The Should I Become a Mathematician? Thread

AnTiFreeze3

I'm not sure that I understand this step (even though it looks very simple). I did it a different way, so the way I got to that answer was different. What I'm not seeing is how you have a-b in the denominator, yet three separate occasions of a-b in the numerator (a² - b²; and a-b), yet when you essentially cancel them out, you are somehow left with a + b + c.

In my mind, when you cancel out the a-b on the bottom with any of the three pairs of a-b on top, you are either left with:

(a-b) +c(a-b), or

(a² - b²) + c

What do I seem to be missing, or not understanding?

micromass

You know that [itex]a^2-b^2=(a-b)(a+b)[/itex]

So

[tex]\frac{(a^2-b^2)+c(a-b)}{(a-b)}=\frac{(a-b)((a+b)+c)}{a-b}=a+b+c[/tex]

AnTiFreeze3

I'm not going to try to right this with brevity like Micro, but instead I want to explain my thought process, because I feel that I may have done something incorrect.

My solution:

{a²(1/b - 1/c) + b²(1/c - 1/a) + c²(1/a - 1/b)}
____________________________________________________________________

{ a(1/b - 1/c) + b(1/c - 1/a) + c(1/a - 1/b)

I noticed from the start that the binomials would cancel out, so long as I was able to manipulate the problem and get them next to eachother, so I didn't see a reason to get rid of the fractions, since I knew they would cancel out anyways with their respective opposites. I then simplified a, b, and c to get rid of any multiplication in the denominator, and I used the commutative property to rearrange the denominator:

{ a(1/b - 1/c) + b(1/c -1/a) + c(1/a - 1/b)}
___________________________________

{(1/a - 1/a + 1/b - 1/b + 1/c - 1/c)}

The denominator then cancels out to equal 1, so I am left with:

{ a(1/b - 1/c) + b(1/c - 1/a) + c(1/a - 1/b)}

This next step is where I have broken math. I recognized what the answer should be, but I think that I may have cheated in order to get to that final result. As a result, I did this:

{(a + b + c)(1/a - 1/a + 1/b -1/b + 1/c - 1/c)

Then, similarly as before, the fractions cancel eachother out, so I was ultimately left with this:

a + b + c

I didn't peak at Micro's answer, and actually came to the correct answer myself. Regardless of that, I still feel as if that last step isn't allowed. Is it even possible to solve it correctly using the process that I used?

EDIT:

I messed it up in the first step, which is why I ended up in a situation where I couldn't correctly solve it.

jbunniii

Can you explain what you did to get to this step? It is clearly not correct, because:

Actually, it cancels to 0, not 1.

AnTiFreeze3

I already mentioned that I messed up the first step, and that that is what threw off my whole solution. Thanks though.

EDIT: Although, if you are curious as to what was going through my mind, I embarrassingly forgot that I needed to simplify it before I could just eliminate a, b, and c. The rest of my problems stemmed from that.

I think it was coincidental that my answer ended up being a + b + c, even after making two big mistakes. Or maybe it wasn't coincidental, and I have just inadvertently invented a new form of Algebra where you break rules until you get the answer.

mathwonk

Very impressive micromass and Antifreeze! nice solutions!

micromass and Antifreeze are very strong, but we can also make progress using some basic principles to help us.

Here is a hint for other possible solutions: Generalized factor theorem: if f is an irreducible polynomial, and if f = 0 implies g = 0, then f divides g. (This is a basic result in “algebraic geometry”, and generalizes the basic result that x-r is a factor if r is a root.).)

For instance, suppose a-b = 0, then what about a^3(c-b) + b^3(a-c) + c^3(b-a), does it vanish too? Then what?

Now how did we guess to try a-b=0? Recall the "rational root theorem"? It says you look for roots of form X-r by trying factors r of the "constant term.

As miromass observed, we can rewrite the top of the fraction after simplifying,

as a^3(c-b) - a(c^3-b^3) + bc(c^2-b^2). Think of this as a polynomial in a.

thus the constant term has prime factors ±b,±c, ±(c-b),±(c+b).

(also other products of these factors, possibly.)

So we should try setting a equal to those factors. e.g. a=b iff a-b = 0.

Dowland

I don't want to start a new topic for this question, so i post it here:

How important is (euclidean) geometry in the higher (that is at the university) mathematics education? I'm currently in high school and feel that I've barely touched the subject, only simple computations with area, proportions, and some volume problems, together with a few "angle games".

I'm thinking of maybe getting the following book: http://www.amazon.com/Elementary-Geo.../dp/0201508672

But maybe it's all too much, and not so important? I've enjoyed the little euclidean geometry I've done, but if I don't have very much use of it in the basic calculus and linear algebra courses, I'll probably skip it (for now).

