## PCM errata II

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 $x$, lifelong happiness is at least as good as $x$.

(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

$(T^i,R^j)(T^{i'},R^{j'})=(T^{i+(-1)^ji'},R^{j+j'})$

27.2 line 18

… it is a simple exercise to show that every homomorphism $f:X\rightarrow Y$ that is not identically zero is automatically an isomorphism …

30.1 line 7

… such that $Sv$ is proportional to …

The v here should be bold face.

33.2 line -5

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

40.1 Sixth line of section 6.4

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

42.2 line -12

It should say “$x^2+y^2+1=z^2$“.

Two lines later it should say “$x^2+1=z^2$“.

In line -2 it should again say “$x^2+1=z^2$“.

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 “$\Lambda^T G \Lambda=G$“.

57.2 Middle of column, it should say

… natural properties (for instance $f(g+h)=fg+fh$ for all functions $f$, $g$, and $h$) …

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

… which leads to the formula $r(n)=\frac 12(n^2+n+2)$.

line -6

replace “$0^2+4^2+4^2+4^2$” by “$0+2^2+2^2+2^2$

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 $\epsilon(t)$ is at most $At^{131/208}(\log t)^{2.26}$ for some constant $A$.

64.2 At the very end of section 6.3 it should say

… as many as $\log n/\log\log n$.

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

$\frac{\sin \sqrt{x}}{\sqrt{x}}=(1-\frac{x}{\pi^2})(1-\frac{x}{4\pi^2})(1-\frac{x}{9\pi^2})\dots$

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 $x$ to the point $(x/(1+x^2),x^2/(1+x^2))$, 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

$0, 1, -1, 2, -2, \frac 12, -\frac 12, 3, -3, \frac 13, -\frac 13, \frac 23, -\frac 23, \frac 32, -\frac 32, 4, -4, ...$

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 $x$ and $y$; this involution is required to satisfy the $C^*$identity $\|xx^*\|=\|x\|^2$.

176.2 Middle of column.

Where it says “where $\gamma(t_0)=a$ and $\gamma(t_1)=b$.” it should say “where $\gamma(t_0)=a$ and $\gamma(t_n)=b$.”

180.2 line 5

Instead of “$\Phi_* x$” it should say “$\Phi(x)$“.

Earlier in the same line it would be better to write “$T_{\Phi(x)}Y$” instead of “$T_{\Phi(x)}(Y)$“.

183.2 line 16

… written either as 0.1 or as 0.02222…

184.2 lines 12-14

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

187.2 line 7

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

194.2 line -8

The final $\frac{\partial}{\partial x_1}$ should be $\frac{\partial}{\partial x_d}$ (so that it matches the $\alpha_d$ 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 $e^{x_2y_3}$. (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 $(\frac 12, \frac {\sqrt{3}}{2})$. I.e., it’s half the square root of 3 and not the square root of half of three.

232.1 bottom of column

$= \exp(X + Y + \frac 12 [X,Y] + \dots)$,

***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 $-cf'+6ff'+f'''=0$. If we add …

240.2 line 8

It should say

… is the matrix $A^*$ defined by $(A^*)_{ij}=\overline{A_{ji}}$.

249.2 line 11

It should be

$=10(10q^2+2qr)+r^2.$

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

266.1 line -21

It should say “$n^{-2}(n\sigma^2)=\sigma^2/n$,” instead
of “$n^{-2}(n\sigma^2)=\sigma^2/2n$,”.

272.2 line 4

It should say “generated by the polynomial $ad-bc-1$.”

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

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 $u$, defined on some interval $I=[a,b]$, such that $u(a)=P$ and $u(b)=Q$. The length …

314.2 line -5

It should say “when one does include the”

323.2 display in line -9

Where it says $\sqrt{D}$ it should say $\sqrt{|D|}$

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 $a\pm ib$ 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 $x/2$s that appear here should be $x/4$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 $S_n$ to $S^n$ (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 $\Gamma$ 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 $\Gamma$ acts properly discontinuously (this means that for any compact set $K$ there are only finitely many $g\in\Gamma$ such that $K$ and $\gamma K$ 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 $\gamma^1, \gamma^2, \gamma^3, \gamma^4$ that satisfy …

496.1 line -19

A few small changes needed here. It should say

However, if we apply $F$ ten times to $x_0$ and $x_0'$, then their respective eleventh digits have shifted leftwards and become the first digits of $x_{10}=0.00111...$ and $x_{10}'=0.10110...$. These two …

523.2 line -21

Replace “slide rules” by “log tables”.

