s/o math sequence for accelerated student?

[songsparrow]

elementary number theory (like why 5 is a sum of 2 squares but 7 is not), can be studied at almost any time after basic arithmetic and geometry, and are fun and stimulating.

 

This sounds interesting - any suggestions for resources to use?

[kiana]

This sounds interesting - any suggestions for resources to use?

 

 

AOPS has written a textbook that would be a good start. For following up, I enjoyed this book very much: http://www.amazon.com/Elementary-Number-Theory-David-Burton/dp/0073383147

but I would definitely definitely definitely get an older edition. You can get the fifth edition used for 3.42. Before you advanced to Burton's book, though, I would say you should have algebra 2 under your belt.

 

Some number theory textbooks have more prerequisites than others. If you run into a specific book at a used-book store or something, I can look it up.

[Reya]

Has anyone ever come up with a post-Algebra sequence that doesn't go through the typical high school stuff first? Right now, DD is saying that she wants to go back to PS for high school, and if she does, she'll have to have at least two high school math classes she hasn't yet taken before she's eligible for Dual Enrollment, and will have to be able to pass the exit exams for whatever high school classes we're listing on her transcript before high school so she won't have to start with Algebra I. Because of this, DH (who's our resident mathematician-I'm certified as a math teacher, but he went a lot farther in math than I did) really wants to take the next few years and focus on other areas of math, then go back in Middle school and do the first several years of the traditional US sequence, sitting for the end of course exams locally so she gets high school credit. He likes the Elements of Mathematics sequence as laid out on the website, so we're going to try the intro course after Christmas, but I'm also looking for other ideas. We've taken a scattershot approach to math overall for years, where we've had a couple of main resources and a bunch of additional ones, so the idea of putting too much faith in a web-based class that isn't even fully published yet is downright terrifying (especially since some of the areas of math he wants DD to cover are ones I haven't actually taken a class in myself).

 

 

EMACs is pretty awesome, if it fits your kid! Also, I REALLY didn't find the first books hard at all. I've had advanced math, but I wasn't a math major or anything.

[mathwonk]

This sounds interesting - any suggestions for resources to use?

 

 

yes, I'll look for some on my shelf to supplement the ones suggested here. But if I were doing it myself, I would not start with a book at all. The beauty of number theory is how elementary it is, it is just about studying the whole numbers and everyone knows them to some extent.

 

I.e. first I would explain what a prime number is, and ask the child to find all the primes up to 100. Next I would explain what a square is, and ask him/her to try to write each prime as a sum of two squares.

 

There will result a list of successes and list of failures and then I would ask him or her to try to guess what the successes have in common and similarly the failures. Only after this is done would I even mention how to prove the resulting fact.

 

This is how real math proceeds,1) investigation of phenomena, then 2) conjecture, then finally 3) logical analysis and proof.

 

Ok, I found the book our son used at TIP in his number theort class:

 

Elementary Number theory by Underwood Dudley:

 

http://www.amazon.co...nderwood dudley

 

 

I have house guests and have to go soon, but I also want to comment on sequences of math courses at some point. I.e. there are methods and there are examples in math. The methods can be applied to almost any type of examples.

 

Fundamentally there are three types of math methods: algebra, geometry, and (infinitesimal) analysis. algebra deals with the structure and behavior of binary operations, like adding and multiplying. Geometry deals with shapes, and analysis deals with approximations of finite quantities by an in finite number of other quantities.

 

so algebra is not intrinsically about x's and y's, rather the algebra of polynomials is a particular example of the study of algebraic operations applied to polynomials. there is also an algebra of rigid motions in space, and an algebra of permutations of objects in various orderings.

 

There are also many kinds of geometry e.g. Basic Euclidean geometry deals with the idea of congruence, where two circles of different radius are considered different. next the idea of similarity is introduced and any two circles are now considered equivalent. In topology not only are all circles considered equivalent but also they are all equivalent to all ellipses, and even the boundary of a kidney bean, or even to any closed plane curve at all that does not cross itself.

 

One can study any one of these three branches of math without knowing the other, so there is no reason at all to necessarily study calculus (a branch of analysis) before studying linear algebra or abstract algebra. This is done just to keep students moving along in different fields sort of uniformly, and sometimes to please the engineering department or for other political reasons to please the faculty who teach these subjects.

 

Once the subjects are learned to some extent, they can be profitably combined, and there are at higher levels hybrid versions of the subjects, like differential geometry, or algebraic geometry, and analytic number theory, but it helps to know the pure subject first.

 

 

But when it comes to making application of the ideas to specific examples, then of course one needs a knowledge of those examples. Calculus e.g. is usually applied to geometry, to find areas and volumes, and at that point it is very helpful to know some geometry. In the other direction, the ancient geometers knew things which are also fundamental ideas of calculus, so understanding calculus is easier if one has had a thorough grounding in geometry as it was understood especially by Archimedes.

 

E.g. a basic fact about area and volume, is that fact that two plane figures which meet every horizontal line in line segments of the same length must have the same area. Similarly two solids which meet every horizontal plane in plane areas of the same size, must have the same volume. using this principle, (known incorrectly historically as the Cavalieri principle since it was known to Archimedes), Archimedes determined the volume of sphere and also of the region common to two perpendicular cylinders. these volume problems are much more difficult and less intuitive to do using calculus than by Archimedes method, but few students today ever see it done as he did.

