PCM errata III

A quick note to say that I’ve just been asked by PUP to send all the corrections I have, so I have sent the corrections that are listed in comments on the post PCM errata II. Further errata should therefore ideally be pointed out here. I think this means that a reprint with corrections should appear in the next few months, but I’m not sure of the exact time scale.

Once again, many thanks to those who have notified me of errors. Particular thanks this time round go to Axel Boldt, who sent a very long list. He was to PCM errata II as Joseph Myers was to PCM errata I. But others too made observations that will lead to significant improvements when the reprint comes out. It has been disconcerting to see just how many mistakes remained after the effort we spent on eliminating them, but I suppose in a book this size it is inevitable, and I imagine that there are several more lurking there. So do please continue to let me know of any errors that you find. At some point in the not too distant future I’ll try to merge the second list with the first to make it easier for people to see which mistakes have already been spotted. For now, after the break, here is a copy of the instructions I have sent to PUP.

If you look carefully, you will see that I haven’t acted on every single comment, because I’m supposed to confine myself to easily implemented corrections that will have a minimal effect on page and line breaks. But I’ve taken account of almost all of them, and when I have not written about one, it is with regrets.

16.2 line 13

It’s been pointed out to me that I use the subset symbol without having
said what it is. That’s not ideal, but I think it’s not enough of a problem to
be worth the hassle it would take to fix it.

43.1 line 20

It would be better to say “a point on the line at infinity of $P’$” rather than
just “the point at infinity of $P’$”.

44.2 line 14

“ball” would be a better word than “sphere” here. In fact, this is an
essential change because of the wording earlier in the sentence.

126.1 Weierstrass quotation

Someone points out that it looks odd to date this 1988, but I think it’s
our convention that we refer to the date of the edition that is being
quoted and not the original book. Is that right? If so, then no need for
any action here.

149.2 line 17

Not sure how, but a “with with” slipped through the net.

163.2 line -3

Where it says SO(2m) it should say SO(m).

164.2 line 10

“motion through space-time” would be better replaced by “motion
in space-time”

180.2 lines 24-26

There are a couple of lower case phis that should be upper case.
(All phis round here should be upper case.)

186.9 line 9

In this heading, it should say $C_c^\infty[-1,1]^*$ rather than just
$C^\infty[-1,1]^*$. The font of the suffix c should be as it is in the
ensuing paragraph.

208.2 line -3

It should say “equal angles, just like the more familiar Euclidean rotations.”

218.2 line 6

In the definition of the ellipticity condition it should say \sum_{i,j}a_{ij}\xi^i\xi^j
In other words, the sum over i and j is missing.

265.1 line -6 and 265.2 line 2

In both places it should say \binom nk p^k(1-p)^{n-k}

That is, the power of 1-p should be n-k rather than just k.

287.1 line 13

The final psi in this definition of the Laplacian shouldn’t be there.
In other words, it should end with $\sum_{j=1}^n\frac{\partial^2\psi}{\partial q_j^2}$.

293.1 line 12

It should say \cos^{n-2r}\theta(1-\cos^2\theta)^r at the end. (In other words,
there is a theta missing.)

299.1 line -5

The words “a diffeomorphism” should be replaced by the single word

303.2 line 19

The words “with a_r=b_r=0 for r>\frac 12 n” should be omitted.

313.2 line 9

Ideally, after the sentence that ends “vector space” I’d like to add
the following sentence:

Moreover, for every sufficiently small region $U$ of $X$, the inverse
image of $U$ is homeomorphic to $\R^n\times U$ (this property is
called \textit{local triviality}).

347.1 line -10

Where it says (b,q)=1 it should say (b,p)=1.

378.1 line -22

Am waiting for an email from the author before I know what to do here.

457.1 line -16

“derivatives of $f$” should be “derivatives of $u$”

470.1 line -3

It should say $B_{a+ct}(x_0)$ where it currently says $B_{a+t}(x_0)$.

478.2 line 11

The following rewriting is needed to remove an unfortunate ambiguity.

