Foundations Rigorous Alternatives to AoPS for PreAlgebra-PreCalculus?

[Chandller]

Hi PF,
If one were so inclined to self-study PreAlgebra - PreCalculus, with textbooks of the very same caliber and rigor of the Art of Problem Solving books, no matter the cost, which textbooks would be the best choice? Thank you in advance.

Chandller

 

[mathwonk]

have you looked at Euler's Elements of Algebra? available free online as a download. Very old, but very high caliber and very elementary explanations, starting from ground zero.

this format is a little tedious, but mauybe you can find better ones:
https://archive.org/details/ElementsOfAlgebraLeonhardEuler2015/page/n57/mode/2up

Another book I have enjoyed but on a rather higher level than euler's is the lectures of LaGrange.
https://www.gutenberg.org/files/36640/36640-pdf.pdf

On p. 20 or 21 for instance he shows how to do subtraction by doing instead addition! O iahev never seen this anywhere else until today when I glanced at it to see whether to recommend it to you. So I lways learn something from Lagrange, but Euler is more suited to your purposes of a rigorous, complete and elementary, course of arithmetic, logarithms, and algebra.

To compare levels of rigor, the AOPS series seems to be written by bright math majors from schools like Princeton, whereas the books suggested here are by historically renowned research mathematicians of the first caliber. Euler's book is the only one I know of wherein such a great mathematician has undertaken to treat such an elementary subject. (Euclid's Elements is also a fine elementary treatment of plane geometry, but Euclid, although a great teacher, is not considered a top caliber research mathematician.)

 

[Chandller]

Hi Mathwonk,
I have heard of that book but have never really looked through it. Seems like it would be a great starting point. Other idea's?

have you looked at Euler's Elements of Algebra? available free online as a download. Very old, but very high caliber and very elementary explanations, starting from ground zero.

 

[mathwonk]

A nice, modern book is the Elementary algebra book by Harold Jacobs, that I used with my own young son, but it is out of print. It seems publishers prefer producing whatever is used in the state of Texas public school system, over what is good quality, since they make more money that way. I will look for a link to Jacobs.

here you go, if you want this, you might jump on this price, $15, but I see the condition is not so great, i.e. only "acceptable":

https://www.thriftbooks.com/w/elementary-algebra_harold-r-jacobs/248035/item/4398210/?mkwid=%7cdc&pcrid=395931773744&pkw=&pmt=&slid=&plc=&pgrid=80257881302&ptaid=pla-838159019196&gclid=EAIaIQobChMIzK7q0ta96AIVGh6tBh1z2AArEAQYAyABEgLSMfD_BwE#isbn=0716710471&idiq=4398210
Jacobs covers less than half what euler does, but it is really easy to learn from, and is entertaining, as it was aimed at children, so has lots of cartoons, which i myself enjoyed.

 

[mathwonk]

here is a link to the famous abebooks site for used books, where other copies, some revised editions, are available. I myself usually prefer the first edition of a book, as not dumbed down to sell more.

 

[Chandller]

Hi Mathwonk,
I had actually put up a thread about a PreCalculus book by Warren Esty and I mentioned that I had been out of college for a while, that I was a non-traditional Applied Mathematics student at college Trig level, and that I was having to take a break from school for a while and wanted to review/redo Pre-Algebra through PreCalculus with rigorous textbooks to secure my Mathematical Foundation for my future in Mathematics and TurboDiesel gave me some spectacular suggestions, especially geared for Applied Mathematics/Engineering. I am actually taking Applied Mathematics (It is the only Math my college offers.) but I really love Pure Mathematics, and I really love Contest/Problem Solving Mathematics. I had also mentioned Hower's Pure Mathematics Preparation Thread/List, which is spectacular, as well, and what I wanted to ask in my post here was, what would be a textbook list, that would take me through PreAlgebra through PreCalculus, that had that same rigor and attributes as the AoPS books. Finally, their cost is of no concern, as scanned and older versions can usually be found pretty easily. Thank you again for the help.

 

[mathwonk]

My suggestion of euler was indeed intended to answer exactly your question, of mimicking at least the rigor of the AOPS series, but since it is impossible to mimic something exactly, i made a best approximation. that is why i gave you the comparison of the credentials of the AOPS authors as well as those of the authors i recommended. My own acquaintance with the AOPS series is with their geometry, from which I myself learned something, but in which I also noticed a major error of lack of rigor. So I recommended books whose rigor, or at least depth, is likely to exceed that of AOPS.

I wonder why you don't just use the AOPS series if that is your ideal. As far as mathematical quality is concerned, I believe Euler is unexcelled. His book starts from scratch and goes up through cubic and quartic equations, which AOPS does not come near, to my knowledge. But if rather than just understanding the mathematics deeply, as presented by a great master, you want a presentation that resembles what contemporary courses will present, you should study a more recent book. In particular, you should not choose the book I think best for myself, but the one, or ones, that speak to you most clearly and compellingly, and only you can make that choice. good luck.

