Very many thanks to all who have written in pointing out errors in the Princeton Companion. I am told that it won’t be too long before the next printing, so I have finally been forced to collect together the errata in a systematic way. I am copying the list I have just sent to Princeton University Press so that if anyone finds an error now they can easily check whether it has already been spotted. (I was asked if I would do this some time ago — now at last I have.) The list appears after the break. From now on if people point out further errors I will add them to the list, with some indication of whether they have yet been corrected.

The secondary purpose of this post is to suggest that you should wait a bit if you are thinking of buying the book. I don’t want to hit sales too hard, but I’m guessing that not all PCM buyers are avid readers of this blog so I might as well reward those who are. Of course, you could take the attitude that the error-strewn version is a collector’s item: if so, hurry while stocks last.

**The errors that have so far been spotted.**

I will use the notation A.B to stand for page A column B.

An erratum with three asterisks in front of it denotes one that is problematic because it is more than just a minor change, or will affect pagebreaks.

Back flap: I suggest “illustrating a logarithmic spiral” instead of the phrase about the Fibonacci sequence.

14.1 Displayed statement (4′) should read

(4′) For all , lifelong happiness is at least as good as .

(I hope that still fits on one line.)

14.2 line -17

The reference to sentence (6) should be to sentence (9).

24.2 First displayed equation should read

27.2 line 18

… it is a simple exercise to show that every homomorphism that is not identically zero is automatically an isomorphism …

30.1 line 7

… such that is proportional to …

The v here should be bold face.

33.2 line -5

It should say “rate of change of ” rather than just “rate of change “.

40.1 Sixth line of section 6.4

It should be n-dimensional rather than d-dimensional.

42.2 line -12

It should say ““.

Two lines later it should say ““.

In line -2 it should again say ““.

43.1 line 13

Change “with a “point at infinity”” to “with a “line at infinity””

***43.1 line -6

Ideally, we would omit everything from “(This is quite hard to imagine” to “but a projective plane.)” since what I wrote in those brackets is unfortunately wrong. But omitting it will affect many pagebreaks. I can’t immediately think of any text that could replace what we have at present. We may need to discuss this.

43.2 line -1

The very last equation of the column should be ““.

57.2 Middle of column, it should say

… natural properties (for instance for all functions , , and ) …

62.1 In the paragraph headed (i), it should say

… which leads to the formula .

line -6

replace “” by “”

63.2 At the end of section 6.2 it should read

… was proved by Huxley in 2003 (the latest in a long line of successive improvements), is that is at most for some constant .

64.2 At the very end of section 6.3 it should say

… as many as .

In other words, ms need to be ns.

75.1 line -21

The number 181 needs to be added to the list of numbers here (in the appropriate place).

121.1 line 4

This displayed equation should read

124.1 line -7

Replace “in every bounded infinite set” by “for every bounded infinite set”.

131.1 line -13

It should be DBCF rather than DBFC here.

169.1 Middle of column.

… if one maps the real number to the point , then …

That is, the final “squared” comes inside the bracket.

171.1 End of penultimate paragraph.

This list is missing some numbers. It should go

and then continue as before. If it’s a problem to add a line here, then we could end the list with “4, -4, …” instead of ending at “5, -5, …”

***172.1 line -20

A new half-sentence is needed here. It should say

… for any elements and ; this involution is required to satisfy the –*identity* .

176.2 Middle of column.

Where it says “where and .” it should say “where and .”

180.2 line 5

Instead of “” it should say ““.

Earlier in the same line it would be better to write “” instead of ““.

183.2 line 16

… written either as 0.1 or as 0.02222…

(instead of 0.22222…)

184.2 lines 12-14

It should read

… sense: if is less than , then the -dimensional Hausdorff measure of is infinite, while if is greater than , then it is 0.

187.2 line 7

It should be “octahedron” rather than “octagon”.

194.2 line -8

The final should be (so that it matches the in the exponent).

***202.2 End of column

I would like to add the following sentence if possible.

… coefficients quickly. A method for doing this was discovered by Cooley and Tukey in 1965 (though it turned out that Gauss had anticipated them over 150 years earlier).