Thoughts on that?

(Sorry for possible language errors, english is not my native, hope it's all readable )

mathwonk

its important, that's a good book: here' a cheaper one:
http://www.abebooks.com/servlet/Sear...metry&x=74&y=9

and here's the all time great original version from euclid:

http://www.amazon.com/s/ref=nb_sb_no...lid+green+lion


and an excellent companion volume:

http://www.abebooks.com/servlet/Sear...ts=t&x=52&y=12

dustbin

Mathwonk, after finishing Elementary Geometry from an Advanced Standpoint and Principles of Mathematics, what would you suggest next? I'm about halfway through A&O and Chartrand's proof book, which I should have finished up relatively soon, since most of my time has been devoted to my summer calculus class.

mathwonk

the natural continuation would be a strong calculus book like spivak or apostol. since you are already taking calculus that makes sense only if your course is at a lower level.

other basic topics are topology and abstract algebra.

dustbin

Great, that will be my plan then! I just wanted to make sure there wasn't some other basic text I should work through after these. My calculus course is taught from Stewart and is almost purely computational, which is at a significantly lower level. I have done some supplementary work/reading from Apostol, but it does not line up 100% with my course in a manner that I can concurrently work through Apostol, though. I may be taking an honors, proof-based intro to linear algebra course this fall, though.

RJinkies

Hi dowland...

Edwin Moise's book Elementary Geometry from an Advanced Standpoint is one of the classics of the 60s like Coxeter's Introduction to Geometry. Both are lively and fun texts, yet they both go pretty deep. Moise doesnt make it dry and boring, and he does help out with proofs as well.

I'd say from the late 60s, geometry isnt really essential for a degree anymore, but if you wanted one text for a whole year to tackle, it was Coxeter, or maybe Moise as a second choice as the one and only 'offering'...

one interesting book was Altschiller-Court.
I think it's Modern Pure Solid Geometry from 1935, which has some of the weirder problems around. Dover has reprinted two of his books, and well the 1935 one was a 60s 70s Chelsea reprint...

The Dover reprints are:
a. College Geometry
b. Mathematics in Fun and in Earnest (recreational mathematical)

and moise should be remembered for writing a good calculus book as well as a good geometry book, as well.

-------

Hi dustbin

- Mathwonk, after finishing Elementary Geometry from and Advanced Standpoint and Principles of Mathematics, what would you suggest next?

a. Some of the New Mathematical Library titles from the 60s and 70s on geometry are good elementary and not so elementary books to collect. Originally started about 1961 by Random House and then reprinted by the MAA from about 1975-now. Sure wish they didnt update them, I think the cryptology one got a new look and more material, but i like the 1960s look of the series... It's about 40-46 books now. and 5 of the books are on geometry, two by coxeter.

------
other books:

b. Introduction to Geometry - Coxeter - Wiley 1960?/1969 Second Edition.
c. Fundamental Concepts of Geometry - Addison-Wesley/Dover - Bruce E. Meserve
[touches n some topology at the end]
d. A Course in Modern Geometries - Judith N. Cederberg - Springer
e. The Four Pillars of Geometry - John Stillwell - Springer
f. Lines and Curves: A Practical Geometry Handbook - Victor Gutenmacher - Birkhauser 2004
g. Geometry - Michele Audin - Springer [not an elementary textbook]
[if you took Differential Geometry with DoCarmo and Spivak [and Coxeter] then you can safely run through this book]
h. Geometry: Euclid and Beyond - Robin Hartshorne - Springer
[after the 1960s, two authors that stand out in geometry are Jacobs and Hartshorne]
i. Geometry for the Classroom - C.Herbert Clemens - Springer
[mathwonk uses clemens and hartshorne together as a substitution for Jacobs]
j. Modern Geometries - James R. Smart [5 editions of this one]
[a difficult text in places unless you took geometry in the 1960s]
[mathwonk's written a few things about this book]
k. Geometry: A Metric Approach with Models - Richard Millman and George Parker - Springer 1981/1991
[mathwonk's written about this one as well - it can get technical getting into things Euclid overlooked]
[MAA tosses this a 1 star recommendation - Geometry: Surveys]
l. Foundations of projective geometry: Lecture notes - Robin Hartshorne
m. The Foundations of Geometry and the Non-Euclidean Plane - G.E. Martin - Springer 1975
[clear and complete, explained beautifully]
[MAA - 1 star recommendation - Geometry: Euclidean and Non-Euclidean Geometry]
n. Transformation Geometry: An Introduction to Symmetry - George E. Martin - Springer 1982
[MAA - 1 star recommendation - Geometry: Polyhedra, Tilings, Symmetry]
o. Geometry - David A. Brannan and Esplen and Gray - Cambridge 1999
[one needs a first course in geometry before tackling this one]
[modern British approach - often used with Rees - Notes on Geometry - Springer]
p. Notes on Geometry - Elmer G. Rees - Springer 1983
[brannan and rees are sometimes used together]
q. Elementary Geometry - John Roe - Oxford 1993
[clean simple introduction to Euclidean Geometry and Differential Geomtry]
[people use Stillwell and Roe together]
[accessible if you already read one easy geometry textbook]
r. Lectures on Analytic and Projective Geometry - Dirk J. Struik - Addison-Wesley 1953/Dover 2011
[mentioned in the classic Parke III - under: Geometry: Analytic Geometry]
s. Beyond Geometry: Classic Papers from Riemann to Einstein - Peter Pesic - Dover
[Very interesting]
t. Geometries and Groups - V. V. Nikulin - Springer 1987
u. Geometry: Seeing, Doing, Understanding - First Edition and Third Edition - Harold R. Jacobs - WH Freeman - an 800 page monster
[mathwonk liked the first and second editions more of Jacobs, the third edition was an easier textbook, and the opinions are still mixed if the book is better or worse off]
[Jacobs did a kickass Elementary Algebra book - WH Freeman 1979 with an Escher cover, as well as Geometry:Seeing,Doing, Understanding. As well as the awesome and friendly text - Mathematics: A Human Endeavor]
[one flaw with Jacobs is that you don't really get taught proofs and thats probably best done with the more elementry but *rigorous* text - Geometry by Moise and Floyd
v. Geometry - Moise and Floyd
w. Euclidean and Non-Euclidean Geometries: Development and History - Marvin J. Greenberg
[half the book is accessible to most folks]