 

differential calculus is abstractly a way of going from a height function to a slope function, i.e. from knowing a curve to finding its tangent lines at various points. to apply it to specific types of functions, one needs to know some concrete classes of functions. thus one can apply calculus to polynomial functions, and also to trig functions if one knows about these and also to exponential functions if one has studied those. but if one only knows polynomial functions one can certainly do differential calculus on those alone without knowing trig. this is a nice way to begin.

 

The connection between integral and differential calculus is the fact that the height of a curve is also the slope of another curve, the curve which is the graph of the area of the original function. I.e. the slope of the area function is the height function. Also the slope of the volume function of a solid is the area function for a horizontal slice of the solid.

 

In application this means that one can go from knowing a formula for the area of all horizontal slices of a solid to finding a formula for the volume of the solid. This is an improvement on Archimedes since all he knew how to do was to compare the volumes of two different solids by comparing their slice areas. Thus he had to bootstrap by knowing the volume of one solid and then use that to find the volume of another solid with the same slice areas. He could not work just with the one solid itself. Still he was very clever and was able to show as his crowning achievement that the volume of a hemisphere is the difference of the volumes of a cylinder and a cone. Since he knew both those two volumes he got the volume of a hemisphere, hence also a sphere.

 

In modern calculus courses we challenge students to compute the volume of the common region between two perpendicular cylinders, but Archimedes saw that this problem was analogous to the previous one and showed it equals the difference of the volumes of a cube and a square based pyramid, and just subtracted to get it. I spent most of my adult professional life torturing students with that tricky calculus problem before learning that Archimedes knew how to make it easy what 2,000 years ago?

 

We to often teach today simply what is in the books, rather than teaching from a larger perspective of what has been accomplished over the centuries. Of course it takes us a lifetime to learn this material that broadly, and then we struggle to find an outlet for it. I am grateful to have this forum myself to toss out suggestions based on over 50 years of learning and enjoying math.

[mathwonk]

here is a brief discussion on the theorem of two squares, aimed at students who have seen some "modular arithmetic" or :"clock arithmetic". I.e. in this arithmetic we count by twos or threes or fours or (in clock arithmetic) by twelves, etc. In mod 4 arithmetic, we repeat every time we reach 4. so 5 is 1 mod 4, and 7 is 3 mod 4, and 95 is 3 mod 4. Also 4 is considered as 0, since adding 4 does not change a modular number, so 3 is also -1 mod 4, etc... In particular -2 has a square root, mod 5, since 2^2 = 4 = 0-1 mod 5. anyway here is the argument, oops all the exponents fell down, so a2 means a^2, etc....: and every b followed by a parenthesis seems to be a wry smile.

 

modular arithmetic itself is a great topic for beginning number theory.

 

actually it seems now i can attach this document as a pdf file. no it won't let me.

 

 

Fermat’s Theorem on Sums of Two Squares

The secret to understanding which integers are sums of two squares lies in introducing complex (“imaginaryâ€) numbers. All numbers exist in our minds of course, hence are imaginary so we no longer use this old fashioned term for them, except informally or out of habit. Just as real numbers correspond to points on the line, complex numbers correspond to points on the plane, so just as it takes two real coordinates to describe a point in the plane, it takes two real numbers to describe a complex number.

 

We define a complex number to be an ordered pair of real numbers (a, B). We add them by adding each coordinate separately, i.e. (a, B) + (c,d) = (a+c,b+d). The multiplication is more complicated, and we define (a, B)(c,d) = (ac-bd, ad+bc). In fact our new multiplication is commutative and has all the other properties you would expect of a multiplication (associative, and especially distributive).

 

The definition can be made easier to remember as follows: notice that in our multiplication the complex number (1,0) is a multiplicative identity, since (1,0)(a, B) = (1a-0b,1b+0a) = (a, B). Thus it is natural to call this number “1â€, i.e. to set 1 = (1,0). On the other hand the complex number (0,1) has the interesting property that when you square it you get -1, i.e. (0,1)(0,1) = (0-1,0+0) = (-1,0) = -1. This is news, since we never had a number whose square was -1 before.

 

If we denote this interesting complex number by “iâ€, i.e. if we set i = (0,1), then every complex number can be written in one and only one way as a linear combination of 1 and i as follows: (a, B) = (a,0) + (0, B) = a(1,0) + b(0,1) = a1 + bi = a+bi. From now on we drop the clunky notation of (a, B) for a complex number and use a+bi instead, where a and b are real numbers.

 

Now the multiplication is easy to remember, because all you need to know is that i^2 = -1. I.e. now (a+bi)(c+di) = ac + adi + bic + bidi = ac +adi + bci + bdi^2 = ac +adi + bci -bd = (ac-bd) + i(ad+bc), and this is the same answer we got before for this multiplication, namely (ac-bd, ad+bc).

We are interested only in complex numbers a+bi where a and b are both integers, and we call them “Gaussian integersâ€. They are the complex version of the integers.

 

The useful property they have in relation to the problem of the sum of two squares is this: if n = a^2 + b^2, where n, a and b are natural numbers, then we can factor n as n = (a+bi)(a-bi) in the Gaussian integers. To see this, just use the old “difference of two squares†formula. I.e. now that -1 is a square, a sum of two squares behaves exactly like a difference of two squares, so we have n = a^2 + b^2 = a^2 - b^2(-1) = a2 - b^2i^2 = a2 - (bi)^2 = (a+bi)(a-bi). Thus a number n like this which is a sum of two squares, even if it is prime in the usual integers, is no longer prime in the Gaussian integers.