From this we wish to dedcue that the derivative of $S$ at $\phi$, in
some appropriate sense, is zero, and hence …

479.2 line 7

The sentence in parentheses needs to be rewritten as follows:

(For instance, for energy the one-parameter group of …

479.2 line 19

The displayed formula is missing two factors of \frac 12, which
should appear in the places where they appear in formula (58).
I think it would also be appropriate to have a dx at the end of
the integral, for consistency with (58).

490.1 line -14

\nabla_i A_0 should be \partial_i A_0

490.2 line 2

\partial_{x^i}^2 should be \partial_i^2

512.1 line 22 It would be slightly better to write

… a \textit{unitary} operator is an invertible …

but if this causes problems with line or page breaks then it is not essential.

520.1 line -16

The Z inside C^*(Z) should be blackboard bold, as it is four lines earlier.

521.2 middle of page

Where it says $T^*T-T^*T$ it should say $T^*T-TT^*$. (A mistake
that it doesn’t require any mathematical knowledge to spot …)

524.1 line -7

The word “period” should be replaced by the phrase “angular frequency”

527.2 line -4

There is a factor \frac 12 missing before the x^2.

543.1 formula (2)

There is a left square bracket missing at the beginning of the first line.

548.1 line 18

A valid-sounding objection has been raised to this formula. I’m not
sure what to do about it and may consult the author. But I think we’ll
have to leave it for the time being, unless I get back in touch about it.

639.2 line -10

It would be good, but not essential if it causes problems, to change
“all the true statements” to “all the true first-order statements”

645.1 line -1

Where it says R(x_i,z) it should say (x_i\sim z) (if \sim gives the
tilde symbol used just above). And where it says \neg R(y_i,z)
it should say \neg(y_i\sim z)

681.1 line -1

It should say “is at most $6x$” rather than “is $6x$”.

714.1 First sentence of V.25

Apparently I misstated the conjecture, which is embarrassing. Here
is an alternative first sentence.

The Poincar\’e conjecture is the statement that a COMPACT [III.9]
smooth $n$-dimensional manifold that is HOMOTOPY EQUIVALENT
[IV.6.2] to the $n$-sphere $S^n$ must in fact be homeomorphic to $S^n$.

This requires a further change at the end of the paragraph. Where it says,
“Thus, the Poincar\’e conjecture asks” it should be changed to “In three
dimensions, the Poincar\’e conjecture asks”

852.1 line -20

It should say

… which takes the value 1 between 0 and $\frac 12$ and $-1$ between
$\frac 12$ and 1.

67 Responses to “PCM errata III”

  1. Richard Says:

    I am constantly amazed at how many typos persist in what I’m writing, despite repeated reading! Whenever I think I’m finally getting things cleaned up, it seems that a gremlin puts them back in as I sleep. When I’m reading linearly through my writing I tend to start reading what should be there rather than what actually is there. My typo detector seems to work better if I am reading pages randomly.

  2. Vicky Says:

    What happens about typos in errata?! (478.2 above) No doubt this will already have been noticed, though.

  3. Michael T Fischer Says:

    pg 262, 2nd to last paragraph, gives the impression that one can compute the nth hexadecimal digit of PI without any prior knowledge of the preceding n-1 digits, this may be technically true, but in fact all the preceding terms in the sum (mod 1), need to be considered.

  4. Jakob Says:

    For those of us who bought the collector’s item first version of the book, will there at some point in the future be available a nicely formatted list as, say, a PDF file, with all typos discovered up to that point?

  5. Axel Boldt Says:

    Regarding the spacetime sentence in 164.2 +10, changed above: Maybe this is nitpicky, but I think that “motion in/through spacetime” is improper usage. There’s motion through space, but in spacetime there are only static world lines; nothing is moving there. The notion of motion needs an external time coordinate which we don’t have in spacetime. As an analogy: if we describe the vertical motion of a bouncing ball by a graph in a (t,x) coordinate system, we wouldn’t say that the ball bounces around in the (t,x) plane.

    • gowers Says:

      I agree that “motion through spacetime” is misleading, for exactly the reason you give. But one can just about defend “motion in spacetime” by saying that, in spacetime, motion means motion in space as the time coordinate varies. To use your analogy, one might say, “In the (t,x) plane, motion corresponds to a curve of non-zero gradient.” At any rate, that was the thought going on in my mind when I came up with that change — it was the best I could do, given that changes to line and page breaks were not welcome.