There you go.....

----
For the truly hardcore and insane you could do your own Harvard 130 - Classical Geometry course on your own in six textbooks:
a. Ryan - Euclidean and Non-Euclidean Geometry, an Analytic Approach
[short text]
b. Yaglom - A Simple Non-Euclidean Geometry and its Physical Basis
[flawed masterpiece]
c. M.K. Bennett - Affine and projective geometry
[great reference
d. Meschkowski - Noneuclidean Geometry
[short book]
e. David Hilbert - Foundations of Geometry
[looks elementary but is very subtle]
f. Euclid - The Elements
[perhaps you heard of this one]
----
----
----

All my notes from the catacombs....

mathwonk

I especially like nikulin (and shafarevich) geometry and groups. a followup to that is a book on geometry of surfaces by John Stilllwell. Another good provocative book is Experiencing Geometry by David Henderson and sometimes Daina Taimina.

Dowland

@ mathwonk, RJinkies

Hi guys, thanks for the responses. Out of pure curiosity, what's so important about euclidean geometry? The mentioned book seems to go very deep, and I suspect there's much unnecessary drilling with profoundly derived techniques, if you know what I mean.

BTW: When I come to think about it, Serge Langs book "Basic Mathematics" includes a part called "Intuitive geometry", which I suspect includes much euclidean geometry. I have ordered the book mainly to learn some algebra, trigonometry, etc. Maybe the geometry the book covers is quite sufficient for now? Do you have any experience with the book? In that case, would you say that the geometry included in the book is enough to have in your luggage when entering the world of university mathematics?

micromass

I guess I kind of disagree with the previous posters. I don't find Euclidean geometry important enough to read an indepth book on it.

Sure, Euclidean geometry is very beautiful and trains people to use logic and proofs. As such, it is valuable. But I feel that most theorems in Euclidean geometry are not used very much in university classes. For example, if you draw angle bisectors in all the angles of a triangle, then the bisectors will intersect in one point. This is a remarkably beautiful theorem. But I have never used it in my entire college education.

However, geometry is still important. And with geometry, I mean here: coordinate geometry. Knowing about equations of lines and planes, inner products, vectors, etc. That is extremely useful stuff in college education. Also, trigonometry is extremely useful. If I were you, I would focus on these two subjects.

Basic mathematics by Lang certainly covers all of these things. So I guess it is good enough. Lang also has a geometry book though that covers more stuff (and that probably covers it in more detail).

Dowland

Thanks, micromass.

By "geometry" above, I was loosely referring to "euclidean geometry". Do you know how well that's covered in Lang's book?

dustbin

Out of curiosity...
I often hear people say that Spivak and Apostol's Calculus texts are basically introductory analysis texts. What is the difference between Spivak/Apostol and books that are specifically titled along the lines of Introductory Analysis or Introduction to Anaysis (such as Rosenlicht)?