Calculus Calculus by Spivak: Understanding Problems in Chapter 1

[PFuser1232]

So recently, I started reading Michael Spivak's "Calculus" to shore up my understanding of the subject, after being told that this text is well-known for its rigor. In chapter one, he lists 12 properties (or axioms) of numbers (distributive law, trichotomy law, and closure under addition, to name a few). I have a problem with the exercises for that chapter. They seem extremely trivial at first sight (for example, prove that $\frac{a}{b} = \frac{ac}{bc}$ for $b, c ≠ 0$), and that's the problem. I can't understand the approach that is expected to be followed while answering such questions. Should I put myself in the mindset of someone who has never learned algebra before, and solve these questions (however redundant they may seem) in a systematic way that employs the 12 properties listed earlier in that chapter? Or should I proceed with whatever procedure I had been using before starting the text?

 

[godingly]

Absolutely. There's only one way to do math these days - and that's rigorously. If you'll study math, many of your first courses (including Calculus) will begin with this (rigorous introduction to number theory), and work their way up to more advanced topics. There's an answer book somewhere, it could certainly come in handy. My advice - don't try to do all the exercises, and once you've proven something - add it to your arsenal, you don't have to prove it again.

 

[slider142]

Yes. The point of these exercises is to demonstrate to the reader that the 12 properties he has listed are sufficient to produce all of the algebraic "rules" that they are familiar with from memorization in elementary classes. During the next few chapters, he will attempt to demonstrate just how much these seemingly simple properties imply about the structure of any system one might model with them. They almost imply calculus itself, and he will attempt to show exactly where calculus begins and pure algebra with these 12 properties falters.
However, be aware that this is not quite "full rigor". It is an introduction to the mindset of an analyst, and is an excellent bridge between rote mathematical formalism of applying memorized rules, and pure mathematics, where one is interested in solving general problems using only logical apparatus. Once you have completed Spivak's Calculus, you might want to check the level of rigor of a pure analysis text like Rudin. But it is not recommended to read Rudin before a gentler text like Spivak, as Rudin tends to put conciseness before explanation.

 

[akki31]

hi everyone, by degree i m an electronics & telecomm engineer, but quite interested in leaning theoretical physics,so i asked my professor about it, he said do not directly start with physics but learn necessary mathematics first, so after searching for the best book to learn calculus on the internet i found out about spivak calculus and i started it recently,,book is really unique and some its problems are very challenging , i have to refer solution manual every now n then but i know it's not always right way to solve the problems ..currently m on chapter 2 'numbers of various sorts' and m able to solve few unsolved problems of it but some are really hard so my query is,is it really necessary to do all the exercises of this chapter (and previous chapter) or i can start with functions and further chapters?

 

[slider142]

...is it really necessary to do all the exercises of this chapter (and previous chapter) or i can start with functions and further chapters?

It is definitely not necessary to complete all the problems! However, I would recommend reading and attempting the problems long enough to understand the point of each question before moving on, as the content of many early questions are used later on in the text (some will be fast; some will seem strange for awhile until you get much further into the text). They are, in other words, part of the narrative, unlike the rote exercises one might find in less rigorous texts. However, being able to complete every question on a first reading would be a monumental task!
In regards to Chapter 1, be sure to at least attempt problems 20-23. It is the attempt to understand, the seeding of certain types of thought patterns, and the determination required to follow through on questionable hunches that matters, more than being able to complete the specific problem. Be prepared to erase a lot; it is why I use a chalkboard. And don't be afraid to leave the book after trying a hard problem and do other, unrelated things. Sometimes the brain gets stuck in one narrow approach, and new avenues of approach will occur to you when you are not even consciously thinking of the problem.

 

[akki31]

It is definitely not necessary to complete all the problems! However, I would recommend reading and attempting the problems long enough to understand the point of each question before moving on, as the content of many early questions are used later on in the text (some will be fast; some will seem strange for awhile until you get much further into the text). They are, in other words, part of the narrative, unlike the rote exercises one might find in less rigorous texts. However, being able to complete every question on a first reading would be a monumental task!
In regards to Chapter 1, be sure to at least attempt problems 20-23. It is the attempt to understand, the seeding of certain types of thought patterns, and the determination required to follow through on questionable hunches that matters, more than being able to complete the specific problem. Be prepared to erase a lot; it is why I use a chalkboard. And don't be afraid to leave the book after trying a hard problem and do other, unrelated things. Sometimes the brain gets stuck in one narrow approach, and new avenues of approach will occur to you when you are not even consciously thinking of the problem.

thanx a lot for such insights...i was in dilemma before starting further chapters having not solved all the problems from exercise