223.1 line -16

In the display here there is a stray comma in the last expression, which should be . (NB the “e” should still be a Roman e — it’s just the comma that needs removing.)

228.1 line 3

It should end with . I.e., it’s half the square root of 3 and not the square root of half of three.

232.1 bottom of column

It should read

… ,

***238.1 line 1

The factor 6 here is not explained. Unfortunately, it may take more serious rewriting to clarify this. (It’s not exactly wrong, but comes from a slightly different form of the original equation.) So I think we’ll have to leave it for now.

However, it should say

… equation . If we add …

240.2 line 8

It should say

… is the matrix defined by .

249.2 line 11

It should be

I.e., there’s a missing q in the bracket.

266.1 line -21

It should say “,” instead

of “,”.

272.2 line 4

It should say “generated by the polynomial .”

277.1 line -4

Reword as follows.

… a system of numbers where we introduce not just one square root of -1 but three, called i, j, and k (together with their negatives). Once one knows …

***278.2 Middle of column, item (ii).

Ideally we should extend the sentence as follows:

… is a real number, and similarly for multiplication on the right.

If the extra line causes problems, then we can perhaps live without this, but it will bother some readers.

297.1 line 21

It should say

… spherical harmonics are eigenvectors of the Laplacian, but they …

310.2 Second paragraph of section 2.1

This should read

Suppose, then, that we are given two points P and Q in the plane. We take as our class of admissible functions all smooth, real-valued functions , defined on some interval , such that and . The length …

314.2 line -5

It should say “when one does include the”

323.2 display in line -9

Where it says it should say

324.2 middle

It should say “published in 1952” rather than “published in 1934”.

325.1 line -11

It should say “It is easy to deduce that is itself a prime in the …”

NB the “i” should be in the same font as the other ones round here when this correction is made.

327.1 lines -8, -6, -5

The three s that appear here should be s.

339.1 line 3

It should say “arithmetic progression” rather than merely “arithmetic” here.

353.2 line 12

“Pollardand” should be “Pollard and”.

355.1 line -16

“Pollardfor” should be “Pollard for” (!)

424.2 line 18

Change “the first three” to “three of the”

425.2 Middle of column.

Change to (it comes at the end of a line).

***444.1 line 18

Change “non-trivially” to “”properly discontinuously””, so it should read

… if a group acts “properly discontinuously” as a set of isometries on …

A better change, but one that would affect linebreaks and pagination, would be

… if a group acts properly discontinuously (this means that for any compact set there are only finitely many such that and intersect) as a set of isometries on …

445.1 line 19

Replace “smooth compact manifold” by “complete Riemannian manifold”.

line 20

Replace “Given any closed path” by “Given any contractible closed path”.

460.1 line 4

It should say

… matrices that satisfy …

496.1 line -19

A few small changes needed here. It should say

However, if we apply ten times to and , then their respective eleventh digits have shifted leftwards and become the first digits of and . These two …

523.2 line -21

Replace “slide rules” by “log tables”.

531.2 line 9

It should say image( ker() rather than

the other way round.

555.1 line -7 it should say

… is by constructing a (preferably nice) bijection …

558.1 line 22

It should read “When there is exactly … ”

568.1 line -13

It should read

… set of integers using …

575.2 line -5

It should say “Two very obvious conditions” here.

595.1 lines 16-17

.. at least . Therefore, the probability … is at most , which is …

645.1 last line

The right-hand side should read

This one will need careful formatting.

695.1 line -6

A mathematical mistake here. It should say

We define the *index* of the fixed point to be the number of turns made by the vector from to , counting this negatively if these turns are in the opposite direction to the way that goes around . (This definition is problematic if for some , but again …

701.2 line 7

It should say “no proof of the formula , where is the …”

706.1 lines 9 and 11

The two capital Fs should be small Fs. That is, we want

“a function ” and we want f(x,y) inside the integral.

708.1 line -16

It should say “tile the plane if and only if the algorithm

fails to halt.”

708.2 line 13

It should say “if and only if the algorithm halted at .”

708.2 line -3

It should say “in the form .”

714.2 line 1

Change to .

