Algebra Math texts that make you fall in love all over again

[ibkev]

I often see textbook recommendations here that about rigour and depth of coverage. Serious books for serious people!

This question is different. I'm looking textbooks that inspire a love for mathematics. And I want to come at it from two different angles:

(1) is there a textbook that was so good, led to such beautiful insights that it was almost a religious experience and cemented your decision to be a student of mathematics?

(2) What textbooks would you recommend a casual, self-studying, student of math, that keeps them motivated and inspired to keep moving forward? Assumptions:

• student has a day job and only so much time to devote
• ideal texts have a big payoff/effort ratio (possibly sacrificing some depth in the process)
• inspires love and interest in the subject
• not discouraging, in the sense of a punishing difficulty level (we all know texts that are viewed as a right of passage!)

The form forced me to choose a prefix but please feel free to respond about any math topic/level you like!

A friend recommended this one:
https://www.amazon.com/dp/0198534469/?tag=pfamazon01-20
He said it is full of diagrams and graphs and is incredibly clear.

 

[Let'sthink]

I think loving or getting inspired by maths or its application is an intellectual exercise, which involves only thinking. it is not like listening to music and dancing and watching things. Those are experiences which consume time and you experience pleasure of a different kind which does not involve much thinking. Thinking involves time so the raw material for extracting pleasure out of maths and its application is in the first instance TIME only. Time and willingness to think are the two essential internal inputs the external inputs such as discussion and books can just help!

 

[smodak]

https://www.amazon.com/dp/0801896940/?tag=pfamazon01-20

https://www.amazon.com/dp/144196052X/?tag=pfamazon01-20

https://www.amazon.com/dp/0679776311/?tag=pfamazon01-20

 

[vanhees71]

Two volumes by Courant and Hilbert on Mathematical Methods in Physics:

https://en.wikipedia.org/wiki/Methoden_der_mathematischen_Physik

 

[Logical Dog]

What is mathematics? richard courant ---> difficult book to follow for me personally but an excellent holistic introduction to mathematics fundamentals.

Book of proof. really rigourous and good intro to basic foundations of maths. FREE HEREhttp://www.people.vcu.edu/~rhammack/BookOfProof/

Elias zakon lecture notes ---> can be found free online, legally.

Discrete mathematics by Norman biggs (I ordered the Indian economy edition on amazon much cheaper).
link for you: https://www.amazon.com/dp/0198507178/?tag=pfamazon01-20

how to think about analysis
https://www.amazon.com/dp/0198723539/?tag=pfamazon01-20

 

[Mondayman]

I am relatively new to mathematics (I only discovered a passion for it at 23), but two books that did it for me were:

Mathematics and the Physical World - Morris Kline

A First Course in Calculus - Serge Lang

I'm sure as I learn more I'll have more books to add.

 

[jonjacson]

Simply outstanding books:

https://www.amazon.com/dp/0486409163/?tag=pfamazon01-20

 

[Stella.Physics]

This one is not pure Mathematics but still a book full of inspiration:

"Gravity: an introduction to Einstein's General Relativity" by James B. Hartle

 

[Logical Dog]

What is mathematics? richard courant ---> difficult book to follow for me personally but an excellent holistic introduction to mathematics fundamentals.

Book of proof. really rigourous and good intro to basic foundations of maths. FREE HEREhttp://www.people.vcu.edu/~rhammack/BookOfProof/

Elias zakon lecture notes ---> can be found free online, legally.

Discrete mathematics by Norman biggs (I ordered the Indian economy edition on amazon much cheaper).
link for you: https://www.amazon.com/dp/0198507178/?tag=pfamazon01-20

how to think about analysis
https://www.amazon.com/dp/0198723539/?tag=pfamazon01-20

Principles of mathematics by allendoerfer and oakley.

Logic for dummies by mark z. (good book)

free legal link to zakons book : http://www.trillia.com/zakon1.html

I=s such as discussion and books can just help!

hello, I would say the authors guide your thinking, when you learn from the best you get a completely different and refreshing viewpoint. :) ...and one day, maybe, you can become like them :P but reading the books is not enough, you must DO the exercises too..but without a good guide to the subject it is difficult for the less motivated people like me to study..I think this is what the op meant, there are a lot of garbage texts out there too.

