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## Criminal offences in maths textbooks

Another vent of frustration:

I hereby nominate the three worst offences committed in maths textbooks as

0. Elementary text books asserting a 'fact' that is clearly wrong and going to lead to confusion at a later date. Example: you cannot square root negative numbers. It takes a huge amount of effort to de-program students who have very fixed ideas when they come to university (since complex numbers are even less likely to be taught at high-school anymore).

1. Using an equals sign when you do not mean equals. Example: an A-level textbook I taught from asserted that the square root of 2 _equalled_ 1.41

2. Asking what the domain of, say, $\sqrt{x^2-4}$ is. The domain is part of the definition of a function - that is just an expression at best. See offence 0. Does it suddenly become a function when you learn about complex numbers?

 Recognitions: Gold Member Science Advisor Staff Emeritus A recent gripe of mine has nothig to do with any text books I know of but I'm sure there will be books out there because this is part of the primary school syllabus. Instead of teaching standard algorithms for multiplying large numbers and for long division, kids are now being told to 'reason' through these problems by breaking them down into smaller problems they know. This way more work and not very useful in introducing mathematical concepts such as algorithms and long division which comes in handy later on.
 Recognitions: Homework Help Science Advisor one annoyance i found when writing my algebra book was texts asserting some correct formula, like that for the discriminant of a general (or even reduced) cubic polynomial, that they say are derivable from some particular algorithm, but omitting the derivation. After a finite but huge number of pages, one learns that the stated algorithm is not actually feasible for calculating the given formula. Hence I made it a rule never to say "a lengthy and tedious calculation (or short and easy one) gives....". If I couldn't do the calculation myelf, reasonably, I showed how to do it on Mathematica, and then also figured out a better way to do it by hand and gave that too, such as the resultant procedure for a discriminant.

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## Criminal offences in maths textbooks

I myself am annoyed by books which give arguments designed for the convenience of the author rather than the student. E.g. in Dummit and Foote, the proof of the decomposition theorem for finitely generated modules over a pid is done in a rather abstract and useless way in the text. Then when a computational version is needed later to actually find a canonical form of a matrix they revert to the standard procedure by diagonalization of matrices using Euclid's algorithm (over Euclidean domains), which they have left to the reader in the exercises.
 Recognitions: Homework Help Science Advisor Another practice I find objectionable is providing answers for exercises. This effectively deprives the reader of the pleasure and instructiveness of doing them himself, and of deriving checking procedures, and deprives the exercises of usefulness to the instructor who may wish to assign them. There are few things as useful as learning to convince oneself that ones solution is correct. There is a good reason that here on PF the standard first answer to a question is: "show us what you have tried".
 Recognitions: Homework Help Science Advisor Errors of course are a fact of life, but authors and editors who decline to acknowledge or correct serious errors in edition after edition puzzle me. E.g. the wonderful book by Shafarevich on Basic Algebraic Geometry has gone through many versions, all containing a false theorem on upper semi continuity of fibres of a morphism. On opage 77, section I.6.3, second edition, vol. 1, the corollary of theorem 7 is false unless one assumes the map is also proper. Unfortunately this false version is also used later in the discussion of tangent spaces, page 92, section II.1.4, where a modified argument is needed. (One can projectivize the spaces involved to render the appropriate map proper.) Many other famous books have appeared for decades with the same serious errors uncorrected. Fortunately the errors are usually in the proofs rather tha the statements, but to students learning the reasons the harm is there. Among the many misleading and misstated arguments, the only error in Lang's famous and excellent Algebra book ever corrected to my knowledge was the false theorem to which a counterexample was actually published in a journal, by perhaps James Cannon. Here is a tiny example of persisting errors. On page 90, second edition of Algebra, after making clear the distinction between products and sums, lang gets it wrong himself. In the second displayed isomorphism, the right side should have a product sign, not a summation sign. Of course for two summands of modules there is an isomorphism between the two modules, sum and product, so in a twisted sense it is still partly correct, but the whole force of the explanation about their different properties is vitiated by getting it backwards here. This is the quality of sanctimoniousness of authors: dear reader, YOU should always be careful to understand and notice this important point, (but i won't bother to myself, not even here in my book on the topic!).

