than the books I had in high school.. Funny or sad, depends how you look at it. And even they have been "watered down" a little due to curriculum changes from the 40's editions.

That really prompted me to ask about the Allendoerfers book, since this whole trouble of finding decent mathbooks and the whole general state of math education is really quite frustrating.

 

[RJinkies]

Mathwonk - Yes, in general I always recommend first editions...As a general rule, the first edition is the authors' own best shot at exactly what they want to say and do, hence it is the best. Usually later editions exist only because the publisher wants to be able to sell more copies...

Mathwonk - I myself cannot think of a single book for which I would prefer a later edition. It is tempting to want that extra chapter, but truthfully I seldom even get through all of the shorter version, and if I do, I almost surely do not need the extra stuff...Thus as always, it is helpful to take a look at the books in a library and see which appeals to you.

-------

So true...

it's pretty rare to see later editions of books, outside of first year physics [when it was actually adding stuff on atomic theory, and a huge ripple of books in the 40s after the atomic bomb]

In calculus,
Dull's Mathematics for engineers had a second edition in 1941... [McGraw-Hill]
Lamb's Infitessimal Calculus - 3ed 1919 [Cambridge]. corrections 1944
Sherwood and Taylor - Calculus - revised edition 1946 Prentice-Hall
Love and Rainville - 5th ed 1954

-----------

Basically when these people did new editions, it was almost always worth buying the newer one and most changes were usually extra chapters at the end and in 80% of cases the book wasnt touched. People usually proof read stuff carefully and didnt change their vision every 5 years for a totally different rewrite...

Advanced Calculus

only Kaplan - Advanced Calculus for Engineers and Physicists 2ed 1951 Ann Arbor Publishers...

[Kaplan was way more famous with Advanced Calculus - 1952 Addison-Wesley]

--------------

So i would say that pre 1960 usually the newest editions were usually the best choice and rarely would an older edition be a problem either, unless you really wanted that extra frill with the two new chapters in the back...

--------

Physics is another world, Symon's Mechanics i think is great as a 1971 3ed, and it seems double the book from 1960s 2ed...and the 1953 1ed was only like 2 chapters less than the 1960 edition...

and most of the Halliday and Resnick Texts from 1960 into the early 80s, it was basically 30% more problems, than anything else...

---------

Math texts in the 1970s started the horrid trend on occasion, and by the 80s-now it's getting ridiculous... and yes, the books are often better with the first edition...

Often i judge by the cover, the paper, the graphics, and what's extra, or how the rewrite was, and the saddest thing of all, is with these new editions, proofreading is out the window.

I seen some math texts or physical chemistry or electronics books that just get decimated by the students comments when the book suddenly becomes almost unusuable.

-----------

if you really really like a textbook, sometimes it's nice to own all the different editions, and just see what these guys were thinking, or the greedy publisher was thinking...

often i'll run to the old physics books with the 1960s pictures and illustrations than the new stuff. [I try not to look at Halliday and Resnick after 1986], and i prefer the 1960 and 1965 PSSC physics...

and how can you not adore the analog computers and rocket missile cones in the 1964 Dolciani Modern Algebra 2 Textbook? I find the older photos from the 50s to the 70s the best part of those books...

and all the India ink drawings like out of scientific american or a 1960s Addison-Wesley or McGraw-Hill book, and not computer illustrations all the time.My rule is 50-50, go with the old books and the new books both...

and when you hit the 1970s, don't be foolest by new editions...

it can be a war, of the cool cover of a 70s Springer book or the 90s book with 2 extra chapters and crappy illustrations by computer and new nasty tex typesetting..

Often i felt the strength of a book is by how few editions come out...

 

[dustbin]

Dustbin, that's great that you have nearly completed the book. Could you tell a bit more about the experience? Like what kind of math backround you had and do you perhaps now prefer the style it has to other books you have studied previously? Did you think the problems were hard/interesting?

I really like the book now. At first I found it very challenging because I knew nothing of what the opening material is on (proof and logic). For some reason, I also found the way the author wrote somewhat weird. It is very blunt and to the point. There are not elaborate descriptions and lengthy explanations. Some people, like mathwonk stated, may not like the dry/serious writing style... but I for one do. For instance, I LOVE the way Apostol writes. I could sit and read Apostol's material just for his writing style. He can be a little long winded on subjects, but he provides very motivating information on what he is writing about. Allendoerfer just bluntly states things. This took getting used to, but I now quite like it.

Once I worked through an introductory book on proofs (Chartrand and some of "How to think like a Mathematician") I jumped back into Principles of Mathematics. This time around it is a significantly better experience. Some of the problems (the proof problems) are very challenging... others are easy. I've felt that as the book progresses, the problems have gotten easier. This is probably due to my increased comfort with proofs. I've also noticed that there seems to be more computation problems as the book progresses. Honestly, I just skip most of the computation problems because I am very comfortable with that material. I am reading the book for a nice introduction to more formal mathematics. I've found the proof problems very interesting and really like reading his proofs.

My math background was pretty terrible. I grew up in a very small town with few academic opportunities. I took Algebra 1,2 and Geometry in high school. I took several years off from school to work and then started at a community college. I've always been good at math and thinking abstractly, but my preparation was limited. I started out at intermediate algebra in community college and am now taking honors calc 2 and honors linear algebra. I had never seen a proof until sometime this last spring (when I started reading Allendoerfer after becoming more interested in mathematics). I think it is a great place to start out. If you have no knowledge of logic or proofs, it may help to start out with a book that explains the subject in more depth. If you have some familiarity... I say start with this.

