Suggestions for algebra text for 14 year old

[observer1]

Hello

Could someone suggest a good book for my son to read to learn algebra?

I look in the bookstores and the books titled "Algebra..." all involve equations and unknowns and arithmetic manipulations and train or work problems (2 farmers plow three fields in one day, etc)

I am interested in my son learning algebra: sets of numbers, groups, fields, rings, etc.

But I cannot find any such books that teach MATH instead of ARITHMETIC

 

[PeroK]

You could take a look at this resource:

http://www.examsolutions.net/maths-revision/syllabuses/GCSE/period-1/Higher/module.php

This has all the UK secondary school (High school) maths on it.

This would be a prerequisite for studing groups and rings, which would come under "abstract algebra" and is generally university material. If you Google for "abstract algebra pdf", that will turn up a lot of free texts.

 

[observer1]

Hello

My 12 year old son is slightly precocious.

And I would like for him to read some books on Algebra.

However, most of the books I find (in stores or on the internet or in teacher materials), are more concerned with equations and unknowns or rate problems, or arithmetic.

I am more interested in him learning the idea of algebra: set theory, fields, rings, groups, etc.

Can anyone suggest some books I could buy?
Or any internet sites?
(Most of them all reduce the "idea" of "math" into the "machinery" of "arithmetic."

 

[SteamKing]

Hello

My 12 year old son is slightly precocious.

And I would like for him to read some books on Algebra.

However, most of the books I find (in stores or on the internet or in teacher materials), are more concerned with equations and unknowns or rate problems, or arithmetic.

I am more interested in him learning the idea of algebra: set theory, fields, rings, groups, etc.

Can anyone suggest some books I could buy?
Or any internet sites?
(Most of them all reduce the "idea" of "math" into the "machinery" of "arithmetic."

Ya got to learn to walk before you can run with the Big Dogs.

IMO, it's better to learn basic algebra before you toss your boy in the deep end of the pool. All that stuff about fields, rings, groups, etc. is sometimes called 'college algebra' and for good reason: it's typically taught only after a student has had about two years of instruction in 'regular' algebra, not to mention some geometry and trig.

He's only 12; what's the rush? Unless he's going to join a junior mathematics program at the local college, mastering the 'regular' algebra topics is much more useful, in terms of analyzing a problem and setting up a solution, which is the bread and butter of what algebra is for anyways. The rings, fields, groups stuff only becomes useful when you get into analysis, which in any event, is going to be a ways down the road, mathematically speaking.

 

[micromass]

A 12-year old learning ring theory? Even if he can handle it (which would make him extremely gifted), I doubt you'll find any books on his level. The best you can do is look for easy college texts.

Set theory is another deal. I learned set theory in primary school and in Belgium this is a very standard topic there. So there are definitely books on the topic. They're just not in the english language. Perhaps you can look at the "new math" movement in the US and see whether there are books you'd like?

 

[micromass]

And sure, you might have good reasons to teach your child abstract algebra? But let me ask you a few things:

- Does your child know how to solve quadratic polynomial equations and other polynomial equations? If not, entire ring and field theory will make very little sense since it's all about that.
- Does your child know the basics of complex numbers? If not, the statement of the fundamental theorem of algebra will not be understandable and out goes a lot of abstract algebra.
- Does your child know vectors and basic geometry? If not, vector spaces and field theory will make very little sense.
- Does your child know how to find the symmetries of a cube or a dodecahedron? If not, don't bother starting group theory. Does your child even know the concept of reflection, rotation in 2D and 3D? Does your child know the matrices of these operations?
- Does your child know how to prove the fundamental theorem of arithmetic and does (s)he actually grasp the proof? If not, don't bother starting ring theory.

 

[jtbell]

All that stuff about fields, rings, groups, etc. is sometimes called 'college algebra'

At my college it was called "Advanced Algebra". I think at most colleges and universities in the US, "College Algebra" is a remedial course for freshmen who didn't learn basic algebra well enough in high school.

 

[SteamKing]

At my college it was called "Advanced Algebra". I think at most colleges and universities in the US, "College Algebra" is a remedial course for freshmen who didn't learn basic algebra well enough in high school.

Well, I took 'college algebra' in high school waaaay back yonder when as part of the advanced math curriculum. It was not a remedial course then, and it had plenty of the abstract stuff in the course. It was more akin to preparing students to take calculus than to catch up on more basic math.

 

[ProfuselyQuarky]

Heck, I'm in high school and I only have a slight idea on what ring theory is . . .

But, I can do all those things listed in post #4. Does that mean that I could start studying ring theory on my own?

 

[micromass]

Heck, I'm in high school and I only have a slight idea on what ring theory is . . .

But, I can do all those things listed in post #4. Does that mean that I could start studying ring theory on my own?

Yes, you actually can. You just need the basic notations of set theory and proofs, which are quite easy to learn. Then you can start studying most of abstract algebra. You don't even need calculus! There are plenty of topics that can be taught very easily in math high school but aren't.

If you wish to study abstract algebra, take a look at Pinter's "A book of abstract algebra" which is very cheap but a very good introduction which should be understandable for motivated (very important!) high schoolers. https://www.amazon.com/dp/0486474178/?tag=pfamazon01-20 I'm not saying it will be easy, but it's definitely doable.

 

[ProfuselyQuarky]

Yes, you actually can. You just need the basic notations of set theory and proofs, which are quite easy to learn. Then you can start studying most of abstract algebra. You don't even need calculus! There are plenty of topics that can be taught very easily in math high school but aren't.