 

[Chandller]

I actually just got a hold of the ebook and at first glace, I don't see any Trigonometry in Elements of Algebra. I am not seeing any Geometry, either. With those not included, what would be some great books, of the same caliber as Euler's book, that would fill those holes, so one could be completely prepared for Calculus?

 

[mathwonk]

I think you would be better off with the actual AOPS series. If you like the Esty book, although I have no access to it, th review at amazon is very positive, and he does have a PhD in math and is a college professor of math. good luck in your search.

 

[Chandller]

Hi Mathwonk,
Thank you. And, thank you for your help.

Chandller

 

[Chandller]

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.

 

[mathwonk]

The most important prerecquisite for calculus is algebra, and Euler covers that, possibly in too much detail for your purposes, and at too much length. Perhaps Jacobs would be better. As to preparation for solving problems and taking contests, there is a difference of opinion as to whether practicing solving sample test problems is as useful as spending time understanding the material deeply. I would say for most contests practice is useful, but for tests with more difficult problems, deep understanding to be able to attack unexpected problems is better.

Euler does not discuss trigonometry or geometry. Euclid is my favorite source for these, but you will not easily detect any trigonometry in Euclid, unless you understand very well what trigonometry is. As taught today, it consists merely of a collection of formulas and exercises involving applying those formulas to problems involving triangles. In essence however trigonometry is merely the relation between angles, i.e. lengths of arcs of circles, and lengths of sides of triangles having those angles, or subtending those arcs.

E.g. the famous law of cosines, tells you how to solve for the third side of a triangle when given the other two sides and the angle between them. It should be understood as a generalization of the famous Pythagorean theorem, and in fact is proved in Euclid shortly after that wonderful result. To discuss it without bringing in circles, one only needs to know that if two segments form an angle, and we drop a perpendicular from the end of one segment onto (an extension of) the other segment, forming a right triangle, then the distance of the foot of that perpendicular from the vertex of the angle equals the length of the first segment times the cosine of the angle.

Hence Propositions 12 and 13 of Book II of Euclid state the two cases of the law of cosines which generalize the Pythagorean theorem of Prop. 47 Book I.

There are no formulas here, and hence no visible "trigonometry", but these are my favorite versions of the law of cosines, since I myself prefer geometry to formulas.

In fact since the values of trig formulas like sin, cos, and tan, cannot be readily calculated by hand, except for angles which are multiples of π/6 and π/4, (including some rational multiples, thanks Vanadium!), trig calculations cannot in general be carried out at all by hand, and can only be approximated by calculators, or by using infinite series, as Euler did, in his "precalculus" book: Introduction to the analysis of the infinite. I.e. Euler considered the study of infinite series to be prerequisite to study of calculus. Euler's precalculus book however is quite challenging and not rigorous in the sense of modern treatments, i.e. it lacks a modern theory of convergence, so will not be compatible with what one sees in schools.

His results however are amazing, and correct, e.g. his accurate computation of the decimals of π on the first page of chapter VIII, to over 100 places. (An error was introduced in about the 113th place, presumably by the publisher, since the remaining digits are correct.)

So to Euler "precalculus" meant those topics he thought one should master before calculus, whereas to day it means whatever you find in a common book with that title. E.g. there are no "vectors" in Euler or matrices, as those had not been invented yet.

So it depends how much leisure you have. If you are prepping for some contemporary courses you may want to stick to contemporary books, but if you want to explore the mind of a master, some pages in Euler may be helpful, e.g. even the first few paragraphs of Elements of Algebra, wherein he explains what quantities are and how a choice of units allows numbers to be used to measure them, and hence the power of mathematics in science.

Enjoy! and thanks for your generous message.

 

[Vanadium 50]

In fact since the values of trig formulas like sin, cos, and tan, cannot be calculated by hand, except for angles which are multiples of π/6 and π/4, trig cvalcul;ations cannot in general be carried out at all by hand

I disagree:

\sin\left(\frac{\pi}{60}\right) = \frac{\sqrt{8-\sqrt{3}-\sqrt{15}-\sqrt{10-2\sqrt{5}}}}{4}

And indeed there is a closed-form solution for sines and cosines of any multiple of π/60.

In high school we had to do, I think, all multiples of 6 degrees. Numerically, so we didn't have radicals in radicals. It was a long problem, but not a particularly hard problem. I am not sure it was particulalrly rigorous. Just lengthy.

 

[mathwonk]

As Vanadium remarks, with some effort, one can compute cos and sin also of π/10, as well as sums and differences of the basic three, such as π/10 - π/4 + π/6 = π/60, and there are also half angle formulas and so on, that extend these basic examples. nice!

comments on general trig:

Trigonometry is about measuring angles. How do we assign a number to measure the size of an angle? And how do we even specify an angle?