***lines 2-5 This parenthetical remark should be removed, though it will change the pagination.

756.1 Middle of column.

For consistency with the rest of the book, we should say “going on to find seven more proofs over the years.” However, we should not make this change unless Jeremy Gray is happy with it.

767.1

It would be better to entitle this article “Ernst Eduard Kummer”

789.1

The first word of the column should be spelt “Theologie.”

813.1 line 5

Omit “closed”, so that it reads “formally real fields”.

832.1 At the end of the first full paragraph, add words so that it reads

… can take any value greater than 1.

***889.2

The description of CBC and OFB modes is oversimplified. The CBC description gives the impression that two successive cleartext blocks are added before encryption (so a 128-bit repeat in cleartext would result in a 64-bit repeat in ciphertext, similar to the ECB problem described in the previous paragraph)

when actually each cleartext block is added to the previous ciphertext block before encryption. In turn this reads like the description given of OFB mode; OFB does not feed the input cleartext through the block cipher at all, but starts with an initialization vector and repeatedly enciphers that to generate

a keystream, that is then added mod 2 to the cleartext to generate a ciphertext.

I think this is going to be too complicated to change this time, but I’ll keep a record of it.

909.1 line 14

“Becky has five siblings” should read “Desta has five siblings”.

919.1

“” (in the second paragraph) should read ““.

920.2

In the display the should be removed inside the “exp”, and the “” in the square root on the bottom should be ““.

Four lines later, there is again a stray , and again the “” on the bottom needs to be a ““. And this time, we also need to change the “” to a ““.

984.1 line -2

“quatre-vingt” should be “quatre-vingts”

997.2

In equation (7), “” should read ““.

1014.2 The remark in brackets that looks as though it explains Robinson’s Non-Standard Analysis has nothing to do with it and should appear after the following entry on the Langlands program.

March 18, 2009 at 11:33 am |

It’d be really nice if the errata was made available as a PDF. Preferably typeset in a way that blends nicely with the rest of the book and which makes it printable on both US Letter and A4 (that is, don’t make the margins too narrow).

By the way, it’s an amazing book!

March 18, 2009 at 9:29 pm |

I know that this is off topic but I am part of a mathematics podcast

Combinations

and Permutations that I think that your readership may find

interesting. On past episodes we have covered whether mathematics is

discovered or invented, math in pop culture, and most recently

we discussed the four color theorem and some new ideas for mathematical journals. You can find the podcast on iTunes or through our host. Give us a listen, you

will not be disappointed.

March 19, 2009 at 5:07 am |

A minor error I found is an historic inconsistency. In 756.1 line 21 it says that Gauss found 6 proofs of quadratic reciprocity (it says 5, plus 1 mentioned earlier). In 719.1 line 3, it says they were 8 proofs. I think the correct number is 8, 6 published during his lifetime, and 2 posthumous.

March 19, 2009 at 8:47 am

Thanks — but that one’s already on the list above!

March 19, 2009 at 10:37 pm |

[…] This is an excerpt from The Princeton Companion to Mathematics. By the way, the most recent errata for PCM are here. […]

March 23, 2009 at 12:35 am |

512.1 line 22

Should read “Just as in the finite dimensional case, a unitary operator is an invertible operator U …” (I think the first “operator” is missing).

March 25, 2009 at 1:16 am |

347.1 line -10

Should be “(b, p) = 1”.

March 26, 2009 at 6:42 pm |

520.1 line -16

C*(Z) font for “Z” is wrong

521.2 middle

Definition of essentially normal operators should read “T*T – TT* is compact”

March 28, 2009 at 6:35 pm |

543.1 formula (2)

First equation is missing a left bracket “[“.

March 29, 2009 at 5:16 pm |

852.1 middle

Definitions of Uj(x) and H(x) have opposite signs, so equation

Uj(x) = H(2^j x) fails.

April 16, 2009 at 4:56 pm |

208.2 line -3: angles are sent to equal angles, not to opposite ones.

April 17, 2009 at 3:02 am |

16.2 line 13: subsets have not been discussed and the subset symbol has not been introduced.

April 18, 2009 at 12:54 am |

43.1 line 20 “the point at infinity” also needs to be changed to “a point on the line at infinity”.

