# Errata

This page contains the errata for the first edition of A Programmer's Introduction to Mathematics. I have only included mathematical errors, or grammatical errors that included significant changes to the text to clarify what is meant. Thanks to those readers who submitted simple typos, but those are omitted here for brevity.

Chapter Page Erratum Credit Notes
Polynomials 9 Graph 2.1 seems to contradict sentence 5 of the first paragraph on page 9 Patrick Stingley I added an additional clause to use Figure 2.1 to demonstrate why more nuance is needed
Polynomials 20 Fix the indexing from zero in 'n+1' points Arman Yessenamanov This prompted me to make a few other changes to the prose to explain indexing from 1 vs 0, and convert the chapter to always index from 1 for consistency. This was also pointed out by Francis Huynh in a separate manifestation of the problem.
Polynomials 21 (x_1,x_2) -> (x_1,y_1) Arman Yessenamanov
Polynomials 22 Correct the statement of Theorem 2.4 to say 'degree at most n' instead of 'degree n'. Harry Altman Many easy examples show it need not be exactly degree n. I fixed a few other instances of this throughout the chapter.
Polynomials 22 Rephrase g(x) differently from 'another such polynomial' K. Alex Mills I used this opportunity to clean up some of the language in that proof.
Polynomials 25 a_{d+1} -> a_d Abhinav Upadhyay
Polynomials 27 Clarify confusion around 'decoding to any possible plaintext' Francis Huynh
Polynomials 29 Ex. 2.6, 'k real roots' should be 'n real roots' David Wu
Polynomials 29 Replace the pi+e and pi*e part of the exercise with a 'lookup online' exercise. Timo Vesalainen Add forward reference to vector spaces.
Polynomials 29 product and sum ... [are] algebraic David Furcy Clarified the sentence by splitting it into two and hinting that one is much harder.
Polynomials 29 Exercise 2.6: 'k real roots' -> 'n real roots' Sriram Srinivasan
Sets 46 a \in a -> a \in A David Furcy
Sets 47 Elevate the definition of inverse to a full definition. K. Alex Mills
Sets 49 Add that f is not a surjection in the set-theoretic rephrasing of the single-elimination tournament proof K. Alex Mills
Sets 49 Clarify confusion around the use of 'preimage exactly 2' in the double elimination example. Wojciech Kryscinski
Sets 50 Correct the use of balls and squares in paragraph starting 'Now we count:' Jason Hooper
Sets 52 X is a set of size (n+1 choose 2) -> (X choose 2) is a set of size (n+1 choose 2) Arthur Allshire
Sets 59 Improve the clarity of Exercise 4.2 by reminding the reader of the complement notation. Jan Moren I also added a joke in a footnote.
Graphs 72 guarantee of the theorem -> guarantee of the proposition David Furcy
Graphs 73 Figure 6.5 has an incorrect coloring of the Petersen graph. David Furcy
Graphs 77 Typo in Name: Wolf[g]ang Haken Maximilian Schlund
Graphs 78 Define the term cycle as a path whose start and end vertex are the same Matthijs Also, add index markers for all uses of the word cycle
Graphs 78 Lemma 6.9. should require $m \geq 2$. Counterexample: $K_1$ and $K_2$. Fidel Barrera-Cruz I added a short continuation after Lemma 6.9 that explains Fidel's erratum and dives deeper into why this proof is not obvious.
Graphs 79 Correct the proof of the five color theorem: when constructing $G'$ by collapsing $w_i$ and $w_j$, the expression $N(w_i) \cap N(w_j)$ should instead be $N(w_i) \cup N(w_j)$ (a union, not an intersection), i.e., the collapsed vertex has as neighbors all the neighbors of $w_i$ and $w_j$, not just those vertices that are neighbors of both. Fidel Barrera-Cruz
Graphs 79 G' should have |V|-1 vertices, not |V|-2. Leandro Motta Barros
Calculus with One Variable 99 from x to some other x′ -> from (x,f(x)) to (x',f(x')) David Furcy
Calculus with One Variable 106 In Proposition 8.5: x_n - 2 is less than epsilon -> x_n - 2 is less than epsilon prime Arthur Allshire
Calculus with One Variable 111 Clarify the paragraph that explains the best approximating line by providing a bad definition first. K. Alex Mills Also added a minor joke about 'artisanal' definitions.