Start from an x,y axis system on the plane, and draw the segment of length one from the origin (0,0) to the point on the positive x xis, one unit away, i.e. with coords (1,0). This is to be the base of our angle.

Now to specify an angle, we want a second side, a segment also of length one, from the origin to a second point (a,b), and take it in the first quadrant. Now we have an angle formed by these two oriented segments, (0,0),(1,0) and (0,0),(a,b). Such oriented segments may be called vectors, if you want.

Now how big is the angle? Notice that as the angle gets larger (remaining in the first quadrant), the coordinate b gets bigger. But if we take the angle twice as large, b does not get twice as big, so b is not a good measure of the angle, in that sense. On the other hand as the angle gets larger, a gets smaller! We can see that actually a and b measure the size of the perpendiculars dropped from the upper end of the angle, to the two axes.

If you know some trig, you know a and b are called the cosine and the sine of the angle. Since they do not vary linearly with the angle, if we want to use these coordinates to specify our angle, a basic question is to ask, how do a and b change s we say double the size of the angle? That is, if the angle is called s, how do we express cos(2s) and sin(2s) in terms of cos(s) and sin(s)? This is solved by some of the basic formulas, the “double angle formulas”.

Thus in this sense, these formulas let you go from knowing the end point (a,b) of an angle to knowing the endpoint of twice that angle.

A more natural measure for the size of an angle, is its “radian measure”, which is the length of arc of a unit circle, centered at (0,0), which is cut off by the two sides (0,0,(1,0), and (0,0),(a,b) of the angle. But this is hard to measure! This radian is expressible as a function of the numbers a, and b, by means of the functions arccos and arcsin. I.e. the arccos function, or equivalently the arcsin function, can be viewed as the arclength function on a circle. If you give me a number a, and draw a vertical line through (a,0) until it hits the unit circle at a point p = (a,b), then the arclength of the part of the circle between (1,0) and the point (a,b), is arccos(a).

Then if, in reverse, we want to start from an angle of say s radians, spanned an arc on the circle reaching from (1,0) to another point p on the circle, and we want to find the coordinates of the point p, we need to evaluate the inverse functions, the cos and sin of s. This is not at all easy to do, except for the simplest of angles, like 30degrees, 45degrees, and their multiples.

So most trig books limit themselves to those angles, but not Euler! He writes down actual formulas for those functions, albeit necessarily infinite series formulas;

E.g. cos(t) = 1 - t^2/2 + t^4/24 - t^6/720 ± …….; sin(t) = t - t^3/6 + t^5/120 ±……

and formulas for inverses of trig functions, like arctan(t) = t - t^3/3 + t^5/5 ±……

And then he calculates an amazing array of results and measurements using these, such as the value of the first 100 or more decimals of π.

To deal purely geometrically with sines and cosines, one needs the law of cosines, and it helps to have a little vector notation, to calculate lengths of segments and cosines of angles, using the Pythagorean formula. E.g. the angle spanned by the arc on the unit circle from (a,b) to (c,d), has cosine = (ac+bd). Anyway, see how much of this is explained in your various books!

As Vanadium stated, one can expand significantly on the basic calculations, using formulas for sin(t/2), cos(t/2), etc...e.g. from his formula one can write down cos and sin of π/(2^n.60) for all n...

Of course the ones one usually encounters on a test are the simplest ones based on the triangles of sides 30/60/90, and 45/45/90. The ones requiring an (admittedly constructible) triangle of sides 36/54/90, are not as readily produced, although they do exist, and arise when constructing a regular pentagon, or if one knows the 5th root of unity is (sqrt(5)-1)/4 +i sqrt({5 + sqrt(5)}/8). I.e. if one has the cosine and sine of π/4, π/6, and π/10, from knowing the sides of these triangles, one can find the cosines and sines of their sums and differences, such as π/10 - π/4 + π/6 = π/60. I don't know if Vanadium has an easier approach in mind?

I guess since Gauss constructed the heptadecagon, theoretically we might be able to also express the cosine and sine of π/17 in terms of square roots? and π/257, π/65537?

 

[Vanadium 50]

I guess since Gauss constructed the heptadecagon, theoretically we might be able to also express the cosine and sine of π/17 in terms of square roots

\cos\left(\frac{\pi}{17}\right) = \frac{1}{8} \left( 30+2\sqrt{17} +\sqrt{136-8\sqrt{17}}<br /> +\sqrt{272-48\sqrt{17} +8\sqrt{32-2\sqrt{17}}\times (\sqrt{17}-1) - 64\sqrt{34+2\sqrt{17}}}\right)^{1/2}

I confess I looked it up. The key, of course was knowing that there was a solution to look up.