 Quote by mathwonk Another practice I find objectionable is providing answers for exercises. This effectively deprives the reader of the pleasure and instructiveness of doing them himself, and of deriving checking procedures, and deprives the exercises of usefulness to the instructor who may wish to assign them. There are few things as useful as learning to convince oneself that ones solution is correct. There is a good reason that here on PF the standard first answer to a question is: "show us what you have tried".
I think it should also be the responsibility of the student to be not tempted by the answers at the back. There are those like me, who like to try a problem many times before looking at the answer, and may not have an instructor for immediate help. Having said that, I don't completely rely on the back-answers to gauge my understanding. It just makes me a tad more confident, that's all.
 Recognitions: Homework Help Science Advisor i knew that would get an opposing view. I believe its still true though. heres another reason i dislike them: Once in college i handed in a particularly short answer to an easy problem, and the grader gave me a zero and insulted me for "copying the answer from the back"! Of course I was completely unaware there were any answers in the back since i had never looked there. since he did not seem to respect me, i lost all respect for that grader from then on. Of course I have since learned he was merely speaking from experience.
 Recognitions: Homework Help Science Advisor of course i will cut an author more slack if he owns up to his intentions in a book. i especially enjoyed the introduction to sternbergs differential geometry. he explained that he gave proofs of hard theorems right away in order to help tie down the definitions. He also said he hoped he had avoided serious errors, but could make no such pretension in regard to sign errors and factors of pi etc, as he could never keep those straight himself. Anybody that honest, and that smart, can be forgiven much, (and it is probably unlikely much will be found to object to).
 I have to say that asking about the domain of a function is the worst. Students often have the idea that a function is a computable expression, which leads to a lot of trouble later, and questions like that seem to intentionally encourage that confusion.

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 Quote by matt grime 1. Using an equals sign when you do not mean equals. Example: an A-level textbook I taught from asserted that the square root of 2 _equalled_ 1.41
I'm curious -- do you extend this to big O notation? I mean, do you object to statements like
$$\pi(x)=\frac{x}{\ln x}+\mathcal{O}(\sqrt x\ln x)$$
 Recognitions: Homework Help Science Advisor Why would I, or anyone else, have a problem with that if it is true? (Or as a statement of a conjecture, as well.)

 Quote by matt grime Another vent of frustration: I hereby nominate the three worst offences committed in maths textbooks as 0. Elementary text books asserting a 'fact' that is clearly wrong and going to lead to confusion at a later date. Example: you cannot square root negative numbers. It takes a huge amount of effort to de-program students who have very fixed ideas when they come to university (since complex numbers are even less likely to be taught at high-school anymore). 1. Using an equals sign when you do not mean equals. Example: an A-level textbook I taught from asserted that the square root of 2 _equalled_ 1.41 2. Asking what the domain of, say, $\sqrt{x^2-4}$ is. The domain is part of the definition of a function - that is just an expression at best. See offence 0. Does it suddenly become a function when you learn about complex numbers? Feel free to add your own.

Why were you teaching from an A-level textbook? were you giving private tuition?

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 Quote by matt grime Why would I, or anyone else, have a problem with that if it is true? (Or as a statement of a conjecture, as well.)
You said you had a problem with misuse of the equals sign, and this is such a misuse -- but one that is commonly accepted and used. If the equal sign was literal, we could conclude from f(x) = O(x log(x)) and g(x) = O(x log(x)) that f(x) = g(x).

That is, are you complaining about approximations being shown as equalities, or all abuses of notation (even relatively benign ones) involving equalities?
 Recognitions: Homework Help Science Advisor That f(x)=O(x log(x)) has a specific meaning. That equality is being used in a different, and explicit sense, from numerical equality. Personally, I'd prefer that they use ~ to indicate that they are using an equivalence relation, but that is just me. Asserting that sqrt(2)=1.41 is just plain wrong. If they were to have written sqrt(2)=1.41 (2 d.p.), then that would be fine. They didn't.

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 Quote by CRGreathouse You said you had a problem with misuse of the equals sign, and this is such a misuse -- but one that is commonly accepted and used. If the equal sign was literal, we could conclude from f(x) = O(x log(x)) and g(x) = O(x log(x)) that f(x) = g(x). That is, are you complaining about approximations being shown as equalities, or all abuses of notation (even relatively benign ones) involving equalities?
True, it seems like those should be written as:
$$f(x) \in \mathcal{O}(x \ln x)$$
since big-O is really equivalence classes.

And:
$$f(x)=x + \mathcal{O}(\ln x)$$
might be re-written something like:
$$f(x) \equiv x \left(\mathcal{O}(\ln x) \right)$$

But there's a similar thing with constants of integration:
$$\int 2x \mathrm{d}x = x^2 + C$$
so
$$\int 2x-2x \mathrm{d}x = \int 2x \mathrm{d}x - \int 2x \mathrm{d}x =(x^2+C)-(x^2+C)=0$$