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I've taken Calculus 1 and it's time to relearn because I've forgotten some of it. But it's been a couple months since I've done any solid mathematics. I was hoping for a book that would include comprehensive & rigorous text on precalculus, probability & combinatorics, and analysis. I am aware it's too much to ask for a single book and that supplementary material is available through other books, but my problem is that these modern books aren't rigorous enough in their text. They oversimplify everything and bloat the book with unnecessary figures explaining basic things, and their countless excercises also provide no real value as all they require is rote memorization of the given techniques.

My geometry & trig has also gotten a bit rusty over the months.

Where should I begin? What book(s) should I go with? Is there a list?

EDIT: Clarification on my use of the word "rigor": I meant a book that is challenging, a bit formal in nature, and includes useful and fruitful topics not usually found in traditional maths textbooks.

SECOND EDIT: I've found the book I was looking for, after browsing a bunch of similar threads, I've found the precious jewel: https://imgur.com/a/q5kPreP

It was published in 1891, about the time of my grand-grand father! It contains a plethora of information!

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  • $\begingroup$ David Cohen's College Algebra is probably the best of the "modern era" (e.g. past 30-40 years or so) college algebra texts I know of. I don't know about the first 3 editions (1986, 1989, 1992), but I have the 1996 4th edition and it's more than sufficient for your algebra needs (and can be obtained fairly cheaply). $\endgroup$
    – Dave L Renfro
    Apr 22, 2021 at 17:03
  • $\begingroup$ @The_Pleading_one, it seems as though you have answered your own question, which is fine, but answers belong in the answers section. Please edit your question to remove the "I've found the book" part and move that to a separate Answer, below. $\endgroup$
    – shoover
    May 6, 2021 at 19:03
  • $\begingroup$ BTW, the full book is available at the Internet Archive. $\endgroup$
    – shoover
    May 6, 2021 at 19:08

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Frank Ayres First Year College Math (Schaum's Outline). I have/like the original 1958 edition (easy to get used), but the newer edition with co-author is probably OK, also.

https://www.amazon.com/Theory-problems-first-college-mathematics/dp/B0007DPVM2

Since it is a review, it is written economically and clearly. Directed at the student as the customer, not a committee of teachers. Not a ganglion-basher.

Covers first and second year HS algebra, plane and solid geometry, plane trig, analytical geometry, function concept, and very short/easy intro to calculus (pre-calc style).

It does not cover prob/stats. (You'll need something else, preferably not a doorstop but something of a review nature or "for business students" or the like.)

Has example problems and exercises, with answers. But not an annoying amount. And you can decide how much to read versus practice. (But you really DO need to practice. One of the reasons you are having to review is likely lack of drill earlier.)

Other than that, if you have access to your old calculus 1 text, I recommend looking at it. Even if it's not perfect, at least you are familiar with it.

P.s. I am trying to address the real question, I think you are asking. I would just be careful about using the word "rigor". That is probably not what you really need. Or at least not the way many people on MSE or MESE use it (highly abstract, very difficult). Given, the totality of what you said (taken only through Calc 1, need to review pre-calc), you're not the candidate for stuff that is more difficult. You need to work on the basics...need to acquire automaticity in manipulations involving trig, power laws, logs, etc. Any standard review (or even omnibus) text will be "correct enough" for you to get value. What you need to do is disciplined review and practice of the basics. Seat of pants to seat of chair. Door closed. Walkman (or ipod or whatever) off.

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    $\begingroup$ I love your answer, except for the last sentence. Beck in the 70s, I would lay one record (Roberta Flack, Grateful Dead, Doors, Janis Joplin) on the turntable, and let it play over and over. When I finished a session, I had no idea what I had been listening to. It blocked out anything else, and then disappeared. $\endgroup$
    – Sue VanHattum ♦
    Apr 22, 2021 at 14:01
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    $\begingroup$ It's interesting. Heard different things. Some people benefit, others don't. I tend to need monastic silence when studying but others the opposite. Would at least advocate no TV. Music can be more of a background hum. $\endgroup$
    – guest
    Apr 22, 2021 at 14:04
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    $\begingroup$ I used to play the soundtrack to Pulp Fiction when literally (and really literally, not how the Gen Y misuse the word) grinding chemicals during grad study. I would give myself one "song" for a given sample to be mortar-pestled. Seemed to take away from the drudgery and made me more efficient. But it was pretty brainless. Could still have mind wander. But the song count was my egg timer. $\endgroup$
    – guest
    Apr 22, 2021 at 14:07
  • $\begingroup$ For that last sentence, back then when I used to do maths a solid 5+ hours a day without taking any breaks inbetween, I would listen to heavy metal on full volume and it would help me greatly, I would never get bored and it'd always pump me up like caffeine... $\endgroup$
    – The_Pleading_one
    Apr 22, 2021 at 15:31
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    $\begingroup$ You could look at The Art of Problem Solving books. Maybe the new treatment would make it more fun to do the review. It does seem you need to solidify basics, not just learn nuances (on top of sound basics). But since it's second exposure might not be too hard. artofproblemsolving.com/store There would be some reasonable cost though. And you still need to work problems. I'd get the Schaum's too. It's cheap and has basic drill in case TOPS is missing that. I think also the sheer length of the TOPS books (in toto) violates one of your criteria. (Oh well.) $\endgroup$
    – guest
    Apr 22, 2021 at 16:11
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I like William Chen's lecture notes a lot. Clear, nicely structured. He wrote a lot, on a variety of topics ranging from first-year topics to rather advanced stuff.

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