 

[Vanadium 50]

The ones requiring an (admittedly constructible) triangle of sides 36/54/90, are not as readily produced, although they do exist

It's not too bad. You can do it with a regular pentagon, but a 36-72-72 triangle is maybe a bit simpler, if less obviously motivated. Draw an angle bisector of one of the 72 degree angles to the opposite side, and proceed with similar triangles. It helps to know what you're looking for - the golden ratio - which suggests why similar triangles might be helpful. Similarly, using the complex 5th roots of unity also gets you where you need to be (along with a pile of unlikely looking but true formulas like sin(54) - sin(18) = 1/2).

Given all that, I think a more useful concept than "rigor" is the unity of mathematics. To me "rigor" means a certain fussiness about proofs. Nice if you're an i-dotter and a t-crosser, but not so helpful in solving problems.

 

[Stephen Tashi]

If one were so inclined to self-study PreAlgebra - PreCalculus, with textbooks of the very same caliber and rigor of the Art of Problem Solving books

Did you mean "rigor" in the sense of difficulty? - or in the sense of logical precision? I have only glanced at Art of Solving Problem type books. From that perusal, I find their approach inuititive, not logically precise.

It is very difficult to explain elementary algebra in a logically precise way without getting into advanced concepts. It is certainly possible to explain it in a manner that students find clear and "logical" by their own standards without using advanced concepts.

 

[archaic]

A series of two algebra volumes "Algebra: An Elementary Text-Book":
http://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=olbp36404
The second volume introduces the notion of limits and also dives in infinite series, which is taught in calculus 2. It really is a killer book.
Elements of coordinate geometry by S. L. Loney:
https://ia802701.us.archive.org/7/items/elementscoordin02lonegoog/elementscoordin02lonegoog.pdf
Plane trigonometry by the same author:
https://ia802700.us.archive.org/22/items/planetrigonomet03lonegoog/planetrigonomet03lonegoog.pdf

But you will also probably be just fine going through Khan Academy's videos and exercises.

 

[Vanadium 50]

and π/257, π/65537?

Yes. The constructions are ugly (radicals within radicals within radicals) but they exist. If the corresponding regular n-gon is constructable, the sine and cosine of π/n is expressible with radicals (within radicals, within radicals etc.). It is necessary and sufficient that n be the product of a non-negative power of 2 times the product of zero or more Fermat primes.

So the first n that does not work is 7. And π/4294967295 will work, although I shudder to think of what it might look like.

Going back to the "unity of mathematics", the proof of this requires a little geometry, a little trig, a little number theory and a little abstract algebra.

 

[TurboDiesel]

Everyone here loves math. It makes me warm and fuzzy. :-)

I am currently a student, not a mathematics PhD, so please forgive any important point I accidentally omit, here. I am not arguing against any alternative approaches suggested here (all of which appear useful), just trying to fit AoPS into the picture that others masterfully constructed.

• First, based on my reading of Esty's book, I suggest you, Chandller, simply finish Esty and follow with the four books I suggested. His approximately 450-page book is carefully pedagogical without wasting words. The text is in-line, meaning you can flow easily through the examples without any jarring reorientation of your eyes. "A" problems seem to be there for whenever you need a bridge between the text and "B" problems. The smiley-face problems take few quick seconds to execute and are there to make sure you get the concept.

You might be able to finish most, or all, of Esty in the time it takes you to settle on an alternative. Give it a try. I dare you. ;-) You can always follow with AoPS in parallel to university mathematics.

Esty should be useful for self study and the four books I recommended will fill in everything else you need for straight A's in early STEM. You could technically skip precalculus and go straight to those books in my recommended order, but it would not hurt to take the extra time to finish a dedicated precalculus text. Apostol (one of those four) is probably the gold standard for 'rigorous' single variable calculus texts.

To spell it out:

1. [For clarity, I include this first option as it appears to be suggested above. I add Zorich, Strichartz, & Pugh.] Euler's "Elements of Algebra" + Euclid's complete* "Elements" + Euler's "Introduction to the Analysis of the Infinite" are good for a brain-building dive into ancient wisdom and true genius. (Consider following with the excellent Zorich + Strichartz + Pugh.**) Do this if your time is unlimited and your passion for mathematics is ever present. This is absolutely beautiful stuff. I do not think this is likely to be a best fit for your needs. (however, after Apostol, it seems to me that Zorich et alia are a very good way to go through the analysis sequence.).
2. Jacobs + Cohen = inspiring "first look" at high school mathematics.
3. Howers' list builds the Jacobs approach out a bit with some books that are very nice and make you think. It is a great way to go. However, since it requires more effort to locate all these resources at decent prices, this path is easier to face seeming obstacles on. If you are easily side-tracked, simplicity of path is worth keeping in mind.
4. My suggested path--of precalculus (of your choice; e.g. Esty) + four book--builds out your mathematics (allowing straight A's), and then makes them rigorous (starting with Apostol). If you work hard at this path, you are basically immune to failure and have a good chance at scholarships. It gives you essentially zero "I don't have the right book" excuses to stop.
5. AoPS Prealgebra through Precalculus bulds up your mathematical thinking (and joy in math) from the ground up to the point where you could reasonably go directly to Apostol or Spivak (this is not an option the vast majority of curricula allow most to achieve).
6. You can mix in a more challenging path with an easy path.
   • E.g. AoPS "Prealgebra" with Jacobs' "Elementary Algebra," AoPS "Algebra" with Jacobs' "Geometry," etc.
   • E.g. Jacobs "Elementary Algebra" with Gelfand's "Algebra" followed by Euler's "Elements of Algebra," etc.