Hope this helps!

 

[dustbin]

You can also check out MAA's list for different subjects. Here is precalc/calc

http://mathdl.maa.org/mathDL/19/?pa=content&sa=viewDocument&nodeId=3226

They have a book on there that is also by Allendoerfer called "Fundamentals of Freshmen Mathematics." Anyone heard of it?

 

[RJinkies]

the big minus to the MAA site is that

a. they only put books up to 1991, newer stuff you need membership or read the magazine in the library to look stuff up...

b. older stuff - if it's out of print and old, they sometimes junk it off the list, which i think is a big minus.. they keep plenty on, but i think it's not enough...If one could access their older lists and newer lists, it would be one of the better ones... though sometimes they do recommend fads like some of the odd computer aided textbooks, or radical experiments [some good, some awful]...

but the MAA list is something that matches Parke's work almost perfectly... though it doesn't get into Physics, Engineering, Chemistry, Electronics...

you don't see Welchons+Krickenberg of the 50s, Dolciani of the 60s, or Munem of the 80s... for algebra...

but you'll get Three stars for the Demana Graphing Calculator books that were a fad..

-------

I just felt that it was a real letdown that the algebra aka
[school mathematics and Precalculus] parts of the list didnt include more titles, older out of print ones and some of the new ones...

like a huge gap of the 1970s...

--------------

Only two 1960s textbooks?
Only two 1970s textbooks?
come on!

and then the list starts rolling from 1981-1991

1960s
-------
- Allendoerfer, C.B. and Oakley, C.O. Fundamentals of Freshman Mathematics, New York, NY: McGraw-Hill, 1965. Second Edition.

- Ayre, H.G.; Stephens, R.; and Mock, G.D. Analytic Geometry: Two and Three Dimensions, New York, NY: Van Nostrand Reinhold, 1967. Second Edition

1970s
-------
- * Usiskin, Zalman. Advanced Algebra with Transformations and Applications River Forest, IL: Laidlaw Brothers, 1976.

- Larson, Loren C. Algebra and Trigonometry Refresher for Calculus Students New York, NY: W.H. Freeman, 1979.

1980s
-------
- Devlin, Keith J. Sets, Functions, and Logic: Basic Concepts of University Mathematics New York, NY: Chapman and Hall, 1981.

- Simmons, George F. Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry Los Altos, CA: William Kaufmann, 1981.

---------------
- one 1965
- one 1967
- one 1976
- one 1979
- two 1981
---------------
- zero 1982 books
- one 1983 book
- three 1984 books
- one 1985 book
- zero 1986 books
- two 1987 books
- zero 1988 books
- four 1989 books [many are later editions of earlier ones]
- three 1990 books
- two 1991 books

you can see when the billionth edition fad came in the mid 1980s also...My issue is considering how crucial things are for the algebra and calculus crowd, it's the place that should be the least neglected...

But then again, i think all unis should offer algebra and chemistry and math and physics from ground zero...

I think that's how Jeremy Bernstein at Harvard got into physics, he didnt take a class before, and poof ended up with a degree... and he turned into one of the better 70s 80s pop science writers and then later an excellent author on Modern Physics [aka basics for Quantum Mechanics]------
the MAA list is creepy though
2 stars for Sherman Stein and Spivak...
[they offer 2 stars for Leithhold's algebra text but don't add his calculus text]
3 stars for thomas and finney
and 2 stars for Priestley's strange historical approach to calculus. [something Morris Kline wouldn't approve of]

ideally, i'd like to see a maa/Parke like list that shows things before during and after the new math... and sadly that's a black hole for recommending books. Some of the texts were pretty experimental and freaky, neat as a reference, awful as a first exposure...

according to the MAA the only books cool enough for three starts after the Parke era would be
a. Thomas and Finney's Calculus [the 1952 edition is in Parke before Finney joined]
b. Apostol
c. Saywer's book What is Calculus About? [NML - New Mathematical Library of the 1960s]
d. Demana's Precalculus a Graphing Approachanyone out there use or browse, Leithhold and Stein's stuff from the 70s?

 

[Mariogs379]

well those are really tough questions. you are at an elite school where very little hand holding goes on, i.e. everyone assumes you know what you want, and they throw the math at you in the best form they can manage, and let it sort itself out.

There are always better people, always. I have been at all kinds of schools, and when I dropped down from ivy schools to state schools i thought well maybe now i'll be the best one here. No, there were still better people, and there always are.

So the choice has to be based on how much you enjoy what you are doing.

If you were hopelessly outclassed and had no chance, of course you should drop out, but that is not at all the case, with your record.

a certain level of talent is needed as a prerequisite, but after that entry level qualifying exam, it is all about effort.

@mathwonk,

Bit of a specific question but I thought you might be a good source of advice. Here's my background/question:

Went to ivy undergrad, did some math and was planning on majoring in it but, long story short, family circumstances intervened and I had to spend significant time away from campus/not doing school-work. So I did philosophy but have taken the following classes:

Calc II (A)
Calc III (A)
Linear Alg. (B+)
ODE's (A)
Decision Theory (pass)
Intro to Logic (A-)

Anyway, I did some mathy finance stuff for a year or so but realized it wasn't for me. I'm now going to take classes at Columbia in their post-bac program but wanted to get your advice on how best to approach this.

