Foundations Rigorous Alternatives to AoPS for PreAlgebra-PreCalculus?

[Vanadium 50]

I've not read Euler, but I have seen people trying to learn physics from the original sources: Newton from the Principia, relativity from Einstein, and electromagnetism from Maxwell. These hardly ever go well, and of the three, relativity from Einstein is the least bad. People are shocked to find that the Maxwell equations are nowhere to be found in Maxwell: they came about a quarter-century later from Oliver Heaviside. In comparison, Maxwell is a mess.

 

[mathwonk]

I plead guilty to having read (some of) Newton, Maxwell, and Einstein, as well as Pauli and others. They may not be the best sources for a mastery of physics, but I enjoy what I find in the works of masters. E.g. here is one of my favorite quotes from Maxwell, on the best choice of units of length, including a remark on the disadvantages of using the meter:

"In the present state of science, the most universal standard of length which we could assume would be the wave length in vacuum of a particular kind of light, emitted by some widely diffused substance such as sodium, which has well defined lines in its spectrum. Such a standard would be independent of any changes in the dimensions of the earth, and should be adopted by those who expect their writings to be more permanent that that body."

I love reading far sighted geniuses. It had not dawned on me previously to aspire to having my writings outlast the lifespan of the planet.

 

[Mondayman]

People are shocked to find that the Maxwell equations are nowhere to be found in Maxwell: they came about a quarter-century later from Oliver Heaviside. In comparison, Maxwell is a mess.

Heaviside's work itself is supposedly pretty ugly. Nahin writes that he would rarely simplify his work, leaving equations full of cumbersome integrals and complicated terms.

 

[Vanadium 50]

Heaviside's work itself is supposedly pretty ugly

Well, he recognized that EM is a vector theory., A huge simplification over the 20 equations of Maxwell.

 

[mathwonk]

@ Vanadium: Yes the proof Andrews gives is quite eye opening. It is reminiscent to me of the usual group theory proof originally due to Gauss in the setting of modular arithmetic, by dividing up a group into cosets of the same size, but the physical significance of the arguments using beads and polygons makes it more visible and tangible, in a way I think students may find quite helpful. Thanks for this reference!

 

[TurboDiesel]

I think these (below) problems are somewhat illustrative of their respective texts. Feel free to present your own experiences. I feel this comparison adds light to the discussion.

Esty holds your hand in looking at this interesting artifact of sound waves, letting you work the calculations into your gray matter. (Chandller has a copy of this self-study-friendly text.)

Without looking at the AoPS solution manual, or playing with its introductory level practice software, you might not realize that AoPS expects a full solution--covering the answer both in words and in a math treatment (a two-column presentation is used in and encouraged by its software). However with this example problem, you can see it is a trivial matter of degrees of formality to move from a full AoPS two column solution to a proof--and this example is from the Prealgebra text. [Also note that the hint is obtained by working out a following problem rather than a preceding problem, this is representative of the AoPS spirit of exploration.]

Looking at the Cohen problem, you can see an interesting application of the math. Pretty fun.

Santos' problems make an almost gratuitous use of proofs. This is good legwork and the below problem might present the flavour of the text.

Allendoerfer and Oakley often require the use of proofs, but the problems feel useful, somehow. As in not just building understanding, but building understanding of useful relationships.

The major standout with Loney, for me, seems to simply be older style language (and if this is the only standout after well over a century, kudos to the guy). The below problem seems typical for Loney.

Somewhat Arbitrary Example Problems:

Esty "Precalculus" (p444 prob.B17):

"When two sound waves are very similar, but not identical, in frequencey, they will reinforce each other at times and nearly cancel each other at other times. For example, if one note is played at 440 cycles per second and a[sic] another note is played at 438 cycles per second, there will be an audible increase and decrease in amplitude twice a second known as a 'beat.' This can be illustrated with a very wide graph of the sum of two sine waves. However, your calculator does not have enough columns of pixels to display such graphs. Near x = 0 the graph of 'sin x + sin (1.01x)' displays reinforcement (graph it and see). Here is the problem: Find, very roughly, the smallest possible c such that this graph displays nearly complete canceling on the interval [c, c + 10]."​

AoPS "Prealgebra" (Note this is Prealgebra: p474 challenge prob.12.43):

