Other Should I Become a Mathematician?

[mathwonk]

keep reading books written by the best mathematicians you can enjoy and appreciate, and try to have actual one on one conversations with mathematicians, as these convey more in fewer words than any other mode of learning.

 

[Wingeer]

Great advice. I recently bought "Disquisitiones Arithmeticae" and I think I will attempt to read it this summer.

 

[Sankaku]

Not entirely. I know I should put in more effort. However I still struggle with the problem sets. Usually I have to check the solutions, and most of the time I think "Aah, of course!". There are seldom things I have to read more than once to grasp. This is maybe a problem of patience, and something I have to work on myself. Still, I wonder if it is normal to struggle with these subjects, if one compare with the same work effort as earlier courses?

It can often be a shock that upper level Math courses require a much larger investment of time than the usual Calculus sequence. The types of questions (proofs, etc.) may be different and require a different part of the brain, but doing more practice problems is very important. Essentially, you should know when you have done enough because you will feel (fairly) confident going into exams. If you don't feel confident, you haven't done enough.

If you can't get there on your own, get someone to help (office hours, TA, friends, PF, etc.).

 

[Wingeer]

Thank you for your answer.
Yes. Then the conclusion is clear. I have not done enough. Although the abstract algebra is really starting to come together.

Yes. I will have to do that.

 

[mathwonk]

with gauss, you will benefit from even one page. so do not obsess about about reading it all or even a certain amount, just read some, and think about it.

 

[Shackleford]

My calculus textbook says the three greatest mathematicians are Newton, Gauss, and Archimedes. What are your thoughts?

I may have already mentioned that. Haha.

 

[muppet]

Personally, my thought is probably that {all mathematicians in history} and the integers don't have the same order type...

EDIT: Ever have a moment when you realize you should get out more?

 

[mathwonk]

i don't know whos the greatest, those are certainly great. I appreciate archimedes especially, and i also like riemann a lot.

 

[Shackleford]

i don't know whos the greatest, those are certainly great. I appreciate archimedes especially, and i also like riemann a lot.

My probability professor talked about Cantor once or twice. He said when Cantor developed set theory and so forth, the mathematicians thought he was crazy.

The professor also talked about Riemann and the Lebesgue and how Riemann knew there was something wrong with his integral up until the day he died.

 

[Dougggggg]

If we are on the subject of great mathematicians I would like to put in a note of Euclid. Granted many of the proofs in "Elements" are rather simple to understand, the logic he did them with at the time was years ahead of its time.

So I have been doing a little bit of thinking about different branches of mathematics and have been wondering what branch I could see myself falling into. I have finished a bunch of the lower level foundational courses like Calc I, II, and an intro course to higher math (learning proof methods, set theory, mathematical logic, etc.). There are many branches I know I will get a taste of before I finish my undergraduate degree but I really want to get a feel for what else that I might really really enjoy. So given a list of things I will get a taste of before I'm done, is there anything that someone could suggest me maybe checking out to see how interested in it I am. If you know of a good textbook on it then even better. Below is that list of things that I will be getting a chance to study.

Real Analysis
Graph Theory and Combinatorics
Operations Research
Linear Algebra
Euclidean and Non Euclidean Geometries
Probability and Statistics
Modern Algebra
Differential Equations


I would be so grateful for some suggestions of other branches that have active research.

 

[jbunniii]

Euler certainly merits a mention, both for the quality and the quantity (76 volumes of the Opera Omnia published to date) of his work. All the more astonishing given that he was completely blind for the last 20 years of his life but still averaged one paper per week during much of that period.

 

[Sankaku]

Real Analysis
Graph Theory and Combinatorics
Operations Research
Linear Algebra
Euclidean and Non Euclidean Geometries
Probability and Statistics
Modern Algebra
Differential Equations

Don't forget things like Number Theory, Complex Analysis and Topology.

;-)

 

[elfboy]

it's too hard..too much to learn

 

[Dougggggg]

Don't forget things like Number Theory, Complex Analysis and Topology.

;-)

Thanks, I was talking to one of my professors Friday and he also recommended Number Theory and Topology as well. He also mentioned Differential Geometry but he said I should wait until I have taken an upper level proof course before I try self teaching myself over the summer. So my reading for this summer is probably going to be mostly philosophy.