They have two terms so I'm taking Real Analysis I in the first term and, depending on how that goes, Real Analysis II in the second term. I'm planning on taking classes in the fall semester as a non-degree student and was thinking of taking:

Abstract Algebra
Probability
(some type of non-euclidean geometry)

Anyway, here are my questions:

1) What do you think of my tentative course selection above?

2) How much do you think talent matters as far as being able to hack it if I ended up wanting to do grad school in math?

3) I'm also having a hard time figuring out whether math is a fit for me. By that, I just mean that I really like math, I'm reading Rudin / Herstein in my free time, but I've spoken with other kids from undergrad and it's clear that they're several cuts above both ability and interest-wise. Any thoughts on how to figure this out?

Thanks in advance for your help, much appreciated,
Mariogs

Hey Mathwonk,

I finished the analysis class (using Rudin). Really interesting stuff though it's made me wonder whether I'm talented enough for more math. I ended up getting a B in the class but had to put a TON of time to get even that. Having said that (and maybe this is silly), I feel like Rudin must be discouraging for a lot of people. Had we used Abbott, I think I'd feel more confident about my abilities.

So:

1) Thoughts on this: http://www.brandeis.edu/departments/mathematics/graduate/certificate.html

2) I LOVED the cardinality stuff / Cantor's uncountability of the reals; though I don't know that analysis is something I'd want to do a ton more of. The reading I've done on my own makes me think algebra/topology is awesome, though!

I guess this question is vague but should I just do the Brandeis program and then I'll really know whether more math is for me? Seems like maybe my interest level in analysis isn't reflective of my interest in math more broadly...

It's probably the only thing where I feel like you *really* begin to understand things instead of just being spoonfed answers or formulas.

Thanks again for all your help on this!

 

[ovael]

I really like the book now. At first I found it very challenging because I knew nothing of what the opening material is on (proof and logic)... **Removed Text** ...If you have no knowledge of logic or proofs, it may help to start out with a book that explains the subject in more depth. If you have some familiarity... I say start with this.

Hope this helps!

Yes, thanks dustbin that was really helpful, I guess I have pretty similar math backround to you.
And actually the book came today, little earlier than I expected. Funny thing I noticed on the cover was that the book is a gift from the US.

It says: "This book has been presented to Finland by the government of the United States of America, under public law 265, 81st congress, as an expression of the friendship and good will which the people of the United States hold for the people of Finland."

So it's a little late but thanks guys! (I guess most/many posters are from US.)

You can also check out MAA's list for different subjects. Here is precalc/calc

http://mathdl.maa.org/mathDL/19/?pa=content&sa=viewDocument&nodeId=3226

They have a book on there that is also by Allendoerfer called "Fundamentals of Freshmen Mathematics." Anyone heard of it?

That looks really useful site, thanks.

 

[mathwonk]

"Hey Mathwonk,

I finished the analysis class (using Rudin). Really interesting stuff though it's made me wonder whether I'm talented enough for more math. I ended up getting a B in the class but had to put a TON of time to get even that. Having said that (and maybe this is silly), I feel like Rudin must be discouraging for a lot of people. Had we used Abbott, I think I'd feel more confident about my abilities."

a B in a rudin class is a strong affirmation of your ability. congratulations!

 

[Mariogs379]

heh, would be surprised if math grad schools thought so...haha. I just picked up Mendelson's "Intro to Topology" and Munkres' book. Looks like really interesting stuff.

Think I should just go for that Brandeis program?

Thanks again for the help!

 

[mathwonk]

the brandeis program looks good to me. i myself went to brandeis right out of college and the excellent teaching there made me realize i did enjoy math, and that math was even more interesting than i had thought. I learned far more in the environment there than I had as a Harvard undergrad, although Harvard's program is wonderful too, especially now. In fact Alan Mayer, the brilliant professor who first magnetized me to algebraic geometry, is still at Brandeis. I recommend you check it out.

 

[Mariogs379]

What'd you do at Brandeis?

Thanks again for the help.

Sounds like I should take a little math this spring. Maybe just abstract algebra? Or throw in a topology class too?

 

[Dowland]

Allendeorfer and Oakley - Principles of Mathematics

There's a Third Edition 1969 McGraw-Hill

and some of the books might be with title changes so the diffences could be minor or substantial...
in the day it was likely the same book tweaked for college students...Allendoerfer & Oakley - Principles of Mathematics - McGraw-Hill 1ed 1955 - 540? pages
Allendoerfer & Oakley - Principles of Mathematics - McGraw-Hill 3ed 1969 - 705? pages
Allendoerfer & Oakley - Fundamentals of Freshman Mathematics. McGraw-Hill 1959
Allendoerfer & Oakley - Fundamentals of College Algebra. McGraw-Hill 1967

I'm interested of the content in "Principles of Mathematics"; I have googled for a detailed table of contents, but can't seem to find any. I would love to read a more in-depth review of the book as well. So if anyone know of any, I would love it if you shared! :)

Also, "Fundamentals of Freshman Mathematics", is that one similar to the above mentioned book? On Google Books, one can read the following about the book:

"Survey of mathematics designed to prepare the student for a course in analytic geometry and calculus."

Sounds like a precalculus book, just like "Principles"?