• Howers list was carefully compiled from another thread including texts, such as those of Harold R Jacobs, which I believe mathwonk endorsed (at least one of) in the original and very long thread. I have not seen a better "first look" at high-school algebra and geometry than those presented in Jacobs' texts ("Elementary Algebra" and "Geometry"). I would propose David Cohen's "College Algebra" and "Precalculus: With Unit Circle Trigonometry" as "first look" texts that fit well after Jacobs.

• A quick point from the author angle. The guy in charge of the curriculum at AoPS has a PhD in mathematics from MIT and was a top 10 finisher in the Putnam. By way of comparison, Putnam winners include Richard Feynman and Kenneth G Wilson--both Nobel Prize winners. I understand Euler published the plurality of science and math produced while he was a living researcher, so it is hard to compare the work of anyone in existence with the sum of his accomplishments.

Another Putnam anecdote is that two Microsoft notables, Bill Gates and Steve Ballmer finished in the top 100.

Understanding the Art of Problem Solving curriculum
A key point is that AoPS inspires joy in learners. Joy inspires self study. On this metric, I believe they have Euler's "Introduction to Analysis of the Infinite" beat as his students notably complained. [EDIT, from below: "I think I mixed up student's reception to Euler's 'Introduction to Analysis of the Infinite' with Cauchy's student's reception to Cauchy's 'Cours d'Analyse.'" See below post for brief comment.]

Regarding rigor, solving devious problems requires mathematical thinking. AoPS's problem solving approach to mathematical thinking makes building out rigor a trivial exercise at the calculus level, instead of the major obstacle it is after many typical alternative approaches that have resulted in most colleges typically using computational texts (e.g. Stewart) instead of Apostol. I.e. there is no "abstract" barrier after finishing AoPS.

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.

Footnotes:
* Much of Euclid's Elements can be very useful in a standard high school geometry class (mine did, anyhow). Later parts require more effort.

** Vladimir Zorich's "Mathematical Analysis I" and "Mathematical Analysis II" (use as first look and reference in Analysis)
https://www.amazon.com/dp/3662569558/?tag=pfamazon01-20
https://www.amazon.com/dp/3662569663/?tag=pfamazon01-20
Strichartz "The Way of Analysis, Revised Edition" (use as a substitute for in-class lectures)
https://www.amazon.com/dp/0763714976/?tag=pfamazon01-20
Charles Chapman Pugh's "Real Mathematical Analysis" (reputedly excellent preparation for graduate analysis)
https://www.amazon.com/dp/3319177702/?tag=pfamazon01-20
Consider supplementing with a look at the gauge integral.

 

[mathwonk]

@Vanadium, re: post #19
Yes, I have seen some of this. Here is what I found in Mike Artin's Algebra: Since constructions involve intersecting lines and circles, algebraically this means solving equations of degrees one and two, so by field theory at each stage the coordinates of the points lie in a possibly larger field extension of degree 2 of the given coordinate field. Since successive extensions have degrees that multiply, a succession of these extensions has degree 2^n, and adding in a sqrt of -1 also has degree 2 again. Then we ask whether a primitive pth root of 1, for say p a prime, lies in such a field, which implies the equation solved by such a root, namely x^p - 1 = (x-1)(x^(p-1) +x^(p-2)+...+x+1) = 0, has solution in a field extension of degree a power of 2. The irreducible part of this equation, for p a prime, is the factor of degree p-1, which thus must divide the degree of the field extension, so for this it is necessary that p-1 is a power of 2, i.e. that p is a Fermat prime. But I don't know the proof that it suffices for constructibility of a regular p-gon that p is a Fermat prime.

I guess the problem is that a solution could lie say in a field of degree 2^n, but one that is not obtained by successive extensions all of degree 2. It is probably a Galois theory result that that does not happen, (whereas the previous result is more elementary field theory, i.e. field theory with no group theory involved). Yes, it seems to follow from the fact that if p is a fermat prime, then the field generated by the primitive pth root of unity has cyclic Galois group of order a power of 2, and then one invokes the main theorem of Galois theory connecting those subgroups with the intermediate field extensions.

