Surface Pro 4: Episode 4

I am continuing to use my Surface Pro 4 for PDF annotation in my first-year lectures.

Although there are many good features, I am still having some problems with unexpected menus opening up.

Some of these are caused by my hand hitting a key point near the edge of the screen (my Bluebeam PDF Revu settings mean I should have no problem with touch in the middle of the screen.) But I think my main problem is caused by the side button on the stylus. This generates a right click. Now I already told Bluebeam not to use right click to activate the lasso while I am inking, which helps.  But you can still get right-click menus appearing due to accidental right clicks. I would rather like to be able to deactivate the side button on the stylus, but I haven’t found any way to do that. (Does anyone know if it can be done?)

There are also some menus which suddenly appear and where I can’t yet duplicate the operation that brought them up. I think that touch is involved. Maybe I’ll get to the bottom of this some time!


Applications of complex numbers

Here is my latest announcement to my first-year students.


In the spirit of “applications of pure mathematics”, I thought I would say something about applications of complex numbers.

According to the Wikipedia page

complex numbers were first introduced by an Italian mathematician, Gerolamo Cardano, during his attempts to solve cubic equations in the 16th century.

You probably all know the quadratic formula. There are similar but more complicated formulae for solving cubic and quartic polynomials. The search for a similar formula for the quintic proved fruitless, and in fact there is, in general, no such formula for solving the quintic. The relevant area of mathematics is Galois Theory. This is off-topic today, but see if you want a flavour. This is not to say that quintics don’t have roots (they do!), just that you can’t always find a formula for them using the coefficients and nth roots etc.

This is all well and good, but inventing some apparently fictitious numbers in order to find solutions where you didn’t have them before may not feel like much progress. But the amazing thing is that “pure” theory of complex numbers, complex functions and complex analysis has applications almost everywhere you look, and not just within mathematics.

If you have studied physics, you  may already have met complex numbers and functions when looking at impedance, phase angles, and oscillating currents. Wikipedia mentions practical applications in many other fields. I’m only going to mention a small number of things today, but you could look at

for more.

In first year calculus, when you study differential equations, you will see some complex numbers come in when looking for solutions. They then go away again, because you want to find solutions using real numbers. But the exponentials of imaginary numbers lead you to use the functions cos and sin in your solutions.

In second-year complex functions you will see how the beautiful theory of complex functions enables you to use “residue calculus” to quickly find the exact values of “improper integrals” that look a little tricky otherwise, such as

\int_{-\infty}^{\infty} \frac{dx}{1+x^4}\,

and many far more complicated examples. In fact this topic is enough on its own for an third-year project! But you could see

for a few more examples.

I think that it is remarkable that the most efficient way to calculate this kind of real integral involves using the theory of complex functions as (mostly) developed in the 19th century, especially the work of Cauchy and Riemann.

I could say much more here, but for now I’ll just mention that these methods become crucial again for calculating the Laplace transform and inverse Laplace transform, which have too many applications to list here! See, for example,

Best wishes,

Dr Feinstein

Applications of “pure” mathematics

I have just posted the following message in my first-year pure maths module’s announcements forum.


Hi everyone,

With its emphasis on abstraction and rigorous logical thinking, you may wonder whether or not the Pure Mathematics you are learning in G11FPM actually has applications in the “real world”.

In fact I did mention some applications in the first lecture and workshop: for example, secure transactions on the internet depend on encryption algorithms, many of which which have their basis in Number Theory, especially the theory of prime numbers. See, for example,

In fact, whichever area of mathematics you work in, rigorous logical thinking is rather important (although this may not always be apparent at undergraduate level).

Have a look at the page

for a rather good discussion of just how important Pure Mathematics is.

However, I will admit that the main reason that I do research in Pure Mathematics is because I think it is beautiful and fascinating, rather than because I expect my work to have applications in the “real world”.