 

[RJinkies]

my guess as i hinted in that messages was i think Allendoerfer just wrote the same books three times, and revised half of them for 15 years too...

being one of the larger figures in the New Math, he just wanted to add some of the 'new ideas' that program was doing in the 50s and 60s

so i'd think a chapter on formalism and lite set theory stuff trying to add some Modern Algebra glitter to the people just starting off in math... [which may or may not be such a good thing]

just plop 1-2 extra chapters or tighten up the first few chapters so it's got harder problems for college math people [where it's rehashing algebra 12 and extras], you know stuff that might not be great to plop down on grade 10-11 students.

just gear the text for people 1-3 years older...
or it's quite likely that chapters get dropped for some titles, and added for the others, and the core book is the same...

all i know for sure is he added 150 extra pages in 15 years...
which could be 4 extra chapters...

My guess:
a. he only did one algebra book, 3 different ways and at least 5 different editions...
b. and he did a calculus book which was less of a splash..
c. Seattle probably has tons of copies at the uni library and I'm closer than you are, if one of us starts synchronizing our watches now...

with a lot of these things, seeing the actually book reviews like in the College Mathematics Journal or the American Physics Teacher and the AJP i think is where lots of the 30s 40s 50s 60s and early 70s books are hiding...

would be cool if someone had a website and showed photos of the books and what some of the praise or criticism was... Allendorfer, Krickeberger, Dolciani, the Courant-Fritz John set... or things like Resnick, Symon PSSC etc etc...

i wonder if Parke kept going on for another 10-20 years and took interest in SMSG and PSSC and how it created a kick start of new and *sometimes* better books after his list... sadly he seemed in a rush to get it out for 1957 and i think half of the 1956 year he looked a most things, and sometimes dropped a Dover title of ike a Russian book that will be out in 1-2 years..or a turn of the century reprint...his priority was his rather busy consulting practice in Mass as an Applied Mathematician with a good 2000+ reference books...

[actually he said that for professional type science people that if a book though it seems pricey if it can save you a day's wasted work, it paid for itself...though that's not really true for students, is it.. lol]

sure wonder what he thought of those programs in his old age.

Morris Kline did slam a lot of stuff with his books why johnny can't add and why the professor can't teach... but i do remember i wasnt all that hyped about his calculus text, and then at the time, Apostol didnt grip me either...[it seemed like a near impossibility to do that many pages in a semester i thought, and well, it's pretty dense, it's pretty hard and lots of proofs, Courant though murky seemed way more accessible, but you sure can't pick up lots of stuff easily or quickly...

[which you can from Thomas and Finner, or Sherman Stein, or JE Thompson or Syl Thompson... or hell, Granville Longley and Smith...]

GLS seemed like the nicest text to breeze through at the library, and so was Courant-John as the two texts, i'd most likely 'oops i lost it' excuses out of a pile of really BLAND 70s calculus texts...

oh a 60s McGraw-Hill -Calculus for Electronics

three calculus books got the thumbs up from me back in the day...along with Feynman's Lectures and the Berkeley Series... I didnt see any great high school or first year physics books that stood out

but i thought the best two texts then were
a. PSSC
b. Frederick J. Bueche's College Physics for Scientists and Enginners
1969 edition [i think it had another title] , 1974 edition 1981 edition, roughly... there was one more 4th edition for sure but i think the look of the book went downhill...

one of the more relaxed and precisely worded texts around. He thought the basics should be really well done, thought it was a top book of the 70s 80s though probably too hard for high school, too easy for some Uni-ersity Physics courses...

Bueche did one schaum's outline, what it was i can't recall [I don't think it was the College Physics one about 1938, or maybe in the 50s he was the editor?] But i think he was a big cheese at the Uni of Dayton in Ohio [Ohio State and Case Western i think are the two main physics places though]

there was also some 50s 60s Addison Wesley books on College Math too, if i recall, it could have Kaplan, who wanted to do an easier book after his higher up calculus text of the 40s 50s..

one author i can't recall,had like a pretty stuff 1950s Trigonometry book that was about 150 pages, it was pretty stiff reading and though a bit difficult, pretty stimulating and seemed like a popular way of doing things for the people after high school and wanted that one scary math class for liberal arts... anyways that author i think did a pre cal book/college math book and a calculs book too.

i think his trig book was circa 54 with Add-Wes but the 'other titles' splash was circa 60-61 if i recall. Two years ago, i knew where that book was *grin*...

 

[ovael]

I'm interested of the content in "Principles of Mathematics"; I have googled for a detailed table of contents, but can't seem to find any. I would love to read a more in-depth review of the book as well. So if anyone know of any, I would love it if you shared! :)

Also, "Fundamentals of Freshman Mathematics", is that one similar to the above mentioned book? On Google Books, one can read the following about the book:

"Survey of mathematics designed to prepare the student for a course in analytic geometry and calculus."

Sounds like a precalculus book, just like "Principles"?