"The diagonals of a rhombus are perpendicular and the area of a rhombus is half the product of the lengths of its diagonals. Similarly, the diagonals of a kite are perpendicular, and the area of a kite is half the product of its diagonals. Is it true that for any quadrilateral with perpendicular diagonals, the area of the quadrialteral equals half the product of its diagonals? Why or why not? Hints: 45"​

Cohen "Precalculus with u.c. trig." (p896 ch11.Test prob.3):

"The distances from the planet Saturn to the Sun at aphelion and at perihelion are 9.5447 AU and 9.5329 AU respectively. Compute the eccentricity of the orbit and the length of the semimajor axis. Round each answer to three decimal places."​

David Santos "Precalculus, An Honours Course" (page 49 Homework prob. 2.8.1)
"Let d > 0 be a real number. Prove that the equation of a parabola with focus at (d,0) and directrix x = -d is x = $\frac{y^2}{4d}$."

Allendoerfer and Oakley "Principles . . ." 2nd Edition (p245 prob.32)
"Show that it is false that $∃_{x}$ [sin 2x + cos 2x = 4]."

SL Loney "Plane Trigonometry" (p467 prob.13)

"The three sides of a triangle are measured and found to be nearly equal. If the measurements can be wrong one per cent, in excess or defect, prove that the greatest error that can arise in calculating one of the angles is 80' nearly."​

I do not have Euler's "Introduction" at hand, so cannot immediately provide an example problem.

 

[vanhees71]

I've not read Euler, but I have seen people trying to learn physics from the original sources: Newton from the Principia, relativity from Einstein, and electromagnetism from Maxwell. These hardly ever go well, and of the three, relativity from Einstein is the least bad. People are shocked to find that the Maxwell equations are nowhere to be found in Maxwell: they came about a quarter-century later from Oliver Heaviside. In comparison, Maxwell is a mess.

I can agree on this judgment about Newton (completely incomprehensible without learning a lot of ancient geometry which is way better understood using modern math, including calculus) or Maxwell (who was not in full command of modern vector calculus, so that you better learn about quaternions to make sense of his writings), but I cannot agree about Einstein. He is a prime example for somebody who really carefully wrote well-understandable no-nonsense papers. Also the few books he wrote are gems, particularly the 4 lectures on relativity.

Of the older authors I also like Dirac, Pauli, Born, and particularly Sommerfeld. Some of their textbooks are way better and even still up to date than more modern texbooks.

Still, I think it's not useful to learn math from too old sources, like Newton for mechanics etc.

 

[mathwonk]

The remarkable Michael Spivak has fortunately taken it as a goal to understand the writings of some of these obscure geniuses and to share his results with us. In volume 2 of his famous Differential geometry series he translates and explains in detail the great paper of Riemann on the foundations of geometry, including curvature I believe (I perused it a few years ago while trying to prepare a course on differential geometry for brilliant 12 year olds). And in recent years he has taken up the large task of understanding Newton's Principia. He gave a talk on the first few basic concepts at UGA and has since written an opus on physics for mathematicians, which I presume is the result of his exploration of Newton. He gave me a preliminary short version, Elementary mechanics from a mathematician's viewpoint, no. 29, in the series Seminar on Mathematical Sciences, from Keio University in Japan, where he lectured on it.

He states that he thinks the hard part of mechanics is not something like symplectic structures, but fundamental concepts like levers! He begins with Newton's definition 1, and analyzes it in great detail, slowly. Nontheless he states "..this book is one of the great classics, probably the greatest book in all of physics, but this doesn't mean one should try learning physics from it! Like many classics it is basically unreadable." Still of course he is obviously making a serious effort to do exactly that. We can now benefit from his years of study of it.

In this same vein, almost everyone says the same about Riemann, that he is essentially unreadable, but I beg to differ. I have read parts of it, and although it is very hard to read, it is not impossible, and what I have gotten out of it exceeds what I have gotten from all other sources I have tried to use to understand the same ideas. (I studied parts of his great paper on abelian functions.) In fact although it may have taken me a day or so per sentence in some cases, nonetheless the amount of insight gained was so great that the ratio of (benefit-insight)/(time spent), is actually greater than that same ratio for modern textbooks on the subject.

