A New Book I’ve been working on a new book about sequences and series. The original plan for my book How to Think Like a Mathematician was to include some few chapters on epsilon-delta proofs. These were written but for reasons of space I left them out and had always intended to put them into a book. That intention is now being honoured. The bare bones of my thoughts on sequences have been uploaded to my website www.sequencesandseries.com. The design is not finished and there are no exercises, specially recorded videos or extra explanations. Also, there is nothing on series yet! Only sequences so far. My plan is to make the website the best place to go for information on sequences and series. Furthermore, I want to make a public commitment to finishing the book this year. So if I don’t, then please feel free to mock me on 1st January 2019! If you are interested in this experiment of writing a book and having some of it online, then please have a look and sign up for updates. The other project I’m working on is I’m gearing up for a proper launch of my latest book Complex Analysis: An...

## Why is it called a matrix?...

posted by Kevin Houston

Here’s a question for which the answer doesn’t seem to be widely known. Certainly, no one I’ve asked so far has known. They all found the answer interesting though! Why is a matrix called a matrix? If we consult www.etymonline.com for the origins of the word matrix we find matrix (n.) late 14c., “uterus, womb,” from Old French matrice “womb, uterus,” from Latin matrix (genitive matricis) “pregnant animal,” in Late Latin “womb,” also “source, origin,” from mater (genitive matris) “mother” (see mother (n.1)). That doesn’t seem to be much help (but does explain the word matriarchy and why surgeons sometimes refer to the matrix). However, surprisingly “womb” is the origin of the word in mathematics. To see this we go back to JJ Sylvester‘s 1850 article Additions to the articles in the September number of this journal, “On a new class of theorems,” and on Pascal’s theorem in which it is used in this context for the first time. He says For this purpose we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding to which may be termed determinants of the pth order. So what is Sylvester talking about here? Well, his interest is in developing what we now think of as Linear Algebra. In particular, he cares about determinants and is thinking of them as arising from a square array of numbers. In the passage above he notes that from an m by n array of numbers...

## It’s coming…...

posted by Kevin Houston

Posts have been a bit scarce round here recently. Something’s coming soon…

## Common Mistakes in Complex Analysis (Revision help)...

posted by Kevin Houston

My new book, Complex Analysis: An Introduction, is nearly finished. To help my students with revision I created a list of common mistakes and this forms a chapter in the book. As a lecturer with many years of experience of teaching the subject I have seen these mistakes appear again and again in examinations. I’m sure that, due to pressure, we’ve all written nonsense in an exam which under normal conditions we wouldn’t have. Nonetheless, many of these errors occur every year and I suspect something deeper is going on. What follows is not intended to be a criticism of my students, who, luckily for me, are generally hard-working and intelligent. Nor is it an attempt to mock or ridicule them. Instead the aim is to identify common mistakes so that they are not made in the future. And if this post seems negative in tone, the a later one is more positive as it delves into techniques that improve understanding. Imaginary numbers cannot be compared The first mistake is the probably the most common: the comparison of imaginary numbers. For example, students write for a complex number. This cannot be right. If were what does it mean for to be less than ? What is usually intended is the modulus of , i.e., . The point is, unlike real numbers, we cannot order the complex numbers. For example, which is bigger or ? This is difficult to decide! Since complex numbers can be identified with the plane ordering them is equivalent to ordering the points of the plane and clearly this can’t be done — at least not in any useful or meaningful way. One last point needs to be made. Although is incorrect, note that expressions like can be true if is...

## Best of xkcd: Educational...

posted by Kevin Houston

The web comic xkcd is well-known for its humour. I’ve also got a soft spot for the educational infographics, such as the following. (Not all the information should be...