Here is table of contents of Allendoerfer's and Oakleys Principles of Mathematics first edition (1955):Preface

List of Symbols

Chapter 1. Logic (p. 1-38)

1. Introduction
2. Definitions
3. Propositions
4. Propositions in Mathematics
5. Quantifiers
6. Symbols
7. Truth Tables
8. Applications of Truth Tables
9. Negation
10. Implications Derived from Other Implications
11. Mathematical Terminology
12. Methods of Proof
13. Methods of Proof (continued)

Chapter 2. The Number System (p. 39-68)

1. Introduction
2. Addition of Real Numbers
3. Multiplication of Real Numbers
4. Formal Properties of Real Numbers
5. Special Properties of Real Numbers
6. Special Properties of Zero
7. Special Properties of Integers
8. Special Properties of the Rational Numbers
9. Decimal Expansion
10. Some Irrational Numbers
11. Geometrical Representation of Real Numbers
12. The Use of Real Numbers in Plane Geometry
13. Distance between Two Points
14. Complex Numbers
15. Solutions of Other Algeabraic Equations
16. Classification of Numbers
17. Congruences

Chapter 3. Groups (p. 69-82)

1. Introduction
2. Groups
3. Examples of Groups
4. Further Examples of Groups
5. Theorems about Groups

Chapter 4. Fields (p.83-102)

1. Introduction
2. Definition of a Field
3. Examples of Fields
4. Theorema based upon Group Properties
5. Additional Theorems
6. Solution of Equations
7. Solution of Quadratic Equations
8. Inequalities
9. Theorems Concerning Fractions
10. Exponents and Radicals

Chapter 5. Sets and Boolean Algebra (p. 103-123)

1. Introduction
2. Sets
3. Relations between sets
4. Union and Intersection of Sets
5. Complements
6. Boolean Algebra
7. The Boolean Algebra (0,1)
8. Electrical Networks
9. Design of Circuits
10. Quantifiers

Chapter 6. Functions (p. 124-158)

1. Functions
2. Special Functions
3. Relations
4. Notations for a Function
5. Rule, Domain, and Range
6. Algebra of Functions
7. Graph of a Function
8. Graph of a Relation
9. Inverse Function
10. Functions Derived from Equations

Chapter 7. Algebraic Functions (p. 159-181)

1. Introduction
2. Polynomial Functions
3. Rational Functions
4. Explicit Algebraic Functions
5. Graphs and Continuity
6. Properties of Polynomial Equations
7. Synthetic Division
8. Roots of Polynomial Equations
9. Rational Roots of Rational Polynomial Equations
10. Real roots or Real Polynomial Equations

Chapter 8. Trigonometric Functions (p.182-224)

1. General Definitions
2. Special Real Numbers
3. General Real Numbers
4. Range and Graph of Functions
5. Addition Theorems
6. Identities
7. Equations
8. Directed Angles
9. Trigonometric Function of Directed Angles
10. Right Triangles
11. Law of Sines
12. Law of Cosines
13. Inverse Functions
14. Complex Numbers

Chapter 9. Exponential and Logarithmic Functions (p.225-235)

1. Introduction
2. Exponential Functions
3. The number "e"
4. Logarithmic Functions
5. Graphs
6. The Logarithmic Scale

Chapter 10. Analytic Geometry (p.242-283)

1. Introduction
2. Mid-point of a Line Segment
3. Directed Line Segment
4. Inclination, Slope, Direction Cosines
5. Angle between Two Directed Lines
6. Applications to Plane Geometry
7. The Straight Line
8. Conic Sections
9. The Circle
10. The Parabola
11. The Ellipse
12. The Hyperbola
13. Applications
14. Polar Coordinates
15. Polar Coordinates Continued
16. Parametric Equations

Chapter 11. Limits (p. 284-329)

1. Introduction
2. Historical Notes
3. Sequences
4. Limits of Sequences
5. Examples of Sequences
6. Theorems of Limits of Sequences
7. Series
8. Limits of Functions
9. Theorems of Limits of Functions
10. Continuity
11. Area
12. Rates
13. Tangent to a Curve

Chapter 12. The Calculus (p. 330- 363)

1. Integration
2. Differentiation
3. Comparison of Integration and Differentiation
4. Rules of Differentiation
5. Second Derivatives
6. Maxima and Minima
7. Related Rates

Chapter 13. Statistics and Probability (p. 364-420)

1. The Nature of Statistics
2. Sampling
3. Presentation of Data
4. Frequency Distributions
5. Characteristics of Frequency Distributions
6. Grouping
7. Averages
8. Interpretation of the Mean
9. Computation of the Mean
10. Standard Deviation
11. Probability
12. Permutations
13. Combinations
14. Binomial Theorem
15. Probability (Again)
16. Empirical Probability
17. Expectation
18. Repeated Events
19. Binomial Distribution
20. Testing Hypothesis
21. Cumulative Normal Curve
22. Normal Distribution
23. Normal Distribution (continued)
24. Distribution of Sample Means
25. The Logical Roles of Statistics

Answers to Selected Exercises

IndexI can't really give in-depth review since I have not yet started studying it, but it seems to be good
"bridge" from basic algebra/geometry/trigonometry/calculus knowledge to higher mathematics.

I'm getting little ahead of myself, but when I'm done with Allendoerfer I will probably get Apostol's Calculus Books.

 

[RJinkies]

sounds like a high school honours course pretty much, or close to a college algebra like class, which got phased out with the space race, they wanted no longer to teach a review of high school and then push you into calculus, but just push you into calculus... and push out more engineers quicker.. and now there's the trend to push calculus into the high schools and then you really zoom into accelerated stuff into uni...