  6. Axel Boldt Says:

    702.2 +14: g_r should be g_k

  7. Axel Boldt Says:

    706.1 +11: for the gradient we need to explain the Lp norm of a vector-valued function, so F is a better notation than f. But it should be explained what norm we use for |F(x,y)|.

  8. Axel Boldt Says:

    716.2 +10: underneath the summation symbol we want n instead of N.

  9. Axel Boldt Says:

    719.2 -2: “has a solution” -> “has a non-trivial solution” [thus excluding x=y=z=0]

  10. pintoj Says:

    65.2 -17,-18: the examples illustrating the upper and lower bounds does not appear to match the formulas given a few lines above it. Maybe it is just some formatting mistakes? One of the formulas is n/e^(c(sqrt(log n)) but the example is showing the value of e^(sqrt(log n)). Similarly, the formula is C(n(sqrt(log log n/log n))) whereas the example is showing the value of sqrt(log n/log log n).

  11. Owen Densmore Says:

    Are we still waiting for the second printing? If so, I think I’ll just order the critter and live with the errors!

    • gowers Says:

      I’m afraid we are still waiting for it. I don’t know when it will appear — all I know is that work has started on preparing a corrected version, but it hasn’t got as far as my actually seeing drafts of corrected pages.

  12. David Bevan Says:


    I did a search for “semigroup” in the text on the official website. Assuming the results are correct, this term is used three times but not defined anywhere.

    — Perhaps the resolution of this would require a bit more than just a second printing.

  13. pintoj Says:

    351.1 16: The definition of Carmichael numbers is missing an important qualification on the integers a. The qualification is that gcd(a, n)=1. In other words, the sentence should end with “where the Fermat congruence holds for every integer a that is coprime to n.” Without this qualification the 3^(560) would be congruent to 1 but in fact it is 375. It would also seem to indicate that a form of the converse of the Fermat little theorem could theoretically be used as a primality test (though useless in practice), namely that if the Fermat congruence holds for every a between 2 and n-1, then n is prime.

  14. priquas Says:

    p 65, column left, line 9: g is the acceleration due to gravity, _not_ the gravitational constant.

    Thanks for this wonderful book!

  15. pintoj Says:

    111.2,-18: “a sequence” should be “an n-tuple” in keeping w/ the nomenclature of that paragraph.

  16. S.H.K Says:

    Any news on the second printing? I’d really love to buy the book, but I’ll wait for the second printing if it’s coming soon.

    • gowers Says:

      I was told it would be “in stock by the end of the month”. I’m not sure when that means it will actually be available, but I think PUP was running out of copies of the uncorrected version, so I think at this point it’s probably worth waiting.

  17. Ted Powell Says:

    14.1 line -14
    (4′) For all x, lifelong happiness is better than x.
    should be
    (4′) For all x, lifelong happiness is at least as good as x.
    for consistency with (4) at line 7 above.

  18. Felix Chen Says:

    Dear Prof. Gowers,

    My local bookstore has a copy of PCM. Wondering if it is the corrected 2nd printing, I asked them to check the string of numbers at the bottom of the title page. For this copy of PCM it’s “4 5 6 7 8 9 10.” This would normally mean it’s the 4th printing. But from what’s been said here we’re only up to the 2nd printing. Furthermore, I asked them to turn to page 149.2, line 17, to check if the repeated “with with” phrase is there. It is. Is there a particular error I can check to confirm if this book is the corrected reprint (assuming it’s now available)? I’m eager to buy it but would rather wait for the corrected printing to come out. Thank you.

    • gowers Says:

      I think the check you’ve already done is enough to establish that it’s the uncorrected version, but it might be worth looking at a few more of the errata. Another you could try is this. In the chronology right at the end of the book there is a one-sentence description of the Langlands programme. The bad news is that in the uncorrected version this description comes after “Abraham Robinson introduces non-standard analysis”, leading to some amusing nonsense.

      Incidentally, there have definitely been more than two printings. I’m slightly surprised that we’re up to four with the uncorrected version, but not in the sense of thinking that that couldn’t be the case.

  19. Felix Chen Says:

    The 4th printing is definitely the uncorrected one, so I’ll wait.

