Taking Maths Further Podcast

This week the topic was types of numbers and infinity. We interviewed Dorothy Ker, who’s a musician and composer. We talked about the way Dorothy uses maths to inspire her creativity, as well as the types of maths that composers and musicians use.

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It may seem obvious to say that there are more whole numbers than square numbers - if you start counting, by the time you reach 20 you’ve counted 20 whole numbers but only 4 square numbers, and the square numbers only get further apart as you go up the number line. The set of all square numbers is contained in the set of all whole numbers, and it’s definitely smaller in some sense, as not all whole numbers are square.

However, since there are infinitely many square numbers, it’s possible to count them in the same way you count the whole numbers. Each square is paired up with its own square root - 1 with 1, 4 with 2, 9 with 3 and so on - so there are countably infinitely many square numbers, and since for any whole number I can find a corresponding square number by simply squaring it, these sets are considered to be the same size.

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It will take 7 days. The number of people doubles each day - so at the end of the first day, only 2 of James’ 250 friends have seen it, on the second day four of his friends have seen it, on the third day 8 friends, on the fourth day 16 friends, and so on until on the 6th day, when the number of friends who have seen it is 128. This means on the seventh day, the remaining 122 friends will also be sent the picture.

This isn’t a hugely realistic model - firstly, websites like Facebook allow you to send things to all your friends at once so it’s not clear why you’d do it two at a time; but also, it’s not necessarily plausible that people will always send it to people who haven’t seen it before, so the way things like this spread in reality isn’t always as fast as this - some people may be counted twice if it’s sent to two random people per day.

This week the topic was standard deviation. We interviewed Clara Nellist, who's a researcher at CERN Geneva, Switzerland. We talked to Clara about her work in the Large Hadron Collider and how she uses standard deviation as a measure of how reliable the results are.

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If the speed of the boat’s engine is x (kph), this means it travels 20km at (x-2)kph and then 20km at (x+2)kph. The total journey needs to take 3.5 hours. What value of x does the boat driver need to use?

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Since speed = distance / time we can rearrange that to get time = distance / speed.

Now expand the brackets to get a quadratic equation: 3.5x^2 - 40x - 14 = 0.

In this episode, we talk about cellular automata - including the Game of Life - and graph theory, and interviewed Jonathan Crofts from Nottingham Trent University about his research on complex networks in neuroscience. Find out more about the Biomathematics & Bioinformatics Research Group at Nottingham Trent.

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One way to colour the squares so they infect the whole grid is to colour the five squares on the diagonal. This will then infect the cells next to the diagonal, which will then infect the next diagonal rows, and so on until the whole grid is infected. In order to infect the whole grid, you need to colour at least 5 squares (and in general, for an n × n grid, you need to start with n squares coloured). This can be seen by looking at the perimeter of the infected area.

If a square needs two or more infected neighbours in order to become infected, then the perimeter of the infected area can never increase - if you imagine a pair of squares which are both neighbours to a third square, they will infect it. The newly infected square will have two free edges which increase the perimeter by 2, but the edges of the two squares it was neighbouring become absorbed and are no longerpart of the perimeter. The whole shape will then have the same perimeter as the two squares you started with.

If you have a square with three or four neighbours, which then becomes infected, doing so will only ever reduce the perimeter of the infected area. As time passes, and more cells become infected, the perimeter of the infected area can only decrease or stay the same.

This means that in order to infect the whole square grid, the arrangement of coloured squares you start off with must have the same perimeter as the whole square, as this will never increase - and the edge of a 5 × 5 square is 4 × 5 = 20. This means you need at least 5 squares to start with, each of which has perimeter 4.

In this episode we interviewed Alison Atkin from the University of Sheffield about her work as an archaeology PhD student, and how she uses statistical sampling to gather data about victims of the plague.

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This week the topic was Bayesian statistics. We interviewed Emma Rixon from Nottingham Trent University about her work as a crime scene investigator and how forensic science uses Bayesian probabilities.

• Introduction to Bayesian Statistics, by computer scientist Kevin Boone.

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