Taking Maths Further Podcast

This week the topic was calculus and differentiation. We talked to Florencia Tettamanti, who’s a mathematician working on fluid dynamics. We talked about how Flo uses calculus to study the motion of fluids like air and water, and what it’s like to be a research mathematician. Interesting links: Basic differentiation, at s-cool Differential equations, at the University of Surrey website Fluid dynamics on Wikipedia NSF videos on Fluid Mechanics - YouTube playlist Puzzle: If your function is given by y = x2 - 6x + 13, what is the minimum value of y, and for which value of x does the function give this value? Solution: If you plot the points x=1, x=2, x=3 and x=4 you can clearly see the curve of this graph and that it seems to have a maximum at x=3, for which the value of y is 4. To see what the graph looks like, you can input the equation into Wolfram Alpha. Another way to see this is to rearrange the equation: x2-6x+13 = (x-3)2+4, and by examining this equation we can see that this is just an x graph, shifted across by 3 and up by 4, so its turning point and hence the minimum will be at x=3 and y=4. If you know how to use calculus, you can find the turning point more easily - if you differentiate x2-6x+13 you get 2x - 6, which will equal zero when x=3, and putting this value back into the original equation gives y=4. Show/Hide

This week the topic was mechanics and friction. We interviewed Dan Hett, who works for CBBC writing computer games for their website. We talked about his work and how he uses a lot of mathematics in modelling how characters move, and making sure that’s done in a realistic way. Interesting links: CBBC games website CBeebies story app (with pop-up book!) Game physics on Wikipedia A-level Mechanics topics at MathsRevision.net Friction and Coefficients of Friction at Engineering Toolbox (with some example values) Coefficient of friction on Wikipedia Puzzle: Susan the Hedgehog runs at 20cm/s across the screen while the run button is held down. Once the run button is released, she slows down with constant deceleration of 8.5cm/s2. Will she stop within 32cm more of screen? Solution: The time taken to stop can be calculated by knowing that every second travelled, 8.5cm/s of speed is lost, so after 20/8.5=2.35 seconds, speed will be zero. We can approximate this deceleration by imagining Susan is travelling at 20cm/s for 1 second, 11.5cm/s for 1 second and 3cm/s for the remaining 0.35 seconds until she stops. This will cover more distance than the actual motion does (as your speed is lower than this for most of the time), but will cause you to travel only 31.6cm - so you will definitely stop within 32cm. (In actual fact, the distance taken to stop will be 23.53cm, because your speed continues to decrease at a constant rate for the whole time. In order to work this out, you need to use a little calculus!) Show/Hide

This week the topic was Fourier analysis. We interviewed Heather Williams, who’s a medical physicist and works with Positron Emission Tomography (PET) scanners, as well as other medical scanning devices. We talked about her work and how maths is important in converting data from the scanner into images that can be used to diagnose patients. Interesting links: PET scanners on the NHS website Being a Medical Physicist on the NHS careers website Central Manchester University Hospitals, Heather's employer Shape of the sine and cosine graphs at BBC Bitesize An interactive guide to the Fourier transform at BetterExplained.com XKCD comic 'Fourier' Puzzle: If a function is made by adding sin(x) + cos(x), what’s the maximum value attained by this function? Solution: This is a periodic function, which repeats every 180 degrees (or π radians). Its maximum value is the square root of two, or √2 = 1.414213..., which it first reaches at a value of 45 degrees, or π/4. The function varies between √2 and -√2, and it looks like a sin curve. The function can also be written as √ 2 + sin(θ + π/4). For a graph of the function, and more detail, input it into Wolfram Alpha. Show/Hide

This week the topic was mathematical modelling and linear programming. We interviewed Rick Crawford from AMEC, who’s a mathematician studying decommissioning of nuclear reactors, and using mathematical models to determine whether it’s safe to continue using a particular reactor given that it may have degraded over time, but without actually building a physical model of it. Interesting links: Nuclear Power Plant at HowStuffWorks Mathematical model on Wikipedia Small angle approximation at John Cook's blog The Endeavour Linear programming at Purple Math Puzzle: A rod sits inside a cylindrical tube of the same height. The tube is 193mm tall, and 50mm in diameter. We assume the rod has zero thickness. What’s the maximum angle away from vertical that the rod can make (to the nearest degree)? Solution: You can imagine the rod as being a line inside a rectangle, since the cylinder is the same all the way round. Then, you need to calculate the angle made by the rod when it’s touching one bottom corner of the rectangle and resting against the opposite side. This will be a triangle whose base is 50mm and hypotenuse is the length of the rod. The angle from the vertical will be the top corner, and the sin of this angle will be the opposite (base of the triangle) over the hypotenuse. So the angle will be sin(50mm/193mm), which is 15 degrees to the nearest degree. Show/Hide

This week the topic was statistical distributions and actuarial science. We interviewed Richard Harland, who works in risk management for an insurance firm. We talked to him about his work as an actuary, and how he uses statistical distributions like the normal distribution to predict the probability of risky events. Interesting links: Normal distribution on Wikipedia Normal distributions at Maths Is Fun Introduction to being an actuary at the Actuarial Institute website Be an actuary website Puzzle: Your factory packages crisps into bags using a machine which isn’t completely accurate and the weight of crisps which ends up in each bag varies according to a normal distribution. The mean weight of a bag is 154g, and the standard deviation is 8g. The bags are labelled as containing 150g of crisps, but 31% of bags produced by the machine are underweight. To what value should you change your mean weight to make sure 95% of bags weigh more than 150g? Solution: On a normal distribution curve, 95% of values will fall within two standard deviations of the mean. This means in order to ensure 95% of crisp packets weigh 150g or more, we need 150g to be two standard deviations away from the mean - so the mean needs to be 150g + (2 x 8g) = 166g. Show/Hide