I think in the earlier days when there was less textbook competition and less crowded unis, you were always better off when a school could teach the most basic of math, or chemistry or physics, if you lacked anything, and the college professors would use better textbooks and it would mesh more with their calculus programs...

one thing i know, that if you take a math or chemistry course where you start on chapter 8 in the first week of classes, it would probably be better to read the first 7 chapters a semester or year earlier, and not miss out on the authors usually well-constructed development.

sometimes the stuff a chemistry book assumes, it's likely you might not have down any of those problems or concepts for 20% of things... and if you just skirted it, you're probably conceptionally shaky that you wouldn't notice somethng, unless it was pointed out to you...

the interesting thing, i found was with a lot of books, algebra texts with a strong new math feel, or books on diff eqs or complex variables, or some organic chem, that *often* chapter 1 where it's suppossedly review, is actually much harder than the new material with chapters 2 and 3...

had an interesting talk with my math teacher about that phenomenon... but i think it was more that sometimes in the 70s kids are less better prepared, and i think the textbook changes and curriculum changes had a fair deal to do with that...

I think i'd cringe at the first 150 pages of allendorfer though, but if it's a slow enough pace, or a second dip into algebra, you can enjoy it a lot more...

I'm guessing that it's probably one of the earlist books in spirit to the new math [before it started AFTER sputnik], without trying to cram too much Bourbaki down the throats of 16 year olds

the PSSC program was Before Sputnik, but the SMSG New math stuff was in reaction to sputnik...

and the Seaborg CHEM books from 1964 are great reading, but I think a lot of the problem solving skills are hidden or absent, or at that time, really pushed in first year chem... things seems a bit more leisurely for chemistry in the days of slide rules for chemistry 11 once upon a time...

sad thing is more all those 60s chemistry texts shows hints of the most exciting stuff going on, and they yanked all that stuff out by the 70s

------

think about using Sylvanius Thompson and JE Thompson's
Calculus Made Easy and Calculus made Simple [1910s and 1930s]
[or Sherman Stein]
if you are going to read Apostol or Spivak

i find an easy book is great parallel reading...

 

[dustbin]

The above table of contents looks, as far as I can tell, exactly like the one in my first edition copy of Principles of Mathematics. So it appears that only their titles may be different? Pretty much all of the page numbers are the same as well (I could only find one or two that were different on the ToC).

I think i'd cringe at the first 150 pages of allendorfer though, but if it's a slow enough pace, or a second dip into algebra, you can enjoy it a lot more...

The chapters on logic were the hardest for me (completely new subjects to me). I did the first 4 chapters, took a break to study a book that focused more on this material in particular, and then came back to it (and re-did it). After my second run-through with it, I handled it very well. The first time, I indeed did work very slowly through it. After coming back to it and going through those chapters again, the subsequent chapters (I've done up to the calculus topics, but not including them) have been much more gentle.

 

[Dowland]

Here is table of contents of Allendoerfer's and Oakleys Principles of Mathematics first edition (1955):


Preface

List of Symbols

Chapter 1. Logic (p. 1-38)

1. Introduction
2. Definitions
3. Propositions
4. Propositions in Mathematics
5. Quantifiers
6. Symbols
7. Truth Tables
8. Applications of Truth Tables
9. Negation
10. Implications Derived from Other Implications
11. Mathematical Terminology
12. Methods of Proof
13. Methods of Proof (continued)

Chapter 2. The Number System (p. 39-68)

1. Introduction
2. Addition of Real Numbers
3. Multiplication of Real Numbers
4. Formal Properties of Real Numbers
5. Special Properties of Real Numbers
6. Special Properties of Zero
7. Special Properties of Integers
8. Special Properties of the Rational Numbers
9. Decimal Expansion
10. Some Irrational Numbers
11. Geometrical Representation of Real Numbers
12. The Use of Real Numbers in Plane Geometry
13. Distance between Two Points
14. Complex Numbers
15. Solutions of Other Algeabraic Equations
16. Classification of Numbers
17. Congruences

Chapter 3. Groups (p. 69-82)

1. Introduction
2. Groups
3. Examples of Groups
4. Further Examples of Groups
5. Theorems about Groups

Chapter 4. Fields (p.83-102)

1. Introduction
2. Definition of a Field
3. Examples of Fields
4. Theorema based upon Group Properties
5. Additional Theorems
6. Solution of Equations
7. Solution of Quadratic Equations
8. Inequalities
9. Theorems Concerning Fractions
10. Exponents and Radicals

Chapter 5. Sets and Boolean Algebra (p. 103-123)

1. Introduction
2. Sets
3. Relations between sets
4. Union and Intersection of Sets
5. Complements
6. Boolean Algebra
7. The Boolean Algebra (0,1)
8. Electrical Networks
9. Design of Circuits
10. Quantifiers

Chapter 6. Functions (p. 124-158)

1. Functions
2. Special Functions
3. Relations
4. Notations for a Function
5. Rule, Domain, and Range
6. Algebra of Functions
7. Graph of a Function
8. Graph of a Relation
9. Inverse Function
10. Functions Derived from Equations

Chapter 7. Algebraic Functions (p. 159-181)

1. Introduction
2. Polynomial Functions
3. Rational Functions
4. Explicit Algebraic Functions
5. Graphs and Continuity
6. Properties of Polynomial Equations
7. Synthetic Division
8. Roots of Polynomial Equations
9. Rational Roots of Rational Polynomial Equations
10. Real roots or Real Polynomial Equations