 

[houlahound]

Gross & Miller, mathematics a human endeavour.

Intro level text: it explained to me why high school math is true. I just kept saying as I read it...aaaah that's why.

 

[S.G. Janssens]

hello, I would say the authors guide your thinking, when you learn from the best you get a completely different and refreshing viewpoint. :) ...and one day, maybe, you can become like them :P but reading the books is not enough, you must DO the exercises too..but without a good guide to the subject it is difficult for the less motivated people like me to study..I think this is what the op meant, there are a lot of garbage texts out there too.

Yes, I agree. High quality books have been essential for me since university, as I have always found it difficult to follow lectures or other forms of oral presentation. (I was and am quite motivated, but lectures often go too quickly for me and sometimes skip important details.)

I am still thinking about what I will reply to the OP.

 

[Logical Dog]

Yes, I agree. High quality books have been essential for me since university, as I have always found it difficult to follow lectures or other forms of oral presentation. (I was and am quite motivated, but lectures often go too quickly for me and sometimes skip important details.)

I am still thinking about what I will reply to the OP.

Not only will some lectures skip important details, but they will often not provide questions and insight like the real authors do that give you the most valuable insights into mathematical objects and mathematics in general because they don't have the time or motivation, or this type of thinking is useless for your course anyway, one must really do a lot of self learning if he wants to get under the "hood" of this subject and/or explore its philosophy. It has been a few months for me since this I found this desire, I am a bottom feeder in mathematics, hundreds or thousands of years behind the boundary of this body of knowledge, but I want to get better and maybe even do a degree, that's why I know these books. :)

Luckily I have a good lecturer who knows a lot about the subject :). If someone is really into math they should get a maths degree! One should not expect to be taught in this manner in a bachelor in engineering or other applied math heavy subjects. Same goes for physics, if you want to deal with high level physics, do physics instead and not a bachelor in other subjects.

 

[houlahound]

Yeah but if you want a job do applied mathematics.

 

[Student100]

Yeah but if you want a job do applied mathematics.

Err...?

To the OP, maybe you can give us a bit of background as to where you're actually at in terms of mathematics?

 

[houlahound]

Just a light hearted jab. My lecturers always used to make jabs where the pure science and math guys boast how smart they are and cite their papers in prestigious journals. The engineers and applied guys would cite their pay packets.

Me personally I did honours level undergraduate (4 years degree) in physics then got a job in coal as a wire logger (underground gamma spectroscopy) and nondestructive testing. The math was basic that any decent high schooler could do it. The company made millions tho and created hundreds of jobs.

 

[Simon Phoenix]

My vote for a book in category (2) would be "Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics" by John Derbyshire
https://www.amazon.com/dp/B004D39PGU/?tag=pfamazon01-20

I loved it - it's aimed at a more general audience but it does a heroic job of explaining the Riemann hypothesis, its significance, and progress towards a proof (or otherwise) up to about 2003. It's a mix of maths and history and roughly speaking the chapters alternate between the two.

It's not really a textbook, sorry, but it is inspiring

 

[Laurie K]

Elementary Calculus by H J Keisler https://www.math.wisc.edu/~keisler/calc.html

 

[Let'sthink]

At elementary level George Gammow's "One, two, three, ... infinity" is also awe-inspiring! The idea of one to one correspondence makes impossible looking propositions become possible and proveable like the number of points on two unequal line segments are equal. When infinity is there anything is possible!

 

[haushofer]

I love the books of Tony Zee, but that's theoretical physics. Inspiring, funny, witty, with historical sidemarks and topics you don't find in other books.

 

[jonjacson]

Elementary Calculus by H J Keisler https://www.math.wisc.edu/~keisler/calc.html

You may be interested:

http://www.17centurymaths.com/

 

[BOAS]

The book that convinced me I was capable of pursuing mathematics (physics) further, was "Calculus Made Easy" - Martin Gardner.

It isn't the best or my favourite book, but it is partly responsible for getting me this far, so it has a special place for me :)

 

[jonjacson]

https://en.wikipedia.org/wiki/Elements_of_Algebra

This one is really nice too.