 

[epsilon>0]

I have a few questions:

How much free time to graduate students and phd's have? Besides mathematics, there are other areas that I would like to be successful in? Is that even possible or is it necessary to prioritize? I've read that you have to want to eat, sleep, and breathe mathematics to be successfull in grad school and beyond, if I did that I know i wouldn't feel fullfilled.

https://www.physicsforums.com/showthread.php?t=148086

I read this thread, but I was wondering if anyone else had any insight.

How important is it to go to a highly ranked school? Do phd's in the top 20 or 30 have an easier time landing an academic position?

 

[kramer733]

I have a few questions:

How much free time to graduate students and phd's have? Besides mathematics, there are other areas that I would like to be successful in? Is that even possible or is it necessary to prioritize? I've read that you have to want to eat, sleep, and breathe mathematics to be successfull in grad school and beyond, if I did that I know i wouldn't feel fullfilled.

https://www.physicsforums.com/showthread.php?t=148086

I read this thread, but I was wondering if anyone else had any insight.

How important is it to go to a highly ranked school? Do phd's in the top 20 or 30 have an easier time landing an academic position?

If you've contributed to any field in science, then that's what matters. It might not be until 200 or 300 years before it gets used in some sort of application but you've still contributed to a pool of knowledge and you're helping humanity understand the world with one more step.

 

[epsilon>0]

If you've contributed to any field in science, then that's what matters. It might not be until 200 or 300 years before it gets used in some sort of application but you've still contributed to a pool of knowledge and you're helping humanity understand the world with one more step.

But you can't really contribute if you are a starving mathematician without a job. Also, for myself the teaching part of academia is important. I noticed that where I went to school, every professor in mathematics had their phd from a top 25 institution.

In professional degrees, it is necessary to go to a top school, if you want to work as an academic or get a prestigious job...is the same true with phds?

 

[tauon]

Don't forget things like Number Theory, Complex Analysis and Topology.

;-)

Speaking of Number Theory. It's one of the courses I have to take but I simply loathe. I already flunked it because I couldn't get motivated enough to study for it. How does one find the necessary excitement for it, what are some interesting results in number theory? And by interesting I mean results with interdisciplinary connections, because I find purely number theoretic results boring as hell.
I kinda feel that, considering all the other maths I'm taking in college, with this number theory course I've reverted back to some level similar to long division in middle-school.
I mean, I don't care that 2^{29} has 9 distinct digits and that you can find which digit is missing (4) without actually calculating the number, I'm completely unmoved by the fact that \phi (n) = \sigma (n) has a finite number of solutions generated by 2, 3 and 5.
In short, I do not like numbers - numbers are for computers. Half the time I was at lectures or seminars (especially seminars) I spent it thinking "you know, I can write code that can solve that much faster than any human can, wtf am I doing here?".
I know standard problem sets are somewhat simplistic and silly for any course, but I feel it's downright ridiculous with number theory. I can't help but think that there are no complex results in number theory, only complicated ones.

Help? :(

 

[mathwonk]

how about mordell's conjecture that if the smooth compact complex surface obtained by smoothing out the zero locus defined by a polynomial with integer coefficients is a doughnut with more than one hole, then there are only a finite number of rational roots?

Or that in the set of all prime numbers, the density of the subsets of those ending in 1,3,7,9 are all equal?

or that a prime > 2 is a sum of two squares iff it has form 4K+1?

or that all primes are sums of at most 4 squares?

and i like euclid a lot too. did you know he described tangents to circles as essentially limits of secants? Prop III.16.

 

[capandbells]

Does anyone know where I can find (or at least find resources to assemble myself) something like a "math roadmap"? That is, some sort of tree that shows the courses one would need to take in order to study other material. Like, if I wanted to study algebraic geometry, I'd need to first study abstract algebra, then commutative algebra first.

 

[dkotschessaa]

I would think that most math departments, like mine, would have something like this they give to their math majors. I have mine hanging on a wall next to my desk that I can stare at so I know what's ahead. I find it oddly inspiring.