The area of mathematics you choose to specialize in is largely a matter of taste. I know that not all of you will find Pure Mathematics to your taste. Still, I will do my best to introduce you to the flavour of the subject. After that it is up to you!

However, I plan to post a few more messages in this thread giving some more applications of “pure” mathematics.

Best wishes,

Dr Feinstein

Polls in classes

I use mobile device based voting via Google forms in my classes, and this has generally worked well. But there can be problems! I have just posted the following message in the forum for my first-year module.


Hi everyone,

I’d like to make a plea for people to take the polls/votes in classes seriously. Well, I think many of you do already, but there appears to be significant distortion introduced by those who don’t.

These polls and the associated discussions between students have an important role in your learning, and provide an opportunity for a form of interaction in the large class we have.

You are already getting most of the benefits if you are attempting the problems and trying to persuade your neighbours that your answers are right. Explaining your reasoning to another student is a very good way to solidify your own understanding of the material! From this point of view, distortion in the vote/poll doesn’t matter so much. However, I would like to be able to gauge how well the class is following at a particular point, and whether I need to explain the question again or give extra hints.

It is hard for me to judge what the situation is if the results are distorted. So please do take the votes seriously, so that I can take appropriate action based on the results.

Best wishes,

Dr Feinstein

Who used a “pure maths” theorem this week in their work?

I often get requests from my students for more examples of the uses of “pure” mathematics in other areas. Of course one possible answer is “it’s all mathematics anyway” and another is “that’s not why I do pure maths”. Then there are all the standard things such as applications of Number Theory in cryptography and internet security, and countless applications of linear algebra. But somehow I would like to have some other kinds of answers along the lines of “I used this theorem last week to help me with this real-world topic”. For example, I was able to use a standard theorem about uniform continuity to clarify a technical detail for my colleague Daniele Avitabile in his recent paper about neural nets (see, even though I know nothing about neural nets.

I would quite like to give more examples of this sort of thing. So, who used a “pure maths” theorem in their work this week?

PDF annotation in Windows

What do others use for their PDF annotation in Windows, especially Windows 10? Here are some quick comments of my own.

Bluebeam PDF Revu is still my favourite so far, but it is expensive! It also needs a lot of processing power, but is fine on modern PCs.

Xournal is free, but appears to have pressure sensitivity issues in Windows 10. Adding extra pages when you need them is also odd if you are working with a pre-prepared PDF skeleton with blanks as I do.

PDF annotator still saved annotations as bitmap instead of vector last time I tried it. Maybe the latest version is better?

Windows Journal has quality issues when converting back and forth between PDF and Windows Journal Note.

OneNote is high quality, but doesn’t seem to be designed for annotation of pre-prepared PDF skeleton files of the type I use in my lectures.

Drawboard PDF comes free with our Surface Pro’s (it’s quite cheap anyway), but I find it is fiddly to select the tool you need. I wouldn’t want to do this live in a lecture! Fine for annotating calmly in the office.

Any comments?

Surface Pro 4: Episode 3

The new session has started, and I am now using my Surface Pro 4 to give my first year G11FPM Foundations of Pure Mathematics lectures.

Generally this is working OK. I still find that my hand occasionally touches something near the edges of the screen and brings up random menus, but hopefully I’ll get better with practice! Also, I hope my handwriting will improve a bit once I am back into the swing of things. Not that it is ever that good, but I think I can do better than I did today!

I am still using a blue plastic box to give me a bit of extra height. At one point today the Surface Pro slipped off the back of the box (it survived). So I am going to use the box in portrait mode from now on, not landscape!

One sad thing today was that the lectern microphone in the room was oversensitive, at least when combined with my loud lecturing voice. This resulted in voice distortion in my videos from the first day’s classes. Sensitivity should be better next time (thanks Terry!). I’m in a different room tomorrow, so we’ll see how things go there.

I quite like the fact that I can still use touch to pan and zoom while I have the pen tool selected in Bluebeam PDF Revu. That does take a bit of getting used to though.