So while I admit that classics are very hard to read, I think they are still worth it. Here is a link to my discussion of this question on mathoverflow (mine is the first answer to the question posed):
https://mathoverflow.net/questions/28268/do-you-read-the-masters/51868#51868

 

[mathwonk]

My derivation of the volume of a 4 ball may have looked unappealingly long, but the summary in the second paragraph, should suffice for physicists here: In brief it is a center of mass argument. Does this scan?:

Just as we can generate a 3 ball by revolving half a 2 disc around an axis through the bounding diameter of the disc, we can generate a 4-ball by revolving (in 4 space) half a 3-ball around a planar axis containing its equator. It follows that the volume of a 4 ball is equal to the volume of half a 3 ball multiplied by the distance traveled by the center of mass of the 3 ball.

Hence we can use Archimedes’ trick to calculate the volume of a 4 ball. Namely he showed that the volume of half a 3-ball equals the difference of the volumes of a cylinder minus that of a cone. Hence the volume of a 4 ball equals the difference of the volumes generated by revolving a cylinder and an inverted cone. Thus all we need to know is the center of mass of these solids, which Archimedes knew!

 

[vanhees71]

In this same vein, almost everyone says the same about Riemann, that he is essentially unreadable, but I beg to differ. I have read parts of it, and although it is very hard to read, it is not impossible, and what I have gotten out of it exceeds what I have gotten from all other sources I have tried to use to understand the same ideas. (I studied parts of his great paper on abelian functions.) In fact although it may have taken me a day or so per sentence in some cases, nonetheless the amount of insight gained was so great that the ratio of (benefit-insight)/(time spent), is actually greater than that same ratio for modern textbooks on the subject.

Maybe you got so much out of reading this classic, because it was hard to read. You have to struggle to understand everything in detail. I also don't say that the geniusses of the past have simply written "bad books" or papers but that in the meantime the notation, methodology, and expression of mathematics and mathematical physics has changed and we are simply not used to the older ones anymore. Of course it becomes more and more difficult the older the source gets. Newton is indeed almost unreadable, but there are some works on "translating" it to make it readable for contemporary physicists and mathematicians. E.g., Feynman has written a little booklet about Newton's treatment of the Kepler problem. Another famous example is Chandrashekar's, Newton's Principia for the Common Reader. Also he took 10 years for this work.

 

[mathwonk]

Thanks for this reference. Indeed the book of Spivak I referenced above, (Physics for mathematicians, mechanics I), seems to rely significantly on Chandrasekar, which he also recommends as "extremely useful".

https://www.amazon.com/dp/0914098322/?tag=pfamazon01-20

(I said this was a recent work, but now I see almost 10 years have passed since its publication. wow.)

 

[Galoisfanatic]