Mine is not available electronically but I did find a few from other universities (I know nothing about the programs. I just googled "Prerequisite flow chart for math majors" since that's what mine was called) and they follow more or less the same logic:

http://www2.sfasu.edu/math/programs/advising/0708MathMajPrereq_tree.pdf
http://www.morris.umn.edu/academic/math/advising2.html

The gist (I've discovered) is that math doesn't "start" until at least two semesters of calculus.

-Dave K

 

[mathwonk]

I am reluctant to give lists of prerequisite books for any goal, as they become very long and make the journey seem inaccessible. Rather if you read even part of one excellent book you are well on your way to some interesting stuff.

If you want to learn algebraic geometry, try walker's algebraic curves, maybe the first couple chapters, then fulton's algebraic curves, as much as you enjoy, or miles reid's undergraduate algebraic geometry, and then shafarevich's basic algebraic geometry.

fulton's book is free online, and shafarevich's book is based on an article in Russian math surveys that is available in libraries for free.

an older book of interest is that of semple and roth, and another good modern introduction is joe harris' algebraic geometry.

if you get an old copy of shafarevich one good feature is that it includes the needed commutative algebra.

there are many other excellent well known books, but most of the books above have low entry levels of prerequisites.

 

[Nano-Passion]

In need of urgent help. =/

I never really payed attention to much math, but I started liking it freshmen college year. I took precalculus in the spring semester and put all my effort in it, likewise I got 100s and aced everything so I was pretty content and that gave me confidence that hard work and effort prevails.

But... whenever I go on the forum I hear many many alien mathematical terms that I have never heard before like topology, etc. I don't really know much math at all aside from pre-calculus and I'm only beginning to self teach myself calculus. I'm pretty much only familiar with the basics of algebra, geometry, functions/pre-calc, and some trigonometry.

What are some sources to further spice up my interest in mathematics and challenge me/get me familiar with other concepts?

 

[jbunniii]

This book may be just what you are looking for:

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

 

[Nano-Passion]

This book may be just what you are looking for:

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

Thanks, I read the reviews and they were amazing. I'm kind of excited but I won't be able to buy it for some time. Any other, but cheaper books?

 

[Ryker]

Thanks, I read the reviews and they were amazing. I'm kind of excited but I won't be able to buy it for some time. Any other, but cheaper books?

But this one's ... 15 bucks

 

[Nano-Passion]

But this one's ... 15 bucks

lmaooo. Why you have to put me on the spot Ryker? =[

Okay well I help support my family so more often than not I end up broke. And I don't get paid until the end of this week but then again I have a lot of stuff to pay for.

But yeah they probably don't get much cheaper then this hehe.

 

[Ryker]

Yeah, sorry, I didn't mean to get at you this way, it's just that, as you mentioned, it's hard to find something cheaper.

 

[zonk]

What geometry book would you recommend for someone who barely remembers high school geometry? I don't remember many of the ratios and facts about circles and triangles. Going through Feynman he uses ratios like this. I also find it hard to do optics problems. And when it talks about how you can or cannot construct certain measurements I have no idea how to do it with a straight edge or compass

 

[jwxie]

Maybe you can find these books at your school library / local library. You'd be surprised

What geometry book would you recommend for someone who barely remembers high school geometry?

I'd say here
http://www.regentsprep.org/regents/math/geometry/math-GEOMETRY.htm

Just pick up a high school geometry book. There is nothing more simpler than a pre book.

 

[zonk]

Would a reading of Euclid's Elements cut it? Does it cover secant ratios? I know it probably doesn't cover the law of cosines and sines but those are things I remember. I'm not finding any other high school level geometry books in my price range; I'll have to import it and pay a hefty price.

 

[sponsoredwalk]

Would a reading of Euclid's Elements cut it? Does it cover secant ratios? I know it probably doesn't cover the law of cosines and sines but those are things I remember. I'm not finding any other high school level geometry books in my price range; I'll have to import it and pay a hefty price.

You could get both of these books online:

Aboughantous: High School Geometry - A First Course

Solomonovich: Euclidean Geometry - A First Course

They would be e-books, not only are they the best geometry books I can find but I seen
them online for around $7 each last time I checked!

 

[Sankaku]

I'm not finding any other high school level geometry books in my price range; I'll have to import it and pay a hefty price.

Do you not have second hand books available where you are? The Elements are great for reference, but probably not the best guided overview.