I'm actually quite familiar with some of the set theory notations already since my Pre-Calc book briefly covers some of those topics. I'll look into the proofs, too.

If you wish to study abstract algebra, take a look at Pinter's "A book of abstract algebra" which is very cheap but a very good introduction which should be understandable for motivated (very important!) high schoolers. https://www.amazon.com/dp/0486474178/?tag=pfamazon01-20 I'm not saying it will be easy, but it's definitely doable.

Yes! Thank you for the suggestion. I don't really care if it's easy, since nothing worthwhile comes easy, anyway. Now I know what I'm going to begin come Spring Break.

 

[jtbell]

At my college it was called "Advanced Algebra".

Correction: it was "Abstract Algebra." Their real analysis course was called "Advanced Calculus" and I probably got the A-words briefly mixed up in my memory.

 

[The Bill]

I'd recommend a reading list leading up to and including the abovementioned Pinter.

Complex numbers aren't really worthy of a full text all their own at this level, so for history and intuition, I recommend Nahin's An Imaginary Tale.

Next, I'd recommend an introductory linear algebra text. A recent but very inexpensive(used) option is David C. Lay's Linear Algebra and Its Applications, 3rd edition.

Pinter's A Book of Set Theory should build enough foundation on the concrete examples from complex numbers and linear algebra to prepare for abstract algebra.

I add my own hearty recommendation to Pinter's A Book of Abstract Algebra. Really, you can get it right away, but make sure the reader knows not to get discouraged if part of it is difficult to get past. When one has difficulty, retreat and regroup by working through something similar in a less advanced source.

Also, there are plenty of good videos online on various topics. Combined with good texts and plenty of work with pencil and paper, attaining a great degree of understanding should be rather straightforward.

 

[observer1]

Thank you everyone. I will look into all of this.

Please allow me to clarify. I am not actually looking for college level or even high school books that fully explain the DETAILS of algebra.

I am looking for books that will prepare his mind for the IDEAS of algebra.

 

[smodak]

"I am looking for books that will prepare his mind for the IDEAS of algebra."I am not 100% sure whether this is t the correct level for a 14 years old, but, I highly recommend the following book. Amazon has a preview that shows the full TOC. Check it for yourself.

https://www.amazon.com/dp/0195105192/?tag=pfamazon01-20

For group, set, ring, and fields - the Pinter books recommended above are quite nice.

 

[ProfuselyQuarky]

I am looking for books that will prepare his mind for the IDEAS of algebra.

By the way, does your son already like math, or are you trying to get him interested in math? There are a bunch of entertaining books about math that cover topics on a recreational basis and make even the most obstinate of people interested (not to say that that interest will continue, however).

Believe me, I read them. And they're not even one of those pop science filled works--they show you real math.

 

[Mark44]

I look in the bookstores and the books titled "Algebra..." all involve equations and unknowns and arithmetic manipulations and train or work problems (2 farmers plow three fields in one day, etc)

I am interested in my son learning algebra: sets of numbers, groups, fields, rings, etc.

But I cannot find any such books that teach MATH instead of ARITHMETIC

Arithmetic is purely numbers -- no letters that represent unknown quantities. The books you've found are algebra, albeit at a fundamental level. They are NOT arithmetic.

I am looking for books that will prepare his mind for the IDEAS of algebra.

Most of the books I've ever seen on modern algebra (or abstract algebra) go heavily into the details, including definitions and theorems. I'm not aware of a lot of books that are more cursory in nature.

 

[micromass]

I am looking for books that will prepare his mind for the IDEAS of algebra.

What are the ideas of algebra anyway? I don't think group/ring/field theory really qualify as the ideas of algebra. They are actually high-level abstractions of more elementary ideas. I don't think it makes much sense to teach somebody abstractions without first having covered the elementary ideas and its details.

Maybe your son will appreciate it, but I'm very skeptical...

 

[The Bill]

In that case, I suggest two books by Ian Stewart: Why Beauty is Truth: A History of Symmetry, and his Concepts of Modern Mathematics.

 

[mathwonk]

please forgive me, but in my opinion also it is foolish to teach ring theory before teaching the arithmetic of the integers, the basic example of a ring. i strongly suggest, first teach the classic examples then teach, much later, their abstractions. have you looked at euler's elements of algebra? he was one of the greatest algebraists of all time, and wrote an algebra book, but no it does not define a ring.

let me see if i can say something more intelligent and less snotty. is it more useful to view actual painting by french impressionists or to read essays by people on "abstract impressionism"? this is a rough analogy, since the integers are the painting and the definition of ring is the essay.

another piece of advice born of experience: brilliant young kids prefer attacking actual problems more than memorizing theory. they want to do, not to simply listen. good luck!

PS: if you want to do what you propose anyway, and to hell with our advice, try a number theory book,

this one seems to review well:

https://www.amazon.com/dp/0486682528/?tag=pfamazon01-20

or maybe herstein's abstract algebra book, since it has lots of problems.

or even artin's algebra if your son is at the level of a strong MIT undergrad; unlikely though this is, i include it for future use, since he will only get stronger.

 

[jack476]

Pinter's A Book of Abstract Algebra is a long-time favorite of mine. I know, modern algebra is usually reserved for advanced college level courses, but I can think of no reason why a smart teenager who knows how to use Google when he's stuck wouldn't be able to benefit from it.