Joke: I would hire someone to transcribe the AOPS series by hand including the solution manuals and put there name on the front.
In all seriousness, I’ve spent far far too much time thinking about the same question, and AOPS as a closed system is simply the best for me. It set me up to make a pretty seamless transition to Spivak, and I never really struggled with that book or any other lower division proof based textbook like Rudin or Axler.It’s not all there is though, technically.
The only thing truly on par with the AOPS series in terms of difficulty and pedagogy is “Extending Mathematics” volume 1/2 by T.J.Heard. These are old books that are pretty obscure. They were designed to teach A-level mathematics up to the scholarship level, an old exam standard in the UK. They cover all of HS math including up to CALC 2. However they are extremely difficult to get through on your own. Computational answers are provided in the back, which isn’t really enough for the more demanding problem sets. Pure Mathematics volume 1/2 by S.L. Parsonson is another text from the same country that covers all of precalculus and Algebra. It’s quite good.
Aside from that, there’s Gelfand’s series of 5 books (his geometry text was recently published by Springer, completing the series). These are amazing. The only issue is a lack of computational exercises. Great supplements, however. Do them in order Algebra->Functions and Graphs->Method of Coordinates->Geometry->Trigonometry
Now there’s one more path that ought to be mentioned seriously. Strictly speaking this is superior to AOPS, and would prepare you better for future mathematics, but I consider it a waste of time. I got this from J.E. Littlewood’s “A mathematical education” from his “A Mathematicians Miscellany” book. This describes a state of the art mathematical education in 1900. Its considerably more in depth and rigorous than the AOPS actually. However, I simply would not want to spend 600 pages learning about conics, 700 on advanced trig, etc..
>The tradition of teaching (derived ultimately from Cambridge) was to study 'lower ' methods intensively beforegoing on to * higher ' ones ; thus analytical methods in geometry were taken late, and calculus very late. And each book was more or less finished before we went on to the next. The accepted sequence of books was : Smith's Algebra ; Loney's Trigonometry ; Geometrical Conies (in avery stiff book of Macaulay's own : metrical properties oft he parabola, for instance, gave scope for infinite virtuosity) ;Loney's Statics and Dynamics, without calculus ; C. Smith's Analytical Conics ; Edward's Differential Calculus ;Williamson's Integral Calculus ; Besant's Hydrostatics. These were annotated by Macaulay and provided with revision papers at intervals. ' Beyond this point the order could be varied to suit individual tastes. My sequence, I think, was : Casey's Sequel to Euclid ; Chrystal's Algebra II ;Salmon's Conics; Hobson's Trigonometry (2nd edition, 1897) ;Routh's Dynamics of a Particle (a book of more than 400 pages and containing some remarkably highbrow excursions towards the end) ; Routh's Rigid Dynamics ; Spherical Trigonometry(in every possible detail) ; Murray's Differential Equations ;Smith's Solid Geometry ; Burnside and Panton's Theory of Equations ; Minchin's Statics (omitting elasticity, but including attractions, with spherical harmonics, and of coursean exhaustive treatment of the attractions of ellipsoids).
All of these books are actually available for free as PDFs on internet archive/google books. This sequence is actually vastly superior to the AOPS in terms of breadth, depth, and rigor. It needs to be preceded by Euclid’s Elements as littlewood had done, but that’s it really. Perhaps I have a lazy modern attitude produced by instant gratification and electronics, but this seems completely insane to me. I do not want to work through those 7-8000 pages of material. Maybe that’s what separates littlewood from me, I guess.
Aside from that there’s the “new math” stuff.
Dolciani Modern Introductory Analysis (a precalculus textbook, the newer one introduced basic differential and integral calc).
Unified Modern Mathematics Series. Starts from 8th grade math and works to a rigorous calculus formulation. Sort of the epitome of new math. Won’t reach you addition, instead it has to be in base 8 with modular clock arithmetic mixed in. It’s charming though queer. The first 3 books are available for free on ed.gov. The latter 3 can be purchased from amazon.
Finally, there’s the Cambridge school mathematics project. Occupies the same niche as UMM, perhaps even more rigorously. More like a real analysis/abstract algebra sequence disguised as a book for high school children. Can be grasped by advanced students, and is more rigorous than AOPS for whatever that’s worth.
If there’s any tendency you’d like to incorporate in your education, whether that be difficulty, rigor, etc. I guarantee that there’s some program somewhere that takes that principle to its logical extreme. For me, however, AOPS is the perfect balance of rewarding, challenging, pedagogical, and rigorous.

 

[MidgetDwarf]

For Trigonometry: Look up the author S L Looney. It can be hard to read at points, but I learned a lot from it. Granted, when I read the book I was taking Calculus 3.

I’m sure it’s an easy read now for me. I good reminder is that what one sees as difficult may not be so in the future.

So I would suggest Euler. If Euler is too difficult or not appealing. Simply put Euler on your book shelf and return to it on a later date. Maybe Lang Basic Mathematics fits the bill for a rigorous pre-cal class.

 

[sahilmm15]

Hi Mathwonk,
I just wanted to add that it wasn't my intention to be argumentative in my previous post. I really appreciate you taking the time to answer my post, and also, letting me know of the AoPS books lack of rigor. I edited the post, btw. I guess,

I posted it that way because I have made quite a few similar posts in the past and ended up spending my time having to rephrase my question in hopes it will urge the answer(or) into actually answering the question I actually asked, all while they spend their time defending the incomplete answer that they have provided..lol.

I apologize and now see that you were just trying to help. In all honesty, I was actually really wondering, since Euler's book covers the Algebra portion (I assume, covers it more than enough to be ready for Calculus. Please let me know if it doesn't.), what other book of the same caliber, would be good choices to cover the remaining bases, so one could be confident in their foundational ability and be completely ready, or even more than ready, for Calculus?

Thank you in advance for the help.

Here is a book on Trigonometry. It clears basics first and climbs all the way to up.https://www.stitz-zeager.com/szct07042013.pdf