There is a free high-school book here (the PDF download icon is toward the upper left):
http://www.ck12.org/flexbook/book/3461
I can't comment on its quality, but it might get you started.

The Elements can be accessed here:
http://aleph0.clarku.edu/~djoyce/java/elements/toc.html

 

[zonk]

No, I do not have second hand books and the libraries here are not as good as the ones in the USA. I tried the CK12 book; it's really not that good and I forgot most of it a few weeks after finishing it. I want a book that gives me a solid overview of geometry. I was thinking of Plainimetry but the answers are not in the back, and I figured I might need hints for some of the reasoning.

 

[Nano-Passion]

Are there any math organizations or competitions that I can join? For example in physics they have the society of physics students organization and in high school they had the olympia for math.

 

[dkotschessaa]

What geometry book would you recommend for someone who barely remembers high school geometry? I don't remember many of the ratios and facts about circles and triangles. Going through Feynman he uses ratios like this. I also find it hard to do optics problems. And when it talks about how you can or cannot construct certain measurements I have no idea how to do it with a straight edge or compass

I need a review myself, despite just having been through precalculus and almost finishing a summer semester of calculus. Soon as that's done I'm going to do some "remedial" work which will include basic geometry. First stop: http://www.khanacademy.org/ I might take a study of Euclid as well, but more for the logic than for the geometry.

-DaveK

 

[Arianwen]

Hi! I'm looking for recommendations of maths I can work on over my vacation to (a) keep mentally fit and (b) expand my horizons a little. To give a bit of background, I've done three of the four semesters of Calculus my university offers to undergrads (up to stuff like optimising multivariable functions and iterated integrals; next semester we do line integrals, Green's theorem, stuff like that, I think.) I've also done some linear algebra - pretty much Euclidean Spaces over the real numbers - and a course that dipped into half a dozen topics in discrete maths.

I've probably left things too late for a book to arrive here much before the end of vac., so if there are other sources I can use I'd love to hear about them. A good book is a good book, though and if I can use it later as well I could justify buying it anyway.

Thanks!

 

[mathwonk]

Euclid's elements is the best geometry book. The law of cosines is Props. II.12 and II.13,

if you understand them. There is a beautiful edition of Euclid from Green Lion press in paperback at about $15, as well as free ones online.

I will post some remarks about reading Euclid here in a minute or two.

 

[mathwonk]

Introduction to Euclid:
Philosophy: Euclid does geometry without using real numbers. He uses finite line segments instead of numbers, so he wants to be able to compare them, i.e. to say when they are equal, or whether one is shorter than the other, without assigning a numerical length to them. To do this he uses the concept of a straight line, and the principle of betweenness for points on a line. These concepts are not made quite precise in Euclid, but we can see some of their properties in his language.

About Euclid’s definitions:
Euclid attempts to define all concepts, but without complete success it seems. Indeed some of these ideas were not made clear for centuries after him, but he does make important progress. In particular he tries to distinguish objects of different dimensions, and gives some hint of the modern way of doing this. In definition 1, he calls a “point” something with “no part”, which is an attempt to define a zero dimensional object. We prefer now simply to say we are given certain fundamental objects called points of which all other objects of study will be composed. We don’t define the points, we just say they are given and we give some of their properties.

In definition 2, Euclid defines a “line”, [we would call it a “curve”, allowing it to possibly be straight], as something with only length but no breadth, an attempt to say it has only one dimension. This is not a precise definition, but in definition 3 he says that the extremities of a line are points. This gives a clue to the modern inductive description of dimension. Namely we have some way to recognize the border of an object, and an object should be one dimensional if its border is zero dimensional, i.e. if the border consists of a finite number of points.

The same pattern occurs in definitions 5 and 6, where in 5 a surface is something two dimensional, and in 6, we see that the border of something two dimensional should be one dimensional. This is a general pattern, that a border should have dimension one less than the thing it borders.