Calculus with One Variable 114 for every $n \in \mathbb{R}$ -> for every \$n \in \mathbb{N} Maximilian Schlund
Calculus with One Variable 116 close to x=0 -> when x is close to 0 Tim Wilkens
Calculus with One Variable 118 "On the other hand, f′(x) = −1 + x4 is simple enough to compute" should be 5x^4 - 1 (as is used later on in the proof matthijs
Calculus with One Variable 122 first equation: "(a-x_k)" should be "(r-x_k)" Jan Moren
Calculus with One Variable 123 Should be abs(function_at_x) > THRESHOLD instead of abs(function_at_x - x) > THRESHOLD Nikita V. This didn't affect the correctness of the algorithm or the tests, just the convergence test. I updated the code repository to match, and hit myself in the head a few times for making this obvious mistake
Calculus with One Variable 126 Exercise 8.2.3 - specify n to infty Arthur Allshire
On Types and Tail Calls 131 In the transitive property, replace A x A x A with A K. Alex Mills
Linear Algebra 141 Remove the use of span in the description for Figure 10.4 before it is defined. Tim Wilkens
Linear Algebra 143 On this page there is a call to do exercises 10.1-10.4, however a subspace, used in Ex 10.3, is not defined until 148, after this call Arthur Allshire Fixed by rearranging the exercises
Linear Algebra 148 In the proof of theorem 10.8 "U_1 = V - {u_1}", should be "U_1 = U-{u_1}" Arthur Allshire
Linear Algebra 162 Projection formula incorrect. c should be <v, w> / norm(v)^2 NOT <v, w> / norm(v) Arthur Allshire I had mixed this up previously by confusing the derivation that followed (using only unit vectors) with the formal definition statement. I corrected the definition and updated a few places later in the section that discuss the norms.
Linear Algebra 163 the vector w - proj_v(w) -> the norm of the vector w - proj_v(w) Tim Wilkens
Linear Algebra 170 ignoring all lines that are in the span of v_1 -> considering only those vectors perpendicular to v_1 Tim Wilkens
Linear Algebra 170 In the equation at the top of the page, the upper bound of the summation should be m, not n. Arthur Allshire
Linear Algebra 179 Exercise 10.3 "is a subspace of the codomain of W" -> "is a subspace of the codomain, W" Arthur Allshire
Eigenvectors and Eigenvalues 206 The matrix B for the map (a,b,c) -> (a, a+b, a+b+c) should be reversed Nicolas Krause
Multivariable Calculus 238 d_n should be defined with ||x_n - c||, not ||x_n - x||. K. Alex Mills
Multivariable Calculus 243 Fix the example of the total derivative by adding 2 to f. K. Alex Mills
Multivariable Calculus 248 The partial derivative of f(x_1, x_2, x_3) = x_1^2 x_2 + cos(x_3) w.r.t. x_1 contains the term cos(x^3) twice (once as cos(x_3)), when that term is not in the partial derivative Ondrej Slamecka
Multivariable Calculus 266 Rewrite the sentence 'Note that if a_i has another path to G, they need to be summed.' K. Alex Mills Clarify that G is a function and that G depends on a_i through possibly multiple paths through the computation graph of G
Multivariable Calculus 287 in this post -> in this book K. Alex Mills Too much blogging and I'm on autpilot :-/
Big-O 292 The stated Taylor Series for 'sqrt(1 - x^2)' is instead given for sqrt(1 + x^2) Darin Brezeale
Groups 310 Clarify the definition of a group that has a single generating element K. Alex Mills The book is correct as stated, but I took the opportunity to clarify the difference between a generator (single element generating the group) and a generating set (where a cyclic group has such a set of size one), whose elements are sometimes called generators.
Groups 329 Owen Jones' 'The Grammar of Ornament' is incorrectly cited as 'The grammar of Orient' Rodrigo Ariel Sota Added additional information about the author and title to the first reference.