What Are the Most Important Concepts for Entering Calculus?

[MJC684]

Does anyone know of any good precaluclus-level books that are good for learning how to write basic proofs? I have a lot of very nice 1960-early 70s era texts that ask for proofs in the majority of the exercises but the problem is that not only do I not know how to write proofs, but there are no answers to the proof questions in the back of the book. I intend major in mathematics in the fall and i feel that I should begin learning the skill now - or am I jumping the gun? What are the most important concepts/skills to enter into calculus with? ( I am going to be finishing the second half of Precalc over the summer on my own). What should I study this summer in order to get the most out of time?

One more thing. How important is propositional logic when it comes to beginner proofs? Is logic needed for any level of proof? I ask this because most of the older texts that I own that i have all the proof questions make no mention of needing to know logic to prove what they are asking you to prove. Is it just assumed that you already do know logic or that you should be able to write the proofs just by studying the definitions and theorems?

Thanks for any help.

PS: If proof is the name of the game in college mathematics, then why are all the "modern" precalculus texts like Larson's, Blitzer, Swokowski etc completely lacking in this area? Are do beginning math majors not really need to know that stuff yet

 

[thrill3rnit3]

mathwonk would be the one to ask about older textbooks, but unfortunately he hasn't been active in a while

and I agree, newer textbooks are watered down and have lost the rigor older textbooks had.

 

[EAAL]

I think that knowing the basics of logic is important. It let's you have a feel of how the premises can be used to get to the conclusion, by only taking "valid" steps. Of course, if you read lots of proofs, and practice reproducing them, and with other exercises, you get the feel for it, and don't have to translate it into logical language, but it comes out in a more "natural" way.

 

[axeae]

I'm not sure if it's "precalculus level" but I've heard "how to prove it" by vellerman is a good intro to proof writing book. I used it briefly but I don't really remember a whole lot about it.

 

[MJC684]

yea I have heard similar things about the velleman book. So do you guys think that working through an actual intro to proof writing text is the right way to go? Or is there some other method I should start out with?

 

[m00npirate]

I would recommend "An introduction to mathematical reasoning" by eccles

 

[MJC684]

Also, do you guys feel that concentrating on getting a feel for writing proofs at the this stage in the game (precalculus) is the right move? Or should I be devoting the majority of my time to mastering trig and analytic geometry, mathematical induction, sequences, series, sigma notation etc. ? What would you guys do if you were me?

 

[MJC684]

I've had someone else recommend the same book to me moonpirate. I'll have to check it out.

 

[qspeechc]

There is Allendoerfer (sp?) and Oakley's book "Principles of Mathematics" which gives a very gentle intro to proofs, logic, set theory etc. Covers some more advanced stuff like rings, groups etc, but also pre-calc stuff, like trig. geometry etc. Has half the answers at the back.
I reckon you should start with the above-mentioned book (Allendoerfer and Oakley), and after that on to Vellerman or something like that.

Oh, and it's very important you know your pr-calc stuff well, before you get to university.

 

[Edgardo]

Here's a thread with a collection of pdf's on how to write math proofs and some book recommendations.

 

[MJC684]

i actually have that book by allendoefer - Principles of Mathematics. The chapter on logic does look interesting. I'm gald i bought it