Today we focus more on the relationship between our objects than on the nature of those objects. So in different situations, what are called points or lines or surfaces could be different things, but in all situations the points will be related to the lines in the same way. I.e. whatever the points are, they should border the curves, and whatever the curves are they should border the surfaces, etc…

So today mathematicians tend to ignore Euclid’s definition 1, and to consider definitions 2 and 5 to be clarified by definitions 3 and 6. Unfortunately definition 4 of what it means for a curve to be straight, is not clarified by any additional property, and we will need one in Prop. I.4. The usual one taken nowadays as basic for straight lines is that two different lines which are both straight, can only meet in one point. This is related to Euclid’s 1st Postulate, that one can draw a straight line between any two points, but only if that means one and only one straight line, so this is the usual modern postulate. So to guarantee that two different lines can only meet once, we need more or less the converse of Euclid’s 1st postulate. I don’t know the original Greek, so I do not know if the words “a straight line” used in that postulate mean “exactly one straight line”.

Terminology that Euclid used differently from mathematicians today
Euclid seems to mean by “straight line” only a finite portion of an infinite straight line. Today we call such finite pieces of lines, line segments, or finite line segments. When Euclid wants to speak of an infinite straight line, he speaks of a (finite) straight line being extended indefinitely or calls it explicitly an infinite straight line. So what he calls a line today we call a curve, what he calls a straight line today we call a line segment, and what he calls a line segment extended infinitely in both directions, or an infinite straight line, we just call a line.

Definition 8 describes an angle as the “inclination” made by two straight line segments which meet but are not in the same straight line. It is not clear to me whether they meet at an extremity, but apparently in that case he considers only the convex angle they make together. E.g. the outside of a 90 degree angle is not considered by him as a 270 degree angle. (Since he does not consider 270 degree angles, it is harder for him to “add” two 135 degree angles.) He defines a right angle as one of the angles formed by two lines that form equal angles. Presumably in this case the lines do not meet only at extremities, since they form more than one convex angle.

Definition 15 describes a circle, but again not quite completely. He says a circle consists of a point called the center, together with a collection of line segments all having that center as an extremity, and all having the same length. But he does not say whether all segments of that length are included, as presumably they should be. E.g. a semicircle seems to satisfy the description given, since all line segments from the center to any point of the semi circle are equal to one another.

We assume he meant that a circle is the figure formed by a center and a segment with that center as extremity, plus all other segments having the same center as an extremity, and which are equal to the first segment. Thus he includes the entire region on and within the circle, whereas today we mean by “circle” only what he calls the circumference or boundary of his circle. I.e. we take a center point A and a segment XY, and we consider the circle to consist of all those points B such that the segment connecting B to the given center A, is equal to the segment XY. It follows that two circles with the same center have either the same circumference, i.e. are the same circle, or else their circumferences are disjoint, i.e. have no common points at all. He is not quite consistent since later he says a circle cannot cut another circle at more than 2 points, apparently referring to their circumferences.

Euclid’s five postulates:
Here are the postulates Euclid explicitly stated (slightly paraphrased):
1. Given any two points, one can draw a straight line (segment) joining them.
2. Given a finite line segment, one can extend it continuously in a straight line, (presumably infinitely in both directions).
3. Given any point as center, and any other point (distance), one can describe a circle centered at the first point and with the other point on the circumference.
4. All right angles are equal.
5. If two lines cross a third line so as to make interior angles on one side total less than a straight angle (two right angles), then the two lines meet on that same side of the third line.

Note Euclid clearly assumes in postulate 5 that a line has two sides. Also there is nothing here asserting that parallel lines exist - rather this has the opposite flavor, guaranteeing that certain lines are not parallel. So this is not the usual parallel postulate I learned in high school. (Through a point off a line, there passes one and only one line parallel to the given line.)

This postulate will imply there is not more than one line parallel to a given line and containing a given point off that line. In the other direction, Euclid will actually prove there is at least one such parallel, using his “exterior angle” theorem.

The properties that Euclid used most without stating them concern how lines and circles meet each other. In modern mathematics we discuss these in terms of connectivity or separation properties. A set is convex if for every pair of points in the set, the straight line segment joining them is also in the set. E.g. a straight line segment is convex. Then Euclid seems to assume basic facts like the following: removal of a point other than an extremity separates a segment into two convex pieces. Removal of an infinite line from the plane separates the plane into two convex “sides”. Removal of the circumference of a circle from the plane separates the plane into two pieces, one of which: the inside, is convex, and the other: the outside is at least “connected” [in what sense?].