 

An example is 13, since 13 = 4+9 = 22 + 33 = (2+3i)(2-3i). Now notice that 13 is congruent to 1 mod 4. We have already proved that a number congruent to 3 mod 4 cannot be written as a sum of two squares. We claim that if n is a prime integer congruent to 1 mod 4, then n can be written as a sum of two squares. All we have to do is prove that n is not prime in the Gaussian integers, since then it can be factored as a product of two Gaussian integers, n = (a+bi)(c+di). Taking absolute values and squaring gives n^2 = (a^2+b^2)(c^2+d^2). Since n is prime, then n = a^2+b^2.

 

Thus writing n as a sum of two squares is equivalent to factoring n in the Gaussian integers. So the problem becomes one of showing that a prime integer n which is congruent to 1 mod 4, factors as a Gaussian integer. I.e. our problem is to show that a prime integer of form 4k+1 is no longer prime as a Gaussian integer.

 

Now look at the property of having a square root of -1. There is a square root of -1 in the Gaussian integers, namely i^2 = -1. But there is already a square root of -1 in some modular number systems Z/n too. For example, if n = 5 then 2^2 = 4 is congruent to -1 mod 5. If n = 13, then 5^2 = 25 is congruent mod 13, to -1. In fact when n is an odd prime, then -1 has a square root mod n exactly when n is congruent to 1 mod 4. This is not so hard to prove, since solving equations in a modular system like Z/n is relatively easy because there are only finitely many numbers to deal with.

 

So if n is an odd prime of form 4m + 1, then -1 has a square root mod n, and we claim then n is not prime in Z, which will prove it can be written as a sum of two squares. So let k be an integer such that k^2 is congruent to -1, mod n. Now consider the Gaussian integers mod n, i.e. numbers of form a+bi where a and b are integers mod n. Then in the mod n Gaussian integers, k^2 = -1 = i^2 , so 0 = k^2 - i^2 = (k-i)(k+i) = k^2+1.

 

Since k^2+1 = 0 mod n, this means that the ordinary integer k^2+1 is divisible by n. But notice that n does not divide either factor (k-i) or (k+i) in the Gaussian integers, since by definition of division, n would have to divide both parts, k and i, and n does not divide i. Thus in the Gaussian integers we have found a product that n divides, while n does not divide either factor. Since prime numbers in the Gaussian integers have the same properties as in the usual integers, this means that n is not a prime in the Gaussian integers, which is what we wanted to show.

 

To summarize the argument: if n is congruent to 1 mod 4, then -1 has a square root mod n, say k, and then k^2+1 is divisible by n in the usual integers, hence also in the Gaussian integers. But in the Gaussian integers, k^2+1 = (k-i)(k+i), and n does not divide either factor, so n is not prime in the Gaussian integers, hence n factors as n = (a-bi)(a+bi) = a^2 + b^2, thus n is a sum of two squares. QED.

[mathwonk]

if you want to look at my notes on this and other topics, look at my web page:

 

http://www.math.uga.edu/~roy/

 

 

and look for notes from math 4000.

 

 

the notes from part two there, lectures 6-9 maybe, explain how modular arithmetic mod 4, tells you that

 

a number like 332245678899935629134 is not divisible by 4, because all you have to do is look at the last two digits. I.e. since 34 is not divisible by 4, neither is the whole big number. do you see why? basically it is because the rest of the number namely:

 

332245678899935629100, is equivalent to 0 mod 4, so can be thrown away. do you see why?

[onaclairadeluna]

EMACs is pretty awesome, if it fits your kid! Also, I REALLY didn't find the first books hard at all. I've had advanced math, but I wasn't a math major or anything.

 

 

The book O chapters (at least the first few) are really not that difficult. They do provide a good foundation for the rest of their books though. It gets pretty meaty pretty fast when you get into the books with numbers. When my son did these books he flew through the first several chapters in a month or two. My daughter is only around Singapore 4 ish and it's actually a little tough for her (coming up with her 7 times tables so she can divide mod 7 or dividing by 29 and the like is a challenge for her still. They say that you need to have completed everything up to pre algebra but I think a bright child could start a little earlier.

 

In case anyone has missed it the first chapter of Book 0 is now available as an online course and if you enroll now it is free. The online course is similar to the books. The books have more examples but the online component has these cool little aplets where you can play their games against the computer. This is coming in very handy for my daughter who is still wrapping her head around the math.

 

http://www.elementsofmathematics.com/

 

I really wish they would make the more advanced books available online. (3,4, 5....) The math in the later books gets very cool.

[mathwonk]

a comment on calculus versus linear algebra. there are two types of calculus, differential calculus and integral calculus. they are essentially inverses of each other. differential calculus inputs global information and tries to output local information, and integral calculus does just the opposite. E.g. if you know how to weigh segments of a cable whose weight varies due to different material composition in different parts, differential calculus would try to compute the density of that cable at one point along the cable, from knowing the weights of pieces of cable containing that point. On the other hand, if you know the composition of the cable at every point, and hence its density everywhere, integral calculus would try to calculate the weight of a segment of that cable, from those densities.

 

More geometrically, in differential calculus one tries to calculate the slope of the tangent line to a curve at a point, and integral calculus tries to calculate the area under a curve from knowing the heights of the curve at every point. The reason these are inverse to each other is that if you graph the area as a function, then the slope of that area curve at a given point, is the height of the original curve at that same point. Equivalently, if you graph the slopes of the original curve as a new curve, then the area function of that new curve will equal the height function of the original curve.