531.2 line 9

It should say image($\delta)\subset$ ker($\delta$) 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 $n\geq 1$ there is exactly … ”

568.1 line -13

… set of integers $\{1,2,\dots,W\}$ using …

575.2 line -5

It should say “Two very obvious conditions” here.

595.1 lines 16-17

.. at least $\frac 34$. Therefore, the probability … is at most $(\frac 14)^k$, which is …

645.1 last line

The right-hand side should read

$\exists z \left(\bigwedge_{i=1}^n x_i\sim z\right) \wedge \left(\bigwedge_{i=1}^m \neg (y_i\sim z)\right)$

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 $x$ to be the number of turns made by the vector from $y(t)$ to $f(y(t))$, counting this negatively if these turns are in the opposite direction to the way that $y(t)$ goes around $x$. (This definition is problematic if $f(y(t))=y(t)$ for some $t$, but again …

701.2 line 7

It should say “no proof of the formula $\phi(p)$, where $\phi$ is the …”

706.1 lines 9 and 11

The two capital Fs should be small Fs. That is, we want
“a function $f$” 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 $q$.”

708.2 line -3

It should say “in the form $y^3+3hy+k$.”

714.2 line 1

Change $S_2$ to $S^2$.

***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

$f_6$ 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

$Y_i = \theta_2 + n_j$” (in the second paragraph) should read “$Y_j = \theta_2 + \eta_j$“.

920.2

In the display the $\frac{1}{2}$ should be removed inside the “exp”, and the “$n$” in the square root on the bottom should be “$\pi$“.

Four lines later, there is again a stray $\frac 12$, and again the “$n$” on the bottom needs to be a “$\pi$“. And this time, we also need to change the “$\theta_2^2$” to a “$\phi_2^2$“.

984.1 line -2

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

997.2

In equation (7), “$\Delta_k$” should read “$\Delta_{2k}$“.

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.

### 66 Responses to “PCM errata II”

1. name Says:

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!

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3. Alejandro Zuleta Says:

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.

4. Career advice from the Fields medalists and some other mathematicians « Academic Career Links Says:

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

5. Name Says:

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).

6. Name Says:

347.1 line -10

Should be “(b, p) = 1”.

7. Name Says:

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”

8. Name Says:

543.1 formula (2)

First equation is missing a left bracket “[“.

9. Name Says:

852.1 middle

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

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

10. Axel Boldt Says:

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

11. Axel Boldt Says:

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

12. Axel Boldt Says:

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

13. Axel Boldt Says:

44.2 line 14: “sphere” -> “ball”

14. Axel Boldt Says:

149.2 line 17: “with with” -> “with”

15. Axel Boldt Says:

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

16. Axel Boldt Says:

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

17. Axel Boldt Says:

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

18. Axel Boldt Says:

180.1 line 14: divergence of curl is 0.

19. Axel Boldt Says:

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

20. Axel Boldt Says:

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

21. Owen Densmore Says:

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

22. Axel Boldt Says:

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

23. Axel Boldt Says:

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

24. Axel Boldt Says:

287.1 line 13 formula has one psi too many

25. Axel Boldt Says:

292.1 line 12 formula needs an additional theta

26. Axel Boldt Says:

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

27. shaivaingast Says:

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)

28. Axel Boldt Says:

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.

29. Axel Boldt Says:

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.

30. Axel Boldt Says:

457.1 line -16: f should be u.

31. Axel Boldt Says:

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

32. Axel Boldt Says:

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.

33. Axel Boldt Says:

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

34. Axel Boldt Says:

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

35. Axel Boldt Says:

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

36. Axel Boldt Says:

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

37. Axel Boldt Says:

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

38. Axel Boldt Says:

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.

39. Axel Boldt Says:

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

40. Axel Boldt Says:

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

41. Axel Boldt Says:

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.]

42. Ashutosh Mehra Says:

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!

43. Axel Boldt Says:

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

44. Axel Boldt Says:

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.

45. Axel Boldt Says:

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.

46. Axel Boldt Says:

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

47. Axel Boldt Says:

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

48. Ian Leary Says:

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.

49. PCM errata III « Gowers’s Weblog Says:

[…] 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 […]

50. Jacques Labelle Says:

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.

51. Bora Eryilmaz Says:

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

52. Amar Doshi Says:

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.

• gowers Says:

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.

53. Amar Doshi Says:

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

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

[…] 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 […]

55. Graham Williams Says:

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

56. Karl Singer, MD Says:

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

57. ALLAN SLEEMAN Says:

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.

58. Maxim Engers Says:

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.

• benkrid Says:

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.

59. Piotr Says:

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