What do we mean by “separates”? We mean the segment joining a point inside to a point outside should meet the border which was removed. So if two points of the plane are on opposite sides of a line, then the segment joining them should meet the line. Thus the line separates the two ides of the plane, and forms the border of both sides. If one point is inside and another point is outside a circle, the segment joining them should meet the circle. We can say something about the shape of a circle if we agree that any (infinite) line containing a point inside a circle should meet the circle exactly twice. And we might be wise to agree that a circle that contains a point inside and a point outside another circle also meets that circle exactly twice.

Some of these facts about how circles and lines meet can be proved, and Euclid does so, but others cannot be proved. In general one can prove that circles and lines cannot meet more than expected, but I do not know how to prove that they do meet as often as expected, without more assumptions than Euclid has made. Today many people assume that lines correspond to real numbers, which does guarantee that lines and circles meet as often as expected, since the axioms for real numbers guarantee this. However most geometry books which make these assumptions about lines do not bother to explain the relevant axioms for real numbers, so to me not much clarity is gained, and perhaps some is lost.

Euclid has one postulate (#5) guaranteeing that two lines do meet under certain conditions, but he was criticized for centuries for including this postulate. It turns out he was right, as this postulate cannot be omitted without broadening the possible geometric worlds he was trying to describe. People were unable to imagine any other geometry than Euclid’s however for a long time where this postulate could fail. A Jesuit priest, Girolamo Saccheri, showed that if we deny Euclid’s 5th postulate then there would not exist any rectangles. This and other consequences seemed so impossible to Saccheri that he concluded Euclid’s axiom must be automatically true, and thus did not need to be stated explicitly. However, there is another plane geometry in which there are no rectangles, called hyperbolic geometry, and unless we assume Euclid’s 5th postulate we cannot be sure we are not in that world instead. Today the results Saccheri correctly deduced , but considered impossible, are regarded as theorems in hyperbolic geometry due to him.

So we regard Euclid’s stated definitions and postulates, plus the ones he used but did not state, as rules for the game we are going to play. They tell us what we can do, and we want to deduce as many consequences from them as possible, without violating the rules.

The problem of congruence
If two triangles have vertices A,B,C and X,Y,Z, a correspondence between their vertices, e.g. A→Y, B→X, C→Z, induces correspondences between the sides: AB→YX, AC→YZ, BC→XZ, and the angles: <ABC→<YXZ, <ACB→<YZX, <BAC→<XYZ.
If a correspondence between the vertices induces correspondences of sides and angles such that every side and every angle equals the one it corresponds to, we call the correspondence a “congruence”.

Notice a congruence must be given by a specific correspondence. It is not sufficient just to say two triangles are congruent, one must say what correspondence induces the congruence. E.g. the triangles ABC and XYZ may be congruent under the correspondence A→Y, B→X, C→Z, but not under the congruence A→X, B→Y, C→Z. Other very symmetrical triangles may be congruent under more than one correspondence, but we should always say what correspondence we mean.

Exercise: Given an example of two triangles that are congruent by more than one correspondence.

The first question we ask in plane geometry is when two triangles are congruent, given only a smaller amount of information. The basic criteria are sometimes called SAS, SSS, ASA, and AAS. E.g. “SSS” is shorthand for the fact that if a correspondence of vertices induces a correspondence of sides such that all corresponding pairs of sides are all equal, then all corresponding pairs of angles are also equal, and hence the triangles are congruent. “SAS” refers to the fact that if two corresponding pairs of sides are equal, as well as the pairs of included angles, then the triangles are congruent. Etc…

Once one knows these basic criteria, most geometry courses proceed in the same way, at least for while. Getting started thus means establishing these basic congruence facts. Some books assume them all, while some assume only a few of them and deduce the others. Euclid proves them all, but only by making some implicit assumptions that he has not included among his axioms. See if you can spot some of those assumptions.

 

[mathwonk]

nano - passion, one of the best introductions to topology is First concepts of topology, by Chinn and Steenrod.

here is one for under $5 from the excellent used book site abebooks:

First Concepts of Topology: The Geometry of Mapping of Segments, Curves, Circles and Disks
Chinn, W.G.; Steenrod, N.E.
Bookseller: Meadowlark Books
(Hawley, MN, U.S.A.)