 

The fact that differential calculus tries to find the slope of a tangent line requires finding the tangent line. That line is the best approximation to the original curve at the given point. Thus to make it useful to study a curve by finding the line that approximates it best, one needs to have studied lines beforehand. this is why understanding lines is useful prerequisite information to studying differential calculus for curves, i.e. functions of one variable.

 

In differential calculus of 2 variables, the graphs are curved surfaces, and one tries to find the tangent plane, or the plane which best approximates the surface. That means that one should understand planes before trying to use them to study surfaces. If not, one can of course memorize some formulas and numbers which generalize the concept of slope from lines to the case of planes, but on will not understand what one is doing without understanding planes and how to represent them first. As the number of variables goes up, the approximating linear objects become more complicated, say linear functions of 100 variables, and iot becomes wise to practice them independently before applying them.

 

To state it algebraically instead of geometrically, differential calculus is the science of approximating non linear phenomena by linear phenomena. hence it is only prudent to understand linear phenomena before trying to use them to understand more complicated non linear phenomena. The mathematical study of linear functions is called linear algebra. This is why a reasonable person would teach linear algebra before differential calculus, at least if the differential calculus course studied general functions of many variables. This began to be common about 50 years ago in math courses but has lagged in some circles due to tradition, and a reluctance of some departments to have their students take more math courses, rather than just get on with their major in a utilitarian way.

 

 

Advanced linear algebra is equally useful in linear differential equations. Indeed the structure theory of linear operators such as jordan form, studied in junior/senior level, or "second", linear algebra courses, is virtually identical to the structure of solutions of differential operators acting on spaces of elementary functions. This fact was unknown to me until very recently, and to most other professors I asked, due to rigid patterns of instruction that prevent us from learning subjects in the logical order, independent of academic-political-practical considerations, like time to degree, etc... This point of view on differential equations is explained in my advanced linear algebra notes on my web page:

 

http://www.math.uga.edu/%7Eroy/4050sum08.pdf

[jennynd]

if you want to look at my notes on this and other topics, look at my web page:

 

http://www.math.uga.edu/~roy/

 

 

and look for notes from math 4000.

 

 

the notes from part two there, lectures 6-9 maybe, explain how modular arithmetic mod 4, tells you that

 

a number like 332245678899935629134 is not divisible by 4, because all you have to do is look at the last two digits. I.e. since 34 is not divisible by 4, neither is the whole big number. do you see why? basically it is because the rest of the number namely:

 

332245678899935629100, is equivalent to 0 mod 4, so can be thrown away. do you see why?

 

 

Any chance you can recommend textbooks for geometry and trigonometryand calculus for young kid...

[onaclairadeluna]

In differential calculus of 2 variables, the graphs are curved surfaces, and one tries to find the tangent plane, or the plane which best approximates the surface. That means that one should understand planes before trying to use them to study surfaces. If not, one can of course memorize some formulas and numbers which generalize the concept of slope from lines to the case of planes, but on will not understand what one is doing without understanding planes and how to represent them first. As the number of variables goes up, the approximating linear objects become more complicated, say linear functions of 100 variables, and iot becomes wise to practice them independently before applying them.

 

 

 

So what would be the best course of action for a young homeschooled student who is ready to study linear algebra?

 

MIT OCW?

EPGY (assuming they would waive the prereq.)?

Other Linear algebra sources?

[mathwonk]

Any chance you can recommend textbooks for geometry and trigonometryand calculus for young kid...

 

 

 

well that's a lot of stuff, but the best introduction to all these topics is in Euclid's elements, which are accessible with a guide or tutor.

 

Then first theorem in trigonometry is the law of cosines, which is nothing but the generalized pythagorean theorem, as it applies to arbitrary, not necessarily right angle, triangles. This occurs in Book II, props. 12 and 13, of Euclid.

 

As to calculus, the first problem is the determination of a tangent line, and this too is done in very modern fashion in Euclid's Book III prop.16, essentially the same way as defined 2,000 years later by Newton.

 

Again, a guide is useful, for which a modest substitute may be my epsilon camp notes on my web page.

 

The fundamental notion of limit, or determining a quantity by an infinite sequence of better and better approximations, underlies all of calculus and was also introduced in Euclid. After the example of the tangent line, essentially as a limit, in later books he proves the volume formulas for pyramids also by a limiting process.

 

 

As to books which are written in a kid friendly way, the geometry book by Harold Jacobs is great, but not as mathematically substantial as Euclid, to which it might serve as an introduction.

 

There are no kid friendly modern books known to me on trig or calculus per se, but one might try the old SMSG texts. anybody else have favorites?? on the other hand, the book we used in high school might be worthwhile, Principles of Mathematics, by Allendoerfer and Oakley.

 

http://www.abebooks....r, oakley&sts=t

 

well i searched used copies, and they are $50. makes me wonder of the cheaper books there called fundamental;s of freshman mathematics, for $4, would be good enough. you could experiment at that price.

 

my favorite user friendly intro to calculus, not deep or rigorous, but fun, is calculus made easy, by silvanus p thompson. The motto: "what one fool can do, another can,", is worth the price of admission.

 

http://www.abebooks....anus p thompson

 

another great layperson's intro to all sorts of math, very well written by the famous mathematician Richard Courant, is "What is Mathematics?" recommended for everyone. It includes calculus.