April 18, 2009 at 11:48 pm |

44.2 line 14: “sphere” -> “ball”

April 23, 2009 at 5:17 pm |

149.2 line 17: “with with” -> “with”

April 24, 2009 at 2:09 am |

164.2 line 10 “motion through spacetime” doesn’t really exist.

April 24, 2009 at 2:48 am |

163.2 line -3: SO(2m) -> SO(m)

April 24, 2009 at 2:57 am |

164.1 line -16: do we want the holonomy group to be all of SU(n), or just a subgroup?

April 24, 2009 at 11:49 pm |

180.1 line 14: divergence of curl is 0.

April 24, 2009 at 11:59 pm |

180.2 lines 24-26: all the lower case phis should be upper case.

April 25, 2009 at 12:38 am |

186.1 line 9: “C^infinity” should be “C_C^infinity”.

April 26, 2009 at 10:18 pm |

Hi Timothy .. boy do I like both these books: VSI and VLI!

Is there a date for when the new printing will be available? Amazon doesn’t seem to note which printing a book is.

— Owen

May 9, 2009 at 12:27 pm

I don’t know but I’ll mention it here when I find out for definite that the new printing exists.

May 5, 2009 at 10:33 pm |

303.2 line 19 condition on a_r and b_r is not needed.

May 5, 2009 at 10:35 pm |

299.1 line -5 “a diffeomorphism” should be “symplectic”

May 5, 2009 at 10:49 pm |

287.1 line 13 formula has one psi too many

May 5, 2009 at 10:51 pm |

292.1 line 12 formula needs an additional theta

May 5, 2009 at 10:56 pm |

218.2 line 6 the formula a_{ij} xi^i xi^j needs a summation symbol (or else Einstein notation has to be introduced)

May 9, 2009 at 9:22 am |

265.1 line -6: p^k (1-p)^k should be p^k (1-p)^(n-k)

265.2 line 2; p^k (1-p)^k should be p^k(1-p)^(n-k)

May 10, 2009 at 10:50 pm |

378.1 -22: “every prime ideal of R arises from a point p of X”. Only the maximal ideals of R arise from points of X; the other prime ideals correspond to higher-dimensional subvarieties of X. The elements of Spec X thus stand in 1-1 correspondence with the subvarieties of X, and Spec X with its Zariski topology encodes not only the points of X but also how they are organized in subvarieties.

May 10, 2009 at 11:35 pm |

313.2 +4: the “more precise definition” should probably mention the local trivialization property. The Moebius strip might be a good example here as well.

May 16, 2009 at 3:07 am |

457.1 line -16: f should be u.

May 17, 2009 at 12:44 am |

470.1 -3: B_{a+t}(x_0) should be B_{a+ct}(x_0)

May 17, 2009 at 5:27 am |

478.2 +11: “its derivative” could be misunderstood as referring to the derivative of phi, rather than to the derivative of S at the spot phi.

May 17, 2009 at 5:32 am |

479.2 +7: energy conservation corresponds to time translation, momentum conservation to space translation, angular momentum to rotation.

May 17, 2009 at 5:33 am |

479.2 +19: Formula should have the same factors of 1/2 as (58)

May 18, 2009 at 11:25 pm |

490.1 -14: \nabla_i A_0 should be \partial_i A_0.

May 18, 2009 at 11:28 pm |

490.2 +2: \partial^2_{x^i} should be \partial^2_{i}.

May 22, 2009 at 6:35 am |

527.2 -2: exp(-1/2 x^2 + lambda x^3) should be exp(-x^2 + lambda x^3)

May 22, 2009 at 6:42 am |

527.2 -2 again: seeing that the formula exp(-1/2 x^2) is used again in footnote 7 on page 528, it’s probably best to leave 527.2-2 alone and change the formula in 527.2-4 to -1/2 x^2 + lambda x^3.

May 23, 2009 at 12:12 am |

524.1 -7: omega is the angular frequency, not the period.

May 24, 2009 at 7:35 pm |

543.1 in the first equation of (2) there’s a leading [ missing.