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[dkotschessaa]

Thanks for this guide to Euclid. I will likely be taking a look at this as soon as my break starts.

 

[Nano-Passion]

nano - passion, one of the best introductions to topology is First concepts of topology, by Chinn and Steenrod.

here is one for under $5 from the excellent used book site abebooks:

First Concepts of Topology: The Geometry of Mapping of Segments, Curves, Circles and Disks
Chinn, W.G.; Steenrod, N.E.
Bookseller: Meadowlark Books
(Hawley, MN, U.S.A.)

Bookseller Rating:
Quantity Available: 1
Book Description: Random House, NY, 1966. Soft cover. Book Condition: Very Good Minus. 1st Edition. 8vo - over 7¾ - 9¾" tall. VG-. Text has a couple underlines in intro; name on ffep; page edges, white covers, have slight fading. Bookseller Inventory # 001453

Bookseller & Payment Information | More Books from this Seller | Ask Bookseller a Question


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I'm confused, I didn't ask for topology I'm only up to calculus. Or should I get this book because its a good introduction to geometry, calc, etc.?

 

[mathwonk]

when you wrote:

"But... whenever I go on the forum I hear many many alien mathematical terms that I have never heard before like topology, etc. "

it seemed to me you were asking to learn about topology.

The book on topology I recommended to you is for the average high school educated adult, with no knowledge of calculus. Topology is more elementary than calculus.

Topology is the study of continuity, whereas calculus adds the concept of differentiability.

 

[Vector Field]

Sorry, to bring up something old, but I read this on the first page. I believe you have given some bad advice. I don't know if it was addressed later and you mentioned someone should ask an Applied Mathematician. That would sort of be me.

You advised someone to stay away from majoring in Mathematics/Economics. This was not entirely good advice. Governments all over the place employ armies of mathematicians to study these things. If a person majors in Applied Mathematics they will also be expected to learn the Pure Math as well, Applied Math doesn't mean you sit around learning mechanical problem solving. You need to analytically solve the problems too.

In looking up a lot of jobs data I have noticed that Applied Mathematicians earn 10,000USD more, on average, than Pure Mathematicians. However, in researching this further it is recommended that you choose a field that best interests you. If the person was quite interested in Economics, then they should certainly major in that. Applied Math without a second discipline is harder to find employment with. It is certainly not impossible, but if there is a particular field you love and find interest with, then you are more likely to find employment in that field as a mathematician if you also do some work in that discipline.

 

[dkotschessaa]

Wow, Mathwonk. You blew my mind. I had assumed (since topology is generally offered some time after differential equations and linear algebra) that topology was necessarily an advanced topic. I am sitting in the math section of my library right now and wasn't 20 feet away from the book you mentioned. And by gawd, no calculus.

I seem to have a weakness in math when it comes to the the graphing side of things. The "thinking geometrically" part of my brain is not developed yet. (My current professor thinks very geometrically so I had a hard time following his thinking at times.) I was going to spend my time off going over basic conics etc., again. But do you think this might be an alternative to strengthen that sort of understanding?

-DaveK

 

[mathwonk]

dear vector field. you are attributing to me it seems, advice given by someone else on page one.

i often give bad advice just not that piece.

 

[mathwonk]

DaveK, well, if you don't know geometry, you might also benefit from studying euclid. i just had a long post erased by this finicky browser where i argued that euclid is the best preparation for calculus.

 

[dkotschessaa]

That was something I was thinking of doing, but then you got me all excited about topology. I should probably stay focused. Topology can wait for me.

-DaveK

 

[Vector Field]

dear vector field. you are attributing to me it seems, advice given by someone else on page one.

i often give bad advice just not that piece.

Oh, it wasn't entirely you. It was an exchange made with another poster. You recommended talking to an Applied Mathematician.

 

[Nano-Passion]

when you wrote:

"But... whenever I go on the forum I hear many many alien mathematical terms that I have never heard before like topology, etc. "

it seemed to me you were asking to learn about topology.

The book on topology I recommended to you is for the average high school educated adult, with no knowledge of calculus. Topology is more elementary than calculus.

Topology is the study of continuity, whereas calculus adds the concept of differentiability.

Wow, I'm surprised. Thank you. Any other book you would recommend? I want to get a general handle around mathematics and spike up my interest.