    By the way, the lady at the bookstore commented that “it’s a gorgeous book.” Definitely, inside and out. I hope the corrected printing hits the shelves soon!

  20. Felix Chen Says:

    Just a quick note: I’ve confirmed directly with Princeton University Press that the corrected printing is out. It’s the sixth printing. I guess it’ll be hit or miss whether Amazon.com has that particular printing among their stock. I sure hope so.

    • Phil H Says:

      ‘Hit or miss’ is indeed correct wrt Amazon. I attempted to find out if they have the sixth printing in stock, but they were unable to tell me.

  21. Felix Chen Says:

    Yes, the 6th printing is in stock at Amazon.com. I got mine several weeks ago.

    • S.H.K Says:

      So i can order PCM from amazon.com and i’ll get the 6th from now on? Or it depends on stock’s level of previous printings? Thanks for the infos

    • Phil H Says:

      Many thanks for that information, it was exactly the kind of response I was hoping for when I posted my comment.

  22. Felix Chen Says:

    From what PUP told me, it’s likely all of Amazon.com’s stock is the 6th printing. Their representative (who’s also mentioned by name in the book’s preface) told me the corrected printing was done in response to low stock. So whereas it’s not guaranteed all of Amazon.com’s inventory is the corrected 6th printing, I’d say it’s very likely. I ordered from them two weeks ago and did receive the 6th printing.

    • S.H.K Says:

      Many thanks for your info, amazon.com helpdesk didn’t know anything about which versions they stock, you’re response has been very useful, i guess it’s finally time to buy the book 😀

    • mtrtmk Says:

      i couldn’t get an answer from Amazon either about which print they currently carry. Can anyone who has ordered the print book from Amazon recently confirm that they received the 6th printing?

  23. Mladen Miksa Says:

    In 6th printing, 43.2 line -13 says “where I is the 4×4 identity matrix”, while there is no I in the equation preceding it.

  24. John Pinto Says:

    On page 681, the ABC conjecture, it states that a,b,and c are coprime, but I thought the condition for the conjecture is that a and b are coprime. If all 3 must be coprime, then are they also supposed to be pairwise coprime?

  25. Owen Says:


    In page 123, right column, line 27-31.
    I have checked Cauchy’s Calcul Infinitesimal, 1823. I think when Cauchy defined definite integral, he assumed the function to be continuous, and he partition the interval between x_0 and X with a set of points x_1, x_2,…,x_n-1 where these points are either always increase or always decrease between x_0, X. I think he did not assume the fuction to be increasing or decreasing in each of the subintervals. I think he only assumed those points that divide the interval are monotonic.
    Please let me know if I am wrong. Thanks.

  26. Matthew Aburn Says:

    Minor typo in the 6th printing:
    306.1 line 19 (equation mid-page)
    F(t) should be F(s)

    This book is so useful for what I am doing – many thanks to you and the team.

  27. SHK Says:

    In the 6th printing:
    page 260.1 line -16 “you might like to while away a few..”
    i guess should be “you might like to walk away a few..”

    • gowers Says:

      I’m not sure I see the problem, unless “while away” is an English idiom that doesn’t exist in the US.

    • pintoj Says:

      “while away” is a perfectly good idiom in the US … I don’t see a problem with that sentence as it is in the PCM.

    • SHK Says:

      My mistake then, English isn’t my first language and that sentence didn’t “sound right” first time i red it. BTW, the book is amazing 😀

  28. Revue de “Petite Initiation aux Mathématiques”, de Timothy Gowers « episodic thoughts Says:

    […] to Mathematics (voir aussi les listes d’errata si vous l’achetez: ici , là  et là), sorte d’encyclopédie moderne des maths pour l’étudiant passionné, une […]

  29. Bora Says:

    34.1 line 9 should be f(x+h) = f(x)+mh+E(h) instead of g(x+h)=… because g is the approximation of f.

  30. Markus Says:

    Is the 6th printing still the most update edition available or the 7th is already out? Thanks

  31. abundantmichael Says:

    Thanks for an excellent book. I have got many insights and much pleasure from it so far. I have the print book and am planning to get the Kindle edition too because it is hard to travel with the hardcopy, it is so big!