May 24, 2009 at 10:39 pm |

548.1+18: equation (19) is mysterious; the left hand side is a complex number that depends on M, the right hand side is an n-by-n matrix that doesn’t depend on M and the notation Phi(V) was not explained. [If rho or Phi depend on M, then this should be reflected in the notation.]

June 1, 2009 at 2:29 pm |

126.1 Line 3 from bottom: The Weierstrass quote at the bottom of the page has the year “1988” attached to it. I think it should be 1888.

And thank you for this terrific book!

June 3, 2009 at 9:22 pm |

646.1 -1: R(x_i,z) should be x_i ~ z.

June 3, 2009 at 9:23 pm

(this is about 645.1 -1, not 646.1 -1)

June 3, 2009 at 9:27 pm |

639.2 -10 maybe “all the true statements” should be clarified as “all the true first-order statements”, since the negation of the Archimedian axiom is true in the first structure but not in the second.

June 3, 2009 at 9:36 pm |

638.1 -15 I don´t think Goedel´s theorem talks about undecidability of theories; it talks about incompleteness. It was Turing who showed (building on Goedel´s trick) that Peano arithmetic and lots of other theories are undecidable.

June 12, 2009 at 3:30 am |

676.2 +24 I don’t see why a factor of 10 can be described as “roughly the same size”.

June 12, 2009 at 5:50 pm |

681.1 -1 “is 6x” should be “is at most 6x”.

July 10, 2009 at 10:17 pm |

I bought a copy of your beautiful book recently, and I’ve greatly enjoyed what I’ve read so far. But I found a rather serious mistake in V.25. Except when n=3, the Poincare conjecture needs some extra hypotheses, otherwise it is false. As stated in the first sentence of V.25, the product of S^2 and S^(n-2) is a counterexample for each n>= 4.

(Article IV.7 contains a correct statement for n=4, and a nice description of why the proof in high dimensions turned out to be easier than in dimensions 3 and 4, but doesn’t contain the statement for general n.)

One way to correct V.25 would be to write `a compact simply-connected smooth n-manifold whose homology groups are isomorphic to those of an n-sphere must be homeomorphic to an n-sphere’. In fact it suffices to know that the mth homology group is the same as that of an n-sphere for each m <= n/2, but this would complicate the statement even more than introducing homology groups.

Another way to correct it would be to write `a compact smooth n-manifold M with the property that for each m<n, any map from the m-sphere S^m to M extends to a map from the (m+1)-ball to M

must be homeomorphic to an n-sphere'. Again, it suffices to know this for each m with m <= n/2. (This is the statement given in IV.7 in the case n=4.) This statement could be shortened by referring to `homotopy groups' as defined in IV.6.2, but the longer and more elementary statement is probably better.

A third way would be to write `a compact smooth n-manifold which is homotopy equivalent to the n-sphere must be homeomorphic to the n-sphere', referring to IV.6.2 for the definition of a homotopy equivalence. This one is my preference, for what it's worth.

July 30, 2009 at 11:10 am |

[…] 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 […]

September 16, 2009 at 1:01 pm |

Great book. A few more misprints or ademptun.

1) p156 An explicite exemple off “a^b= irra. is a = squareroot(2) , b = log base 2 of 9; where a^b=3” (notice that page 223 tell us that a^b = irra. when a = b =squareroot(2).

2) p 169.1 both denominators are (1+x^2)

3) p 171 not 1/4,-1/4,3/4,-3/4, 4/3,-4/3 but 1/4,-1/4, 2/3,-2/3,3/2,-3/2

4) p309.2 last line Z^2 should be R^2

5) p558 line 4 Québecoise should be Québécoise (I am one of the Labelle “frères”)

6)p771 2xT(2) should be 2xT(x)

7) 1902 Longeur should be Longueur; 1901 Absolut should be Absolu

Thx again.

February 14, 2011 at 9:10 pm |

Is the second (corrected) version of the PCM available now?

February 14, 2011 at 9:25 pm

Yes — it has been for quite some time.

August 28, 2011 at 8:13 pm |

51.1 line 11: Perhaps the sentence should contain the word ‘not’:

For example, equations of the form y^2 = f(x), where f(x) is a cubic polynomial in x, may NOT look rather special, but in fact the ELLIPTIC CURVES [III.21] that they define are central to modern number theory, including the proof of Fermat’s last theorem.