And thanks to Turbodiesel for the wealth of precalculus/algebra/geometry/calculus candidates! I have sometimes felt that a major problem for me at least, is actually getting the material up off the page, where it is essentially frozen in place, and into my brain in a warm pliable form. For this I must simply try several different books and work at each until I find one that actually speaks to me, or allows me to understand what it is saying. Sometimes even one sentence in a really good book helps enlighten me. I also often need to put down even an excellent book, and refer to some easier source for background, and then come back, of my goal is really to finish it. I.e. there is hardly any book that suffices for me alone and all the way through. It also helps enormously to try to explain the material to someone else, as for some reason this forces me to understand it better, else I won't have an answer to questions.

 

[vanhees71]

I'm a bit puzzled about the fact that there are many recommendations for ancient textbooks to learn math from. That doesn't make sense to me. It's of course great to read classic text like Euler's algebra or Euclid's elements, but it's not a good idea to learn math from them since there are some centuries of mathematical progress between these books and modern textbooks, from which you learn, and that's not of least importance I'd say, how math is communicated today and not 300 or even several 1000 years ago!

 

[Vanadium 50]

Mathwonk, all my real books are locked away in my work office, which I am not allowed to enter because neither I nor the office has been exposed to coronavirus. Or something like that.

An argument for sufficiency goes like this:

1. If a and b are relatively prime, if π/a is constructable and π/b is constructable then π/ab is constructable.
2. If a is a prime and π/a is constructable, so is π/aⁿ.
3. Show by construction that it is true for a=2,3,5,17,257 and 65537.

The hard part appears to be #2. I would probably attack it by looking at minimal polynomials. (i.e. using fields indirectly).

This is very similar to the question "which angles can be trisected" - getting back to the ancient Greeks and vanhees' comment. Which I agree with on even more fundamental grounds - if one is looking for a textbook, one is looking for a textbook. Not a historical document.

 

[mathwonk]

@vanhees71: I thought that too, until I began to actually read some of those old books. Then I learned that, as far as elementary math goes, e.g. algebra and geometry, the modern textbooks are written mostly by people who are not as deeply aware of the ideas as were those ancient masters. Hence reading at least a standard textbook used in most US colleges, (I'm not criticizing Apostol of course, which is far above the usual standard,) one gets a slice off the top of the iceberg of the old treatment, and presented by an average student, rather than a master.

My first such experience was reading Galileo, who pointed out that the calculation usually made (in calculus!) showing that a projectile moves in a parabola under the influence only of gravity, is flawed since the Earth (which contains our assumed x axis) is essentially spherical and not flat, hence the "vertical" lines we are drawing would meet at the center of the earth! Not one calculus book in my possession had made that point.

Archimedes' derivation of the volume of a sphere was so clear that I was able to immediately generalize it to finding the volume of a "bicylinder" using only the Pythagorean theorem, a problem that today's students find challenging using calculus. It becomes clear that Archimedes completely understood the so called Cavalieri principle, and that many volume calculations use only this, not the fundamental theorem of calculus. Indeed calculating the volume of some solids is made harder rather than easier any trying to antidifferentiate a complicated formula that yields easily to the "Cavalieri" technique, or to the powerful Pappus principle, which is also ancient. Indeed I found that many of the modern calculus books we were forced to teach from in college omitted this powerful principle out of a, to me misguided, desire only to use methods that could be fully justified to an average student, as opposed to presenting techniques that can help them solve problems.

I agree completely with your last point, in many cases, that a modern student often has trouble grasping what is being said in an old text. Euler's text however may be a counterexample to this in my opinion. I readily admit I did not realize myself that Euclid was readable until in my 60's and after a lifetime as a professional mathematician and indeed geometer. I only tried it again after reading an article by the outstanding modern geometer Robin Hartshorne, who taught from Euclid to undergrads at Berkeley, and then wrote a modern text that helps guide the modern student through Euclid. I recommend strongly reading Euclid in combination with Hartshorne's Geometry: Euclid and beyond.

His article is also recommended:
https://www.ams.org/notices/200004/fea-hartshorne.pdf

Pardon this lengthy discussion. It may be true that only a few will enjoy the old books, but for those that may, and do not yet realize it, this is directed at you. bon voyage.

After seeing that Vanadium also is unconvinced, and that I am myself merely an example of a senior mathematician who has learned to appreciate old books, not a student who has learned fom them originally, maybe I should suggest only that after one has studied enough from modern books to understand what the old books were trying to say, than then one is possibly qualified to understand them, and to enjoy them. But if you never pick one up, you will not discover whether you too are someone who will benefit. I can only say that by reading great masters of olden times, I myself have enhanced my understanding of things I thought I had grasped from modern treatments, even treatments written by famous modern mathematicians.