 

[vanhees71]

Another one, I liked very much when I read it the first time is Weyl's "Raum, Zeit, Materie" (Space, Time, Matter), although the mathematicians of his time ahorred it. There is the famous story that Heisenberg went to the famous mathematics professor Lindemann in Munich (who proved that $\pi$ is transcendent) asking about the prospects of studying math. When he mentioned that he had read Weyl's book, Lindemann told him: "You are lost for mathematics." ;-).

 

[S.G. Janssens]

(1) is there a textbook that was so good, led to such beautiful insights that it was almost a religious experience and cemented your decision to be a student of mathematics?

I think this is an interesting thread. Various books that I have encountered during my studies may fit your description, although my experiences were not exactly religious. Most influential was probably One-Parameter Semigroups for Linear Evolution Equations by Engel and Nagel. I never since gained more insight into the topic by reading a single book. (Please do not let the "linear" deceive you. Numerous nonlinear problems can be attacked by perturbing linear problems and, besides, linear problems are of great interest in their own right.) This book should be required reading for anyone who routinely "exponentiates" differential operators using ordinary power series.

More recently, I started reading C.D. Meyer's Matrix Analysis and Applied Linear Algebra. It reinforced my long-standing love for the subject and taught me that new things can be learned about material that I believed I already knew inside-out. It also demonstrated that teaching LA could be fun instead of a punishment.

(2) What textbooks would you recommend a casual, self-studying, student of math, that keeps them motivated and inspired to keep moving forward? Assumptions:

• student has a day job and only so much time to devote
• ideal texts have a big payoff/effort ratio (possibly sacrificing some depth in the process)
• inspires love and interest in the subject
• not discouraging, in the sense of a punishing difficulty level (we all know texts that are viewed as a right of passage!)

It depends a lot on the level and interest, of course. Meyer's book is a good option for a second course on linear algebra, or for a first course at a somewhat higher-than-usual level. In particular, it is suitable for self-study because it comes with a solution booklet by default.

 

[robphy]

interesting insights and methods in each of these...

Bamberg & Sternberg - A Course in Mathematics for Students of Physics
https://www.amazon.com/dp/0521406498/?tag=pfamazon01-20
https://www.amazon.com/dp/0521406501/?tag=pfamazon01-20

https://www.amazon.com/dp/0521269296/?tag=pfamazon01-20

https://www.amazon.com/dp/0961408804/?tag=pfamazon01-20

 

[jedishrfu]

My most recent favorite has been Math 1001 by Dr Richard Elwes. Its a survey in bite-size paragraphs of many areas of math.

https://www.amazon.com/dp/1554077192/?tag=pfamazon01-20

 

[Skywalker1415]

For (2): I'm currently reading Strang's Calculus (free at MIT OCW https://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/) and thoroughly enjoying it. I'm reading mostly on my own, though I have an exam to take. Gives me the feeling of having a friendly and excellent professor explaining the subject to me. Perfect for where I am at, but might be considered too wordy by some.

 

[jonjacson]

https://en.wikipedia.org/wiki/Klein's_encyclopedia

Anything I read written by Felix Klein is fantastic, I wonder if his encyclopedia has ever been translated. I guess no but, ¡I had to ask!

 

[Dennis Plews]

I think loving or getting inspired by maths or its application is an intellectual exercise, which involves only thinking. it is not like listening to music and dancing and watching things. Those are experiences which consume time and you experience pleasure of a different kind which does not involve much thinking. Thinking involves time so the raw material for extracting pleasure out of maths and its application is in the first instance TIME only. Time and willingness to think are the two essential internal inputs the external inputs such as discussion and books can just help!

In thinking about this I came upon the realization, which is hypothetical nature only, that neurons like making new synaptic connections and enhancing neural networks. That is to say, all cells within our body function best when they are doing what they're designed to do (e.g. muscles and bones get stronger when lifting heavy loads). Since neurons make synaptic connections, learning new things should be pleasurable. I know I enjoy learning new things and would love to know if dopamine is released in the process.

 

[dustball]

Try "Mathematical methods of classical mechanics," "Ordinary differential equations" and "Partial differential equations" by V.I. Arnold. "Analysis by its history" by Hairer and Wanner is very nice too. Enjoy :-)

 

[M Quack]

Not a textbook, but it certainly inspired me, has a solid math background, and is good for casual reading:

Doug Hofstadter: Godel Escher Bach: An eternal golden braid.

https://www.amazon.com/dp/0465026567/?tag=pfamazon01-20

 

[sukalp]

What is mathematics? richard courant ---> difficult book to follow for me personally but an excellent holistic introduction to mathematics fundamentals.