[mathwonk]

So what would be the best course of action for a young homeschooled student who is ready to study linear algebra?

 

MIT OCW?

EPGY (assuming they would waive the prereq.)?

Other Linear algebra sources?

 

 

I like Paul Shields little book on Elementary Linear Algebra. I'll find one for you.

 

http://www.abebooks.... linear algebra

 

heres one for $1.

 

 

Oh yes, the SMSG book on linear algebra for high schoolers from the 1960's is good.

 

here are several free smsg books, unfortunately not that one:

 

http://ceure.buffalo.../SMSGTEXTS.html

 

 

heres one:

 

http://www.abebooks.com/servlet/SearchResults?bi=0&bx=off&ds=30&pn=yale&recentlyadded=all&sortby=17&sts=t&tn=introduction+to+matrix+algebra&x=61&y=10

[mathwonk]

heres the easy direction on sums of two squares. evert integer is either divisible by 4, or has remainder 1,2, or 3, when divided by 4.

 

every squared integer has remainder 0 or 1, after division by 4. since a multiple of 100 is divisible by 4, the only integers which are possibly squares are those whose last 2 digits have remainder 0 or 1, on division by 4.

 

in particular a number ending in 43 is not a square.

 

 

thus a sum of 2 squares must have remainder 0,1,or 2, on division by 4. thus integers of form 4k+3, like 43, 19, 23, 75, ... cannot be sums of 2 squares.

 

the hard direction, due to fermat, say that any prime integer of form 4k+1 is a sum of 2 squares, or maybe is a square.

 

but it is best to let the child guess the pattern by trying examples, before telling them about this dividing by 4 business.

[wapiti]

Oh yes, the SMSG book on linear algebra for high schoolers from the 1960's is good.

 

here are several free smsg books, unfortunately not that one:

 

http://ceure.buffalo.../SMSGTEXTS.html

 

by any chance does this help? http://static.cemseprojects.org/smsg/

 

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[jennynd]

 

 

 

well that's a lot of stuff, but the best introduction to all these topics is in Euclid's elements, which are accessible with a guide or tutor.

 

Then first theorem in trigonometry is the law of cosines, which is nothing but the generalized pythagorean theorem, as it applies to arbitrary, not necessarily right angle, triangles. This occurs in Book II, props. 12 and 13, of Euclid.

 

As to calculus, the first problem is the determination of a tangent line, and this too is done in very modern fashion in Euclid's Book III prop.16, essentially the same way as defined 2,000 years later by Newton.

 

Again, a guide is useful, for which a modest substitute may be my epsilon camp notes on my web page.

 

The fundamental notion of limit, or determining a quantity by an infinite sequence of better and better approximations, underlies all of calculus and was also introduced in Euclid. After the example of the tangent line, essentially as a limit, in later books he proves the volume formulas for pyramids also by a limiting process.

 

 

As to books which are written in a kid friendly way, the geometry book by Harold Jacobs is great, but not as mathematically substantial as Euclid, to which it might serve as an introduction.

 

There are no kid friendly modern books known to me on trig or calculus per se, but one might try the old SMSG texts. anybody else have favorites?? on the other hand, the book we used in high school might be worthwhile, Principles of Mathematics, by Allendoerfer and Oakley.

 

http://www.abebooks....r, oakley&sts=t

 

well i searched used copies, and they are $50. makes me wonder of the cheaper books there called fundamental;s of freshman mathematics, for $4, would be good enough. you could experiment at that price.

 

my favorite user friendly intro to calculus, not deep or rigorous, but fun, is calculus made easy, by silvanus p thompson. The motto: "what one fool can do, another can,", is worth the price of admission.

 

http://www.abebooks....anus p thompson

 

another great layperson's intro to all sorts of math, very well written by the famous mathematician Richard Courant, is "What is Mathematics?" recommended for everyone. It includes calculus.

 

Thanks a lot.i ordered both books. I like the fun calculus. That's the type of book I am looking for my DS

[mathwonk]

by any chance does this help? http://static.cemseprojects.org/smsg/

• 
  

 

yes, thank you!

[mathwonk]

Thanks a lot.i ordered both books. I like the fun calculus. That's the type of book I am looking for my DS

 

 

as i recall from thompson's book:

 

 

 

"and fleas have fleas that also bite 'em, and so on ad infinitum!"

 

 

how's that for user friendly limit theory?

[mathwonk]

Topology would be an interesting "off-the-beaten-path" course. I'm not sure what the pre-reqs would be beyond basic H.S. geometry as it's not a course I ever took. Isn't there a math professor on this board? That would be the person to ask.

 

 

here is a concrete answer: the most basic and important result in all of topology, is the fact that for any convex polyhedron, the sum V-E+F of the numbers of vertices minus edges, plus faces, always equals 2, regardless of the number of individual faces. very kid can appreciate this, but proving it would be a challenge, not however outside he reach of a really sharp, creative kid.

[mathwonk]

Actually all the basic topics in mathematics, geometry, algebra, and analysis, have their origins in Euclid, which is thus the best beginning book in mathematics. Here is an essay on how Euclid's treatment of similar triangles leads to the concept of real numbers, as number that can be approximated by fractions, i.e. rational numbers. This will possibly try the patience of the most dedicated reader.