Edit: Perhaps I should modify my claim that elementary algebra has not advanced much. That may be true, but of course algebra in general has advanced since Euler's time, e.g. the proof that a general quintic cannot be solved by radicals was not yet available to him I believe. But although things may be different say in physics, in mathematics the fact that algebra has advanced does not mean that algebra textbooks have also advanced along with the research. Indeed, and this again may be a phenomenon entirely peculiar to the United States, over the past 125 years or so,the typical mathematics textbooks used in most schools in the US have steadily declined in quality. So there is hardly anywhere a high school algebra textbook at as high a level as that of Euler. The same goes for geometry texts in comparison to Euclid, which apparently was universally used as a textbook for hundreds of years world wide. However around 1900, and progressively since then, it began to be abandoned in the US, not as outdated in content or quality, but as too hard for most US high school students.

Indeed if one goes back roughly 132 years one encounters a much higher quality textbook of high school algebra, that of Charles Smith, A treatise on Algebra, 1888, than virtually any used today in the US. I learned of this book from an article I have referred to on this forum before, written by my friend the outstanding high school teacher Steve Sigur: "The decline of American mathematics textbooks", in the Paideia School newsletter. He discussed and documented in his article, the steady decline of the high school algebra texts used commonly in the US over some 100 years, which he attributed to our focus on universal education, requiring the level to be lowered as we expanded the audience. He pointed out e.g. that only a small proportion of Americans attended high school in 1888, perhaps less than 10%, if I recall. I will try to find a copy of Steve's excellent article. Here is Smith's book:

https://archive.org/details/atreatiseonalge00smitgoog/page/n351/mode/2up

But I want to make the point that I do not advocate the use of old books simply because they are old. It is the books written by masters that are outstanding, and in mathematics at least, due to its immutable logical structure, these books do not become outdated scientifically. I.e. the mathematical content does not become replaced, only augmented. Hence if one can find textbooks written in the modern period and also written by masters of the subject, they are likely to be as good as the ancient masterworks. E.g. it was common in the Soviet Union not long ago, for top mathematicians to write elementary math books, and hence those might be excellent sources, but I am not so familiar with them myself. Here is one very elementary algebra book co-authored by Gelfand.

https://www.amazon.com/dp/0817636773/?tag=pfamazon01-20

There is also a linear algebra and geometry book co-authored by Manin, but I have not read it:
https://www.amazon.com/dp/2881246834/?tag=pfamazon01-20

There is also a book, that I have browsed in, co-authored by Shafarevich, on Geometries and Groups, that is excellent, and introduces spherical geometry:
https://www.amazon.com/dp/3540152814/?tag=pfamazon01-20

So if you are uninterested in books by old masters, maybe try some books, by modern masters.

enjoy!

 

[TurboDiesel]

Sorry guys, I think I mixed up student's reception to Euler's "Introduction to Analysis of the Infinite" with Cauchy's student's reception to Cauchy's "Cours d'Analyse." I don't recall the book I read it in (maybe Stillwell?) so can't immediately confirm which. I am updating that point accordingly (above) and will post a reply if I relocate the source. I do know Euler was known for explaining things very well, when he tried ("Letters to a German Princess").

Incidentally, Euler's only peer, in contributions, I am presently sure of is Archimedes. I have to wonder if the only reason they respectively didn't solve much more mathematics is because they didn't have the sounding board it took others tens or hundreds of years to create in response.

I thought that too, until I began to actually read some of those old books.

My HS geometry teacher used the first few books (chapters) of Euclid for a small part of my course. I did not personally find those early chapters challenging and it was pretty interesting to see that thoughts could be so clearly communicated across time.

And thanks to Turbodiesel for the wealth of precalculus/algebra/geometry/calculus candidates! I have sometimes felt that a major problem for me at least, is actually getting the material up off the page, where it is essentially frozen in place, and into my brain in a warm pliable form. For this I must simply try several different books and work at each until I find one that actually speaks to me, or allows me to understand what it is saying. Sometimes even one sentence in a really good book helps enlighten me. I also often need to put down even an excellent book, and refer to some easier source for background, and then come back, of my goal is really to finish it. I.e. there is hardly any book that suffices for me alone and all the way through. It also helps enormously to try to explain the material to someone else, as for some reason this forces me to understand it better, else I won't have an answer to questions.

Thank you mathwonk,

In my few posts here I try to connect several texts together because this kind of knowledge geolocation is how I personally relate to materials. This is what originally resulted in my discovery of PF, while searching for book comparisons.