Book of proof. really rigourous and good intro to basic foundations of maths. FREE HEREhttp://www.people.vcu.edu/~rhammack/BookOfProof/

Elias zakon lecture notes ---> can be found free online, legally.

Discrete mathematics by Norman biggs (I ordered the Indian economy edition on amazon much cheaper).
link for you: https://www.amazon.com/dp/0198507178/?tag=pfamazon01-20

how to think about analysis
https://www.amazon.com/dp/0198723539/?tag=pfamazon01-20

well it is easy mathematics is a study of quantities represented in form of numerals.
numerals-symbols used to represent numbers.

 

[sukalp]

math text that make us fall in love all over again is physics,chemistry problem solving books helps make us fall in love in maths.
as physics,chemistry are a part of science and laws of nature are written in language of maths .

science needs proof of theory written upon particular phenomenon maths makes that proof

 

[Logical Dog]

well it is easy mathematics is a study of quantities represented in form of numerals.
numerals-symbols used to represent numbers.

teach me your ways sensei

 

[sukalp]

teach me your ways sensei

okay so sensi is
step 1-Break concepts part by part
step 2-Skim the first half concepts
step 3-Then understand the half concepts ,underline notes,think in form of pictures,do lovingly
step 4-Then next do review
step 5-follow step 1,2,3,4 for 2nd half concepts
step 6-review the chapter Revise again and again ,take practical examples,give test ,join quiz,join forums
solve sample papers
done

 

[Logical Dog]

okay so sensi is
step 1-Break concepts part by part
step 2-Skim the first half concepts
step 3-Then understand the half concepts ,underline notes,think in form of pictures,do lovingly
step 4-Then next do review
step 5-follow step 1,2,3,4 for 2nd half concepts
step 6-review the chapterRevise again and again ,take practical examples,give test ,join quiz,join forums
solve sample papers
done

There is no generic algorithm for learning maths effectively.. What you have suggested is what the kids do at as and a level, that's no way of learning math imo. good try nonetheless. You must also remember everyone has a different version of what they believe is "having learned something"...there are many levels to every statement in a subject, then there is the philosophy and history, we think we have learned "enough" because we can convince ourselves and draw the line somewhere. It will take me years to be able to say to myself, yes, I have learned mathematics.

edit: I have slept two hours in the past two days, not had my medicines so may not be as eloquent as usual :P apologies

 

[sukalp]

There is no algorithm for learning. What you have suggested is what the kids do at as and a level, that's no way of learning math imo. good try nonetheless. You must also remember everyone has a different version of what they believe is "having learned something"...

it is steps how to use spacing effect technique for learning it is better than cramming
from this link i have learned

 

[jedishrfu]

Let us not forget the ET Bell book Men of Mathematics and the Lancelot Hogben book Mathematics in the Making which has been the inspiration for many well known scientists and mathematicians.

https://en.wikipedia.org/wiki/Men_of_Mathematics

 

[Logical Dog]

https://global.oup.com/academic/pro...athematics-major-9780199661312?cc=de&lang=en&

Another good book from Ms. Alcock.

 

[MidgetDwarf]

I would say it was Edwin E Moise: Geometry. It made me enjoy mathematics and I learned a lot of things that were left out of my geometry course.

Kleppner and Kolenkow made me enjoy physics. I found it extremely boring before I found this book.

 

[Aufbauwerk 2045]

Although it's not a math book, when I was a student I was inspired to learn more math by reading Einstein's original paper on special relativity, the famous On the Electrodynamics of Moving Bodies. It encouraged me because I was not very far from knowing all the math in the paper, specifically partial differential equations, so I saw the light at the end of the tunnel. The same thing happened when I was learning about general relativity.

I was also inspired by reading Ulam's autobiography. He was a mathematician who was instrumental in the development of the H-bomb. He wrote about how amazed he was when he realized that a few mathematical symbols on a piece of paper could change the world.

In other words, nothing motivates me to learn math more than seeing how powerful it is when it comes to understanding the universe!