• The third and last main topic in mathematics is analysis, the complete approximation of quantities by an infinite sequence of simpler quantities. The basic example is the approximation of an irrational number an an infinite sequence of rational numbers. E.g. the expansion of an irrational number as an infinite decimal is such a case. An infinite decimal is an infinite sequence of longer and longer finite decimals, and these finite decimals approximate the number defined by the infinite decimal. Taken together, the full infinite collection of finite decimals actually determine the infinite decimal exactly. It is easier to give a rational example, since rational infinite decimals are easier to write down, and infinite decimals can also be used to determine rational as well as irrational numbers. We all have seen the infinite decimal expansion of 1/3 = .333333.... where there is an infinite number of 3’s in the full decimal expansion. Each finite decimal say .333, is an approximation to 1/3, and the more decimal places we use, the closer the approximation. The full infinite set of finite decimals does however determine the number 1/3 as follows: among all numbers which are larger than every finite decimal of form .3, .33, .333, ... and so on, 1/3 is the smallest one. Since we want to use language that also applies to infinite decimals which terminate, it is better and just as correct to say: the number determined by an infinite decimal is the smallest number among all those which are not smaller than any of the finite approximating decimals.
  

In this way any real number, including all irrational numbers, can be determined by an infinite sequence of rational numbers. The origin of this method of describing real numbers, as usual, is in Euclid. This deep idea is first needed when Euclid discusses the concept of similarity, or ratio between two different line segments. Even though Euclid does not use numbers to describe the length of a single line segment, still the ratio between two line segments should be a number. Since he does not yet have a concept of numbers, he needs a way to describe when two pairs of line segments are in the same ratio, and to describe this he essentially needs to invent a way to describe real numbers, including irrational ones. So the origin of real numbers is Euclid’s description of when corresponding pairs of sides the of similar triangles have the same ratios.

 

Hence without studying Euclidean geometry first, without real numbers as a crutch, it seems to me impossible to ever understand where real numbers come from, i.e. a real number is essentially determined by a pair of line segments. Of course this is a very primitive way to deal with real numbers, and the modern axiomatic way is more efficient if less intuitive. If you read Galileo’s treatise on the science of motion in his beautiful work “On two new sciencesâ€, you will find calculations with real numbers made using pairs of line segments, since the Arabic notation of representing real numbers abstractly by symbols was not yet available to him. By the way I recommend this work by Galileo as suitable for gifted young children at a certain point. The parent could read it first and decide at what age it will work well.

 

Ok, back to Euclid. Suppose given two triangles with equal corresponding angles. How do we prove that their corresponding sides are all in the same ratios? We also have to say what that means, since we cannot yet assign a real number to each ratio of sides. When we are finished we will indeed have a way to assign a real number to each ratio, i.e. we will have essentially created the concept of real numbers.

Given two triangles with equal corresponding angles, move one of them to lie inside the other. I.e. if angles A,B,C on one triangle are equal to angles A’,B’,C’ of the other, in that order, then place angle A exactly inside angle A’, so that side AB lies along side A’B’, and side AC lies along A’C’. Then since angles B and C equal angles B’ and C’, the third sides BC and B’C’ will be parallel, by one of the basic principles of parallel lines in Book I of Euclid. Hence one of the triangles lies entirely inside the other. I.e. if point B’ lies between A and B, then also point C’ lies between A and C, and triangle A’B’C’ lies inside triangle ABC. Assume this is so. We want to show that the ratio of the two sides AB and A’B’ is the same as the ratio of the two sides AC and A’C’. It is already clear that since B’ lies between A and B that A’B’ is shorter than AB, and because BC is parallel to B’C’, hence C’ lies between A and C, and then A’C’ is also shorter than AC. This accomplishes the first step in our sequence of comparisons. I.e. the ratio of A’B’ to AB is less than 1 but greater than 0, and also the ratio of A’C’ to AC is also less than 1 and greater than 0. We just continue to repeat this for all other fractional lengths and we will be done.

 

I.e. now subdivide side AB in half, as one learned to do in Book I, and draw a line parallel to the base BC through the midpoint of AB until it meets the other side AC. Then we claim that line meets the other side at the midpoint of AC. (This will prove the principle of similar triangles in the special case where the ratio of the sides is 1/2.) To see this, hmm... I need to draw a picture and work this out for myself. Back in a second....make that many seconds. I got so discouraged trying to remember how to do this, that I tried to cheat and look at my own web notes of this course, but fortunately for me, that proof was not there! So I had to think again. Now I believe I have it. It is so interesting that geometry proofs look much easier when done one way than when tried another way. I include this burst of candor to reassure you that math is challenging for everyone, and that although one really should not peek at the answers, as it does not help one learn, I also weaken at times on this point. But I try not to. (In my office the answer book to the calculus text provided by the publisher served only as a doorstop.)

 

Ok, consider triangle ABC, with vertex A and base BC, and draw a line through the midpoint X of side AB, parallel to base BC, and meeting side AC at Y. We want to show that Y is the midpoint of side AC. For this, draw a line through Y parallel to side AB and meeting base BC at Z. I will not go through this, but the idea is to use properties of parallel lines and parallelograms to show that figure BXYZ is a parallelogram, hence triangles XYZ and ZBX are congruent. You show also triangles AXY and XBZ are congruent, hence triangles AXY and YZX are also congruent, so figure AXZY is also a parallelogram. Finally triangles XAY and BXZ are congruent to triangle ZYC, so indeed segments AY and YC are congruent, so Y is actually the midpoint of side AC.