I have valued your perspective on textbooks when relating them with each other. I originally picked up a copy of Allendoerfer and Oakley based on the broad positive reception here to your recommendation. I also looked at Euler's "Elements of Algebra" and it is quite clear, even if from a different viewpoint.

if one goes back roughly 132 years one encounters a much higher quality textbook

I think one contributing factor to this is that Geometry was taught at university for hundreds of years. It wasn't untll 1726 that Harvard required arithmetic for entrance after aquiring a mathematics professor. I believe it it is entrance to universities that resulted in the increase in the length of high school math curricula. See here:
http://jwilson.coe.uga.edu/EMAT7050/HistoryWeggener.html

 

[TurboDiesel]

Moving this addendum to a separate post for clarity:

if one goes back roughly 132 years one encounters a much higher quality textbook

I think one contributing factor to this is that Geometry was taught at university for hundreds of years. It wasn't untll 1726 that Harvard required arithmetic for entrance after aquiring a mathematics professor. I believe it it is entrance to universities that resulted in the increase in the length of high school math curricula. See here:
http://jwilson.coe.uga.edu/EMAT7050/HistoryWeggener.html

 

[Mondayman]

Basic Mathematics by Serge Lang is in the right style but is poorly organized in my opinion. Principles of Mathematics by Oakley and Allendoerfer is probably one of the better options.

This book has had some good reviews, it appears it has precalculus and calculus topics: https://www.amazon.com/dp/0395286972/?tag=pfamazon01-20

It's slim pickings for rigorous precalculus texts. I found working hard through a generic precalc text alongside Khan Academy to be sufficient for learning calculus. I think a combination of Oakley and the AoPS would be a decent start.

 

[mathwonk]

@TurboDiesel : Your post and the linked history on my friend Jim Wilson's page at UGA has caused me to check further into how commonly was the excellent book by Charles Smith used in US high schools in the late 19th century, an impression I got from Steve Sigur's article. It states plainly in the introduction to Smith's algebra book, that he was a professor at University of Cambridge, England, and that his book is aimed at students in the "higher schools" as well as junior students at universities. This makes me doubt that such a book was in fact much in use in high schools in the US at the time. A quick glance through it reveals such topics as the binomial theorem for non integer exponents! (He credits the proof to Euler, by the way.) I myself was challenged to learn the statement of that theorem for positive integer exponents in high school. It also treats number theory including Fermat's and Wilson's theorems. It also, in later editions, has a treatment of the intermediate value theorem, and Rolle's theorem from calculus. I wish I knew what my dad studied in the way of math in his high school, from which he graduated in 1906.

 

[mathwonk]

By the way, here is my explanation of how Archimedes' method leads to computing the volume of a 4-ball, a calculation I have not seen in any calculus book, at least in this way, and which I think is easier than what most modern calculus books present.

Archimedes’ calculation of the volume of a 3 dimensional sphere:
Archimedes said that a ball is essentially a cone whose base is the surface of the ball and whose vertex is at the center, hence whose height is the radius of the ball. Hence the volume of a 3 ball equals R/3 times the area of a 2 sphere. Moreover just as a 3 dimensional cone has volume equal to 1/3 the product of base area time height, the same argument by decomposing a cube, shows that a 4 dimensional cone has volume equal to 1/4 the product of the base times height, hence the volume of a 4 ball equals R/4 times the volume of a 3 sphere. Thus computing the volume of a 3 sphere is equivalent to computing that of a 4 ball.

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 do a calculation of the volume of a 4 ball that Archimedes also could have done. 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. (He did this by inverting the cone and then comparing the areas of corresponding slices, concluding via “Cavalieri’s principle” which he also knew.) Hence the volume of a 4 ball equals the difference of the volumes generated by revolving a cylinder and an inverted cone.

Now the center of mass of a cylinder is obviously half way up the cylinder, and Archimedes knew that just as the center of mass of a triangle is 1/3 of the way up from its base, the center of mass of a 3 dimensional cone is 1/4 the way up from its base.

Thus we can use centers of mass and subtraction to get the volume of a 4-ball. I.e. a cylinder of height R and base radius R has volume πR^2.R, and center of mass at height R/2, so revolving it (in 4 space) around an axis (plane) containing its base gives 4 dimensional volume of 2π(R/2).πR^2.R = π^2.R^4.

The inverted cone of height R and base radius R has center of mass at distance 1/4 of the way from its base, hence distance (3R/4) from the axis plane, and volume (1/3)πR^2.R. Thus revolving it generates a 4 dimensional volume equal to (2π)(3R/4). (1/3)πR^2.R = (1/2)π^2.R^4. Subtracting the volume of the revolved cone from that of the revolved cylinder, gives the 4 dimensional volume of the revolved half 3-ball, i.e. the volume of the full 4- ball, as π^2.R^4 - (1/2)π^2.R^4 = (1/2)π^2.R^4.Volume of the 3-sphere, i.e. S^3, (not S^2).
To get the 3-dimensional volume of the “surface” of the 4-ball we can use Archimedes’ relation (that a ball is essentially a cone whose base is its surface and whose height is its radius), and just multiply the 4-dimensional volume of the ball by 4/R. This gives 2π^2R^3 as the 3- dimensional volume of the 3-sphere of radius R.

 

[Vanadium 50]

Wilson's theorems

To continue to ride by hobby horse on the unity of mathematics, have you seen the geometric/combinatoric proof of Wilson's theorem in Andrews' book? Jeepers, it's pretty.

 

[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