    I am wondering if there is a plan to update the Kindle edition to make the many cross references into hyperlinks. And also to make the equations easier to read (I read on the Amazon review that they are faint graphics that can not be enlarged on the Kindle rather than LaTex fonts that scale).

    • abundantmichael Says:

      I bought the Kindle edition and am very happy with it. The equations are easy to read for me. Though they don’t scale with the text because they are graphic images, which does look weird when they are in line ath symbols

      I would love to see the many cross references be hyper links in a future edition.

  32. Giulio Says:

    Any news on the 7th printing? I’ve found the 6th in a local bookstore but if the 7th will fix the last erratas i might wait for it. Thanks!

  33. Gal Says:

    I have just purchased the 7nth printing. On p.14; 2nd column; equation no.10 was meant to define the prime numbers, however I believe it erroneously includes 1 in the definition. I think there might be an implicit assumption that a!=b; but as it stands 1 satisfy the definition.

    I looked at the comments, but I did not see anyone else mentioning this.

    Thanks for a wonderful, wonderful book!

    • Giulio Says:

      Hi, can you check if in the 7nth printing the few errata left have been fixed? Thanks

    • Gal Says:

      Whatever was pointed out in the comments after October 2010 is still there, from what I’ve seen. If you had any specific thing in mind I can check 🙂

    • Giulio Says:

      No needs to be more specific :D, guess no new errata were fixed after the 6th printing. Thanks alot 😀

  34. K. Jeong Says:

    177.1 -10 and -8. f(v) should be f(x), I think.

    • Dan Schmidt Says:

      I think so too. I noticed the same thing independently and found this comment while trying to confirm my understanding. If so, it’s weird that it didn’t get caught until 2014.

  35. kyeonghoonjeong Says:

    For several times, I left replies regarding errata in PCM. But for some reasons I don’t know about, those are not published here. Anyway, I wrote a PDF file in my blog. Please visit my blog.

  36. smcjw Says:

    212.1 line -13 It should say orthogonal, not orthonormal basis, or adjust for a factor of 1/sqrt(2) in line -14.

  37. benkrid Says:

    p 62 r(n) should be = (n^2+n+2)/2 or n(n+3)/2-n-1

  38. Jeremy Roach Says:

    Page 2. The interior of the unit circle should not be thought of as the circle? I.E. less than or equal should be changed to equals.

  39. benkrid Says:

    p157.1 line 15
    should be a = – b = sqrt(2) (minus b instead of b)

  40. Matt E. Says:

    (11th printing) p40.1 line 3
    if I’m not mistaken this should read “are not concepts of Euclidean geometry”, since shearing and stretching are surely affine concepts.

    • Matt E. Says:

      scratch that, it’s of course correct from the affine perspective

    • Matt E. Says:

      well, OK I still think there’s an ambiguity here. “these” should instead read “the latter” – my problem at first was reading it as referring to the former, which can’t be right.

  41. Matt E. Says:

    (11th printing) p508.2
    extraneous ‘+’ in Fig. 9 caption: “rays that+ land”

  42. Sébastien Says:

    p. 47. 2 lines 3-4: “which will turn out to be the case provided EG-F² is always positive”

    One should also ask for E to be positive.

  43. Sébastien Says:

    p. 61, 2 line 1: impose a_0=0 in the definition of a one-dimensional walk

  44. Sébastien Says:

    p.181 2. If one wants the pathological X to have dimension 0, 0-dimensionality should not be defined by finiteness but as being totally path-disconnected. (Maybe the best solution is to define dimension -1 as being empty.)

  45. Sébastien Says:

    p. 186 2. “compose distributions on both the left and the right with suitably smooth functions”: only one direction is possible

  46. Yunfeng Wang Says:

    400.2 line -11

    “The one that uses the Möbius strip and the torus is called the Klein bottle.”

    No, this’s not the Klein bottle (genus 2), but Dyck’s surface (genus 3).

    In general, glue an n-holed torus with m copies of Möbius strip, you get a nonorientable two-dimensional manifold of genus 2n+m

  47. Sébastien Says:

    p.214, 2., line 26 (last displayed equation in III.31): I do believe that the exponent beta – 1 should be beta – 1/2.

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