Chapter 8. Trigonometric Functions (p.182-224)

1. General Definitions
2. Special Real Numbers
3. General Real Numbers
4. Range and Graph of Functions
5. Addition Theorems
6. Identities
7. Equations
8. Directed Angles
9. Trigonometric Function of Directed Angles
10. Right Triangles
11. Law of Sines
12. Law of Cosines
13. Inverse Functions
14. Complex Numbers

Chapter 9. Exponential and Logarithmic Functions (p.225-235)

1. Introduction
2. Exponential Functions
3. The number "e"
4. Logarithmic Functions
5. Graphs
6. The Logarithmic Scale

Chapter 10. Analytic Geometry (p.242-283)

1. Introduction
2. Mid-point of a Line Segment
3. Directed Line Segment
4. Inclination, Slope, Direction Cosines
5. Angle between Two Directed Lines
6. Applications to Plane Geometry
7. The Straight Line
8. Conic Sections
9. The Circle
10. The Parabola
11. The Ellipse
12. The Hyperbola
13. Applications
14. Polar Coordinates
15. Polar Coordinates Continued
16. Parametric Equations

Chapter 11. Limits (p. 284-329)

1. Introduction
2. Historical Notes
3. Sequences
4. Limits of Sequences
5. Examples of Sequences
6. Theorems of Limits of Sequences
7. Series
8. Limits of Functions
9. Theorems of Limits of Functions
10. Continuity
11. Area
12. Rates
13. Tangent to a Curve

Chapter 12. The Calculus (p. 330- 363)

1. Integration
2. Differentiation
3. Comparison of Integration and Differentiation
4. Rules of Differentiation
5. Second Derivatives
6. Maxima and Minima
7. Related Rates

Chapter 13. Statistics and Probability (p. 364-420)

1. The Nature of Statistics
2. Sampling
3. Presentation of Data
4. Frequency Distributions
5. Characteristics of Frequency Distributions
6. Grouping
7. Averages
8. Interpretation of the Mean
9. Computation of the Mean
10. Standard Deviation
11. Probability
12. Permutations
13. Combinations
14. Binomial Theorem
15. Probability (Again)
16. Empirical Probability
17. Expectation
18. Repeated Events
19. Binomial Distribution
20. Testing Hypothesis
21. Cumulative Normal Curve
22. Normal Distribution
23. Normal Distribution (continued)
24. Distribution of Sample Means
25. The Logical Roles of Statistics

Answers to Selected Exercises

Index


I can't really give in-depth review since I have not yet started studying it, but it seems to be good
"bridge" from basic algebra/geometry/trigonometry/calculus knowledge to higher mathematics.

I'm getting little ahead of myself, but when I'm done with Allendoerfer I will probably get Apostol's Calculus Books.

Thank you, ovael! Chapters 1,3,4,5 looks very interesting!

I was actually thinking of buying it, because I've heard such good things about here on PF. But I'm currently doing "Basic Mathematics" (Serge Lang), "Elementary Algebra" (Harold Jacobs) and "Algebra" (Gelfand) at the same time, and in parallell to my ordinary high school curriculum, so I think it will be pretty overwhelming and unnecessary with another book.

However, good luck with your studies now, ovael! And I'm looking forward to a review of the book sometime in the future. :)


(BTW, sorry for any language errors, English is not my native.)

 

[RJinkies]

dustbin - The chapters on logic were the hardest for me (completely new subjects to me). I did the first 4 chapters, took a break to study a book that focused more on this material in particular, and then came back to it (and re-did it). After my second run-through with it, I handled it very well. The first time, I indeed did work very slowly through it. After coming back to it and going through those chapters again, the subsequent chapters (I've done up to the calculus topics, but not including them) have been much more gentle.

What you experienced, is exactly how i think i would feel if i tackled him too! For me i remember people always finding Dolciani a hard text but if you read Modern Algebra Book 2 from 1964 from the start, it wasnt hard at all, but it got me to realize just how shaky our algebra was with a class with 2 textbooks and dolciani was only used 20% of the time, and just taking random stuff out of it...

I think i usually recommended Schaum's Outlnes [there were about 3 or 4] , Dolciani from the 60s or Munem from the 80s as the quickest fix or way to start off algebra...

I think that's one of the big reasons for the decline in math, we arent going slow enough and as thorough enough, and well we also need textbooks we can start beginning to end.

the more i looked at chemistry texts, i found that if you arent reading it from page one, you're really losing out on a solid foundation of the topic... one of the 1967 classics that was used at Caltech [it was a bright yellow one], the author basically started in on chapter 8 and ran through it.

He said that if you had zero chemistry before, and you could cram 3-4 months into reading that, or if you take the course and really really push it, you could basically coast through the course okay... I thought it was one of the finest textbooks since he listed all the great classics of the early and mid 60s at the end of each chapter, and you could end up with 40 textbooks from 1959-1966 on your reading list lol Why the subject lost most of its charm in the 70s, I'm not sure why but i think the focus narrowed and the enjoyable asides and well as deep explanations of the basics just went out the window... just push the mathematical essentials for what people need for organic or physical chem and forget the rest...
dowland - currently doing "Basic Mathematics" (Serge Lang), "Elementary Algebra" (Harold Jacobs) and "Algebra" (Gelfand) at the same time, and in parallell to my ordinary high school curriculum, so I think it will be pretty overwhelming and unnecessary with another book.

