# You're reading: Travels in a Mathematical World

### You have completed level 8. Game over. Insert coin.

(See QCF.)

That is to say, the university have sent me a degree certificate, and I’ve shown it to the bank. So that’s pretty darn official.

### Why do $0!$ and $a^0$ equal $1$?

The last two weeks my first year mathematicians and I have covered Taylor series.This means that several times I’ve had the conversation that goes “What’s $0!$?” “It’s $1$.” “Oh, erm, right. Why again?” “Because it works.” This may not be a completely satisfactory answer!

One of my students, Callum Mulligan, tweeted this question.

Saying “by definition” or “because it makes a bunch of stuff work” won’t cut it. So how to answer this question? To give a somewhat intuitive understanding of why this should be the case to a first year undergraduate. It may be obvious, but it wasn’t immediately obvious to me how to explain this, so I share some thoughts here.

### Dynamic generation of maths questions

I was recently asked about my MSc dissertation (by someone who may choose to ‘out’ themselves here, but as it was a personal email I won’t name them). In my dissertation, for a Masters degree in computing in 2003-4, I developed a system for pre-processing MathML code using PHP to include pseudo-randomised values in the questions for an e-assessment tool. The title is ‘Asking Questions With MathML: dynamic treatment of XML and pseudo-randomised mathematics assessment’.

The query was from someone who is training to be a maths teacher and is doing some web development. They had seen mention of my MSc dissertation topic on this blog and asked where they could read more about the underlying web technologies. Here, basically, is what I replied.

### Probability of dealing four perfect hands of cards in a world of random shufflers

A couple of months ago (really? Two years?! Man!) I posted about an extraordinary coincidence: in a game of whist at a village hall in Kineton, Warwickshire, each of four players had been dealt an entire suit each. My post ‘Four perfect hands: An event never seen before (right?)‘ discussed this story. What really interested me was that the quoted mathematical analysis — and figure of 2,235,197,406,895,366,368,301,559,999 to 1 — appears to be correct; what lets down the piece is poor modelling. The probability calculated relies on the assumption that the deck is completely randomly ordered. Apart from the fact that new decks of cards come sorted into suits, whist is a game of collecting like cards together, so a coincidental ordering must be made more likely. Still unlikely enough to be worthy of mention in a local paper, maybe, but not “this is the first time this hand has ever been dealt in the history of the game”-unlikely.

Anyway, last week I was asked where the quoted figure 2,235,197,406,895,366,368,301,559,999 to 1 actually comes from. Here’s my shot at it.

### Why maths isn’t like science (and may be evil)

Yesterday I gave a talk to the Nottingham Trent University Maths Society, ‘A brief history of mathematics: 5,000 years from Egypt to Nottingham Trent’. I had a slide in this where I said something about what the Greek style of proof means for mathematics. It has helped me put my finger on something of why mathematics isn’t like science, and I thought I would share it here so I can look it up when I’ve forgotten again.

### Recent history

I am preparing a talk for our undergraduate Maths Society (perhaps ill-advisedly) with the title ‘A brief history of mathematics: 5,000 years from Egypt to Nottingham Trent’. This will be a stampede through some very selected hightlights, starting with some arm waving about pyramids and ending with something modern. In fact, as well as some recent results that have been reported on this site this year, I intend to end with a recent piece of research published in the department (and a colleague has promised a nice picture of a brain for this1).

Relatedly, a colleague who teaches on our Financial Mathematics degree suggested I include Black–Scholes in my talk, as one of the most recent results included in an undergraduate degree. It’s from 1973. Can we do better? I asked Twitter.

1. Something to do with complexity and computational graph theory. []