This week the topic was mathematics and money, and how maths is used in finance. We interviewed Sarah O’Rourke, who’s an accountant working on the problem of moving cash around to where it’s needed in cash machines. We discussed the ways she uses mathematical modelling to predict where demand for cash will be high, and also the other types of work that accountants do, and the different ways to become an accountant. Interesting links: Accounting on Wikipedia Double entry bookkeeping at Dummies.com What is Financial Mathematics? at Plus Magazine Maths games - percentages at IXL Tax Matters at HMRC Money Talks, interactive game at the NI Curriculum website Puzzle: Using only £20 and £50 notes, what’s the largest multiple of £10 you can’t make? In an imaginary scenario where the only notes are £30 and £70, again what’s the largest multiple of £10 you can’t make? Why do you think we use the denominations of currency that we do use? Solution: Using only £20 and £50 notes, it’s not possible to make £10 or £30, but all other multiples of £10 are possible. This can be proven by noting that £20 x 2 = £40, and £50 x 1 = £50, and from here every other multiple of £10 can be made by adding different numbers of £20 to either of these base amounts. If our notes are £30 and £70, we can’t make £50, £80 or £110, but all other multiples of £10 above £110 are possible. This can be proven by noticing that once you can make three consecutive multiples of £10, any other can be obtained by adding £30 notes - and in this case, we can make £120 = 4 x £30, £130 = £70 + 2 x £30, and £140 = 2 x £70 so we can then get £150, £160 and £170 by adding £30 to each, and so on. The notes currently in use (£5, £10, £20 and (rarely) £50) have been chosen so that it’s possible to make any amount that’s a multiple of £5 using relatively few notes. We don’t need a £30, as it can be made easily using £10 + £20. The system is designed to make it as easy as possible to make any amount, while keeping the number of different types of note needed relatively small. Show/Hide

This week the topic was maths and art. We interviewed Edmund Harris, who spoke about his work using different media to engage people with mathematics, including his work creating art with a mathematical basis. We discussed his work with tilings and how he uses maths in his work. Interesting links: Edmund's website Edmund's Blog where he posts some examples of his work Mathematical Imagery page at the American Mathematical Society Wallpaper Patterns at EscherMath Puzzle: Look for examples of tilings in the world around you - patterns which repeat, or have reflection symmetry, and see how many you can find.

This week the topic was vectors and matrices. We interviewed Dave Langers, who studies the human brain, and how it processes hearing signals. We talked about how Dave uses matrices and vectors to store information from brain scans, which allow him to manipulate the information more easily. Interesting links: Matrices, on Khan Academy Matrices at Maths is Fun Matrix multiplying at Maths is Fun Auditory system on Wikipedia Puzzle: Using the vectors (1,3,1,3), (2,2,6,6) and (4,8,8,12), by taking multiples of each and adding them together, find a combination which adds up to (0,0,0,0). You must use all three vectors at least once. Solution: 2 × (1,3,1,3) - (2,2,6,6) - (4,8,8,12) = (0,0,0,0) This can be achieved using other combinations too - for example, if you multiply all the vectors on the left of this equation by the same number, you’d also get a zero vector. Show/Hide

This week the topic was data analysis. We interviewed Judith Elgie from INRIX about her work as a data analyst, and how she uses computers to analyse and predict the movement of vehicles on the roads, to generate information about where traffic jams are and which roads are clear. Interesting links: Introduction to regression lines by least squares, from m4ths.com Simple linear regression resources from Statstutor.ac.uk Traffic reporting, on Wikipedia Traffic information, on the UK Highways agency website Puzzle: A lorry can travel from point A make a delivery at point B, 70 km away, in 1 hour. Today, however, after 30 minutes at the normal speed, the lorry is forced to stop for ten minutes. Resuming the journey, can the lorry arrive on time without breaking the 70mph speed limit? Solution: On a normal journey, the lorry travels 70km in 1 hour, which we can assume is a constant speed of 70km/h. Today, they travel at this speed for half an hour: 70km/h × 0.5h = 35km (this is also half the distance, as you would expect. The other 35 km must be travelled in 20 minutes, i.e. one third of an hour, so to calculate the speed needed we can divide 35 / (1/3) = 35 × 3 = 105 km/h. If we convert this to miles per hour, 105km/h is 65.3mph - so everything is legal. Show/Hide

This week the topic was boolean algebra. We interviewed Robie Basak, who's a computer programmer at Canonical, about his work on the Ubuntu operating system, and how he and his colleagues use mathematics and mathematical thinking in order to write computer software. Interesting links: Canonical Website Ubuntu website Boolean Logic, at HowStuffWorks Domino computer video Blog post from Tanya Khovanova, on logic puzzles Knights and Knaves puzzles and examples, Wikipedia Knights and Knaves puzzles, on Maths is Fun Puzzle: Anna and Bill are residents of the island of knights and knaves. Knights always tell the truth, and knaves always lie. Anna says "We