 

The first time I tried this by choosing Z as the midpoint of base BC, instead of making line YZ parallel to side AB. For some reason that did not work for me.

 

After this, one also shows that two lines parallel to base BC and dividing side AB into equal thirds, also divide side AC into equal thirds, and so on for fourths, fifths, etc....

 

Now we can prove the principle of similarity we started out to do. Namely let X be any point on the side AB of a triangle ABC with vertex A and base BC, and draw a line L through X parallel to BC and meeting side AC at Y. I claim the ratio of AX to AB equals the ratio of AY to AC. Draw another line M parallel to BC and passing through the midpoint of AB hence also the midpoint of AC. Since L and M are both parallel to base BC they are also parallel to each other, so both endpoints of M are on the same sides of line L. I.e. if X is closer to B than the midpoint of AB, then also Y is closer to C than the midpoint of AC. I.e. if the ratio of AX to AB is more than 1/2, then also the ratio of AY to AC is more than 1/2.

 

Continuing in the same way, we subdivide sides AB and AC into thirds by parallel lines, and conclude that if say AX has ratio to AB somewhere between 1/3 and 2/3, then also AY has ratio to AC also between 1/3 and 2/3. Continuing on, we find say that if the ratio AX/AB is between 88/139 and 89/139, then also AY/AC is between 88/139 and 89/139. Continuing the argument, it follows that every rational number that is less than the ratio AX/AB is also less than the ratio AY/AC, and every rational number that is greater than AX/AB is also greater than AY/AC. For that reason we say the ratios AX/AB and AY/AC are equal. Thus two ratios, possibly irrational, are equal if they equal the same rational ratio, or else they lie on the same side of every rational ratio. For this reasoning to make sense, it is tacitly being assumed that given any segment AB and intermediate point X, that some finite number of copies of segment AX becomes eventually longer than AB. This assumption is called Archimedes’ axiom, since some people feel it was not sufficiently clear in Euclid, although he does mention the need for it at one point in his discussion. He also states it explicitly as Definition 4 in Book V.

 

Actually, this similarity argument is not the one given in Euclid, although it uses his ideas. I.e. the way to compare ratios of segments which we explained is Euclid’s, given by him as Definition 5 in Book V, but he uses areas to prove similarity. To do this, he first gives an argument like ours to show that two triangles on the same base, have areas proportional to their heights, in Prop. VI.1. Then in Prop.VI.2 he uses that result to deduce the proportionality of sides of triangles having the same angles. It turns out that the two theories of similarity and of areas are essentially equivalent. The area statement in Prop. III.35 can also be used to deduce similarity for triangles.

 

2,000 years later, Dedekind defined a real number as a “separation†of the rational numbers into two parts. I.e. every real number, rational or irrational, does separate the rational numbers into two parts, and conversely every separation of the rational numbers into two parts S and T, such that every number in S is less than every number in T, defines a real number, the unique real number caught between the sets S and T. Equivalently, a real number is defined by all the rational numbers less than it. Or more simply, it suffices to choose an infinite sequence of rational numbers less than the given real number, such as a sequence of finite decimals, together forming an infinite decimal. It already follows from the discussion given by Euclid, that the assumption of Archimedes axiom suffices to insure that every real number, i.e. every ration of two segments, can be completely determined by the way it separates the rational numbers into two sides, but it is not clear that every such separation does occur for some pair of segments. I.e. one can do Euclidean geometry without assuming that every real number does occur as the length of a segment, or equivalently that every interval on the line has endpoints. This assumption is called the Dedekind axiom.

 

So Euclidean geometry, subject also to the Archimedes and Dedekind axioms, is exactly the usual modern school geometry (as in Jacobs or I believe also SMSG, due to Birkhoff) based on real numbers. But Euclidean geometry is more fundamental than the concept of real numbers, and it leads to that concept rather than the other way around. Moreover it is quite possible to do Euclidean geometry within a smaller system of numbers, large enough to include those ratios that actually occur in geometry such as sqrt(2), but not large enough to include those real numbers that do not occur in Euclidean constructions, such as pi, or even cubert(2). The proof that these last two numbers do not occur was out of reach of Euclid, and the other ancient Greeks, and was achieved in the 19th century after the introduction of algebra and number systems into geometry.

 

One other topic in analysis I would like to discuss, due apparently to Archimedes, is the use of approximations to determine volumes in geometry, a subtle but powerful improvement on Euclid’s method for treating areas and volumes. Euclid did areas and volumes differently, (finite decompositions for plane areas, and infinite approximations for volumes), and Archimedes developed a better and simpler method for areas that also works for volumes, based on a more uniform method of making infinite approximations. Archimedes' approach leads directly to the concept of a definite integral in calculus. In a nutshell, the reason that two triangles with equal base and equal heights have the same area, according to Archimedes, is that they meet all horizontal lines in segments of the same length. The same principle shows that two solids have the same volume if they meet all horizontal planes in plane figures of the same area. (This principle today is usually named after Cavalieri, although known much earlier to Archimedes.) Using this idea Archimedes determined the volume of a sphere to be equal to the difference in the volumes of a cylinder and a cone, both of which volumes he knew. Archimedes considered this the crowning achievement of his mathematical career, and had a figure illustrating it inscribed on his tombstone. His method would even have allowed him to determine the "volume" of a 4 dimensional ball, as I explain in the epsilon camp notes on my UGA web site.