August 29, 2011 at 2:32 pm

Actually the sentence is correct as stated. “Special” here doesn’t mean “remarkable” so much as “particular”. The idea is that these equations look too specific to be of general significance, but in fact that isn’t the case at all.

August 30, 2011 at 12:30 pm |

Ah! Got it. Also many thanks for this terrific book.

October 13, 2011 at 7:52 pm |

[…] Companion 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 […]

November 9, 2011 at 7:43 pm |

Hi,

Greatly enjoying dipping into this volume. I’ve noticed one or two errors in the Lie Theory article (pp.229-2234) as follows:

231.1, line16: The Lie algebras u(n) and su(n) are not the same: su(n) has the extra trace 0 condition.

231.2, para 3: The proposed motivational calculation for the Lie bracket doesn’t work. In fact you have to start with (at least) the degree two Taylor approximations to A and B, ie. let A=I+eX+e^2X^2/2 etc in order for the commutator calculation to work out as claimed. This needs changing in a future edition, as it would confuse a non-expert who tries to check it. In fact, for this reason, it might be better to change the order of exposition in this section a bit, so as to introduce the idea of the exponential first, and then come back to the Lie bracket.

232.1, line 11: It’s incorrect that the exponential map is surjective for a general connected group. It’s true if the group is also compact (standard theorem), but false eg. for the connected group SL(2,R) (see Sepanski: Compact Lie Groups, p.87). What is true, of course, is that the image of the exponential generates the group.

233.1, line -3: whose n+1 coordinates …

Finally, in several places (230.2, 234.1) the notation Sp(2n) is used for the compact symplectic group. Of course, this is just a matter of notation, but most people would denote this as Sp(n) in analogy with O(n) and U(n) (Sepanski, Knapp, Brocker-tom Dieck, Chevalley, Fulton-Harris…, but admittedly not Bump).

These are just minor quibbles. This is a great book!

Graham Williams

December 6, 2011 at 12:08 pm |

On pg 187, in the section on Duality sub section Platonic Solids, it states that if you connect the centers of a cube you will get an octagon. It should be an octahedron

December 6, 2011 at 2:15 pm

I’m glad to say that that one was picked up (see post).

August 6, 2012 at 12:32 am |

I just bought a copy of the sixth printing. I hope that this is the corrected version that you refer to above. However, 3.1 -6 gives a definition of a circle that is actually a disk: there should be an equals sign not an inequality.

But I think that I am going to love the book anyway.

December 21, 2013 at 4:35 pm |

I have an early printing so this may have been fixed (but I didn’t see it in the lists of errata):

112.1 lines -7 and -8

(i) and (ii) clash in defining A(1,1) as first 3 then 2.

January 8, 2016 at 8:07 pm

totaly agree, but it may seems that the next step update the old one(i, ii, iii), it it definitely = 2(A(1,1)=2 as state in the example he gave ), it is not an error if in sequential programming language(like an update of the previous statement), he should explaint it anyway.

July 25, 2022 at 6:03 pm

Also, in (iii) shouldn’t the restriction read “whenever x >= 1 and y >= 1”?

December 19, 2016 at 3:12 am |

363.1, lines -8 to -4 (second last paragraph) “Polynomials are the only functions that computers can work with …”.

I suggest removing this paragraph (unfortunately this will affect page breaks). I don’t think this paragraph is correct, and the rest of the section doesn’t seem to reference the information contained in this paragraph.

Computer processors implement many functions that are not (implemented in terms of) polynomials over any field, such as saturation arithmetic, operations on n-bit words (eg AND, OR, XOR, number of bits set to 1, bit shifting, etc), various comparisons, bit packing/unpacking, integer/floating point conversions, and many others. See for example here:

https://en.wikipedia.org/wiki/X86_instruction_listings

The information that functions like logarithms are computed in terms of polynomials is also not entirely correct. Taylor series expansion is one of the techniques but even then it is only part of the algorithm. For more details, see:

http://math.stackexchange.com/questions/61209/what-algorithm-is-used-by-computers-to-calculate-logarithms