Lang seemed way more approchable with his basic math book and his old calculus book when he wanted to make a simplified course... lots of people don't appreciate his later stuff, till you're closer to grad school with linear and stuff, and a fair number of people get an allergy to him if they try to soak him in too soon. I was surprised when i came across his easier books and i was expecting a terse harsh introduction...

Jacobs did some good stuff with his elementary algebra and geometry books in the 70s 80s, one had an Escher artwork thing on it too... how do like Elementary Algebra by him, and i assume you got 1 of the 5 Green and White Birkhauser Gelfand books which probably go well with the NML series too...

Definately want to hear your thoughts on Lang and Jacobs algebra... i don't think many people at all pick or get dumped jacob's from the MAA list of recommended books for algebra anymore... but what what i recall it was always considered a great text for people with little or no background...i'm not sure if teacher's only picked it because of the Escher artwork or people listened to the MAA more lol

 

[Dowland]

idowland - currently doing "Basic Mathematics" (Serge Lang), "Elementary Algebra" (Harold Jacobs) and "Algebra" (Gelfand) at the same time, and in parallell to my ordinary high school curriculum, so I think it will be pretty overwhelming and unnecessary with another book.

Lang seemed way more approchable with his basic math book and his old calculus book when he wanted to make a simplified course... lots of people don't appreciate his later stuff, till you're closer to grad school with linear and stuff, and a fair number of people get an allergy to him if they try to soak him in too soon. I was surprised when i came across his easier books and i was expecting a terse harsh introduction...

Jacobs did some good stuff with his elementary algebra and geometry books in the 70s 80s, one had an Escher artwork thing on it too... how do like Elementary Algebra by him, and i assume you got 1 of the 5 Green and White Birkhauser Gelfand books which probably go well with the NML series too...

Definately want to hear your thoughts on Lang and Jacobs algebra... i don't think many people at all pick or get dumped jacob's from the MAA list of recommended books for algebra anymore... but what what i recall it was always considered a great text for people with little or no background...i'm not sure if teacher's only picked it because of the Escher artwork or people listened to the MAA more lol

Oops, I meant "Elementary GEOMETRY", not Algebra! I aksed earlier in this thread about the importance of learning euclidean geometry thouroughly and I got some mixed answers, but I finally decided to give it a try (can't hurt and one can always make some extra sparetime for mathematics right

As Regards to Lang's book, I love it! I have worked through approximately half of the book now and it has really given me a new way of looking at mathematics. Basic Mathematics is the first math book I've ever read at the side of the ordinary high school curriculum in my country, and it feels lika a completely different philosophy and attitude towards the subject and the student, which I think every high school student interested in mathematics should have the opportunity to expercience.

Can't write more for the moment, maybe I'll return with some more elaborated comments on the book.

 

[RJinkies]

Damn, it's extremely difficult to find someone with an opinion of Jacobs Algebra text...

shame the MAA doesn't like any basic geometry books before 1968...

Geometry: School Geometry

Loomis, E. The Pythagorean Proposition - NCTM 1968
** Jacobs, Harold R. Geometry, New York, NY: W.H. Freeman, 1974 First Edition.
* Konkle, Gail S. Shapes and Perceptions: An Intuitive Approach to Geometry - Prindle, Weber and Schmidt 1974
* Moise, Edwin E. and Downs, Floyd L. Geometry - Addison-Wesley 1975
** O'Daffer, Phares G. and Clemens, Stanley R. Geometry: An Investigative Approach- Addison-Wesley 1976
* Bruni, James V. Experiencing Geometry - Wadsworth 1977
Kempe, A.B. How to Draw a Straight Line - NCTM 1977
* Fetisov, A.I. Proof in Geometry - MIR 1978
Hoffer, Alan. Geometry - Addison-Wesley 1979
Clemens, Stanley R.; O'Daffer, Phares G.; and Clooney, Thomas J. Geometry - Addison-Wesley 1983
** Jacobs, Harold R. Geometry, New York, NY: W.H. Freeman 1986. Second Edition.

one thing i hear about the 50s 60s geometry texts are they seem to be zombie-like. Some people seemed to like the challenge, but most anyone with high school geometry thinks its pretty useless if you take higher math classes...

as for Lang he gets the most praise for his easy books, but most of his stuff people don't like till they are in 4th year and like all that sterile bourbaki like formalism with linear.. Some people really dislike his book, but after a year or two then mellow and appreciate it more when get higher up...

so it was a shock for me when i found out lang did a great basic book, and once upon a time a pretty clear barebones calculus text...

 

[Group_Complex]

I have wanted to be a pure mathematician since I was about 15. I thought I was quite good at mathematics till I got to university and saw how talented some of my fellow classmates are. For instance many of them do no study besides attending class, and spend most of their time socialising/drinking/gaming yet still out perform me in many assesment tasks. This is quite frankly soul crushing as I spend most of my time thinking about mathematical things and it really highlights my lack of natural ability.

The thing is I still love the subject. Yet everytime one of my cocky friends tells me how well he has done in the latest exam or assignment I feel crushed and betrayed my the subject I love so much. I used to believe that genius was 90% Hardwork, but now I see that most of these sayings are just political correctness gone wrong.

Is it possible to even get a phd let alone tenure in pure mathematics when you lack that "spark" of genius?