[mathwonk]

Any chance you can recommend textbooks for geometry and trigonometryand calculus for young kid...

 

 

 

Sorry to be so slow answering fully. yes the Elementary Functions SMSG book linked by wapiti in post 68 looks good for beginners. trig functions are correctly if informally defined on page 226, and exponential function are discussed in the pages surrounding 147 or so.

 

the problem with the definition of trig functions is that they are really sort of inverse functions of circular arc length. so the problem of defining arc length still remains. i.e. this definition explains correctly that sin and cos are a sort of inverse functions to arc length, but does not tell at all how to evaluate those functions. i.e. even after learning this fact, one has no idea what sin(1) is e.g., or cos(1).

 

this problem is a challenging one, and is only solved in calculus courses where one learns first that the functions sin and cos satisfy the differential equation y'' + y = 0, and that sin(0) = 0, while cos(0) = 1. it then follows that one can write infinite formulas for these functions.

e.g. sin(t) = t - t^3/3! + t^5/5! -+...... and cos(t) = 1 -t^2/2! + t^4/4! - + .....

 

in this sense the exponential function is slightly easier to define, assuming of course we know how to take all possible roots of numbers. here again calculus to the rescue gives the formula

 

exp(t) = 1 + t + t^2/2! + t^3/3! +......, for y = exp(t) which solves the d.e. y' = y, and which also reveals a connection between sin, cos, and exp,

 

namely exp(it) = cos(t) + i sin(t).

 

unfortunately the base of this nice exponential function is a complicated constant, namely e = e^1 = 1+1+1/2! + 1/3! +........ ≈ 2.71828......

 

so this problem of finding actual useful formulas for the "elementary" functions can be used to motivate learning calculus later.

 

actually euler considers this material precalculus and explains it all in an old fashioned but impressive way in his precalculus book, analysis of the infinite.

[mathwonk]

here is another link for some good looking smsg math books, elementary through junior high. does anyone know where to find a good copy of their calculus text? 3 volumes, 1000 pages. the copy i have looks interesting but in terrible reading condition. (it has a few mathematical mistakes but not too many people would notice.)

 

i seem to have omitted the link, i'll look again. try this:

 

http://ceure.buffalo.../SMSGTEXTS.html

 

well unfortunately, after looking at these books on this link, i think some of them are the sort of thing that gave the "new math" a bad name. the first course in algebra, e.g. student text part I and II, look just as boring and vacuous as possible, with hundreds of pages essentially teaching jargon about set theory rather than substantial math.

 

this sort of thing may be faintly useful to a future math major to learn more precise ways of talking about math, but really, 400-500 pages of "open sentences" and "solution sets"? please.

 

i'll try the geometry book..... well i didn't like it either. because it uses the birkhoff approach to geometry wherein one assumes one understands the real numbers first and then treats points on a line as real numbers. of course when you ask what a real number is they tell you they correspond to points on a line. this kind of circular nonsense does not help bright children understand.

 

ironically, these mathematically mediocre smsg books on this link, are physically the best copies i have seen on the web, crisp and bright and clean - maybe nobody read them. they also seem kind of dumbed down. that smsg calculus book is not dumbed down, but i can't find a decent copy.

 

it is true harold jacobs geometry uses the birkhoff approach as well, but his book is a lot of fun, does not belabor the set theoretic jargon, and contains a number of solid geometrical arguments taken from euclid and hilbert and archimedes. i like jacobs more, but again the best geometry book hands down is euclid, preferably with a good guide like hartshorne, Geometry, euclid and beyond.

[kohlby]

I did not want to go the standard math sequence of Alg I, Geom, Alg II, Pre-Calc, Calc since he would run out of math too fast. Also, there's a math maturity that students need. As a former high school math teacher, I found that honors Geometry was one place where the math maturity was a huge issue due to proofs. (And I found high school Geometry boring when I was a student but college Geometry fun. I'd rather my child is ready for the more interesting stuff!) I could easily follow that sequence if I picked basic math courses, like the college-prep level at most high schools. But part of having a gifted child isn't speeding through, but going into more depth. That is why we're doing AoPS.

 

My son is in 4th grade. He started MUS Alg I in 3rd but that's the point where MUS no longer worked for us. So we started AoPS Intro to Algebra in 4th. He's doing well but I noticed his math maturity wasn't quite high enough in a few sections. We got through it, but I still noticed it. Those were sections that they don't do in high school math classes, but I want my gifted child to get the most out of it and understand. So, we're doing

 

Ch 1-12 AoPS Intro to Algebra for Alg I

Then either Intro to Prob or Intro to number theory (I created a post asking which one - still trying to figure it out!)

Rest of AoPS Intro to Algebra for Alg II

Then I don't know. If he needs more math maturity time, then we'll be doing the course we didn't do prior. Otherwise, onto Geometry

Then I still don't know. By this point, I'll know his interests better to figure out where to go from there.

 

*I also may add in a SAT math mini-course in addition to a regular math one year. Another option is a problem solving math class - in which I'd use math competition problems.

[onaclairadeluna]

Ch 1-12 AoPS Intro to Algebra for Alg I

Then either Intro to Prob or Intro to number theory (I created a post asking which one - still trying to figure it out!)

Rest of AoPS Intro to Algebra for Alg II

...

 

Why not do both NT and P and C? FYI I think the number theory book is a tad easier than the probability book. I think my son loved the Probability book more, but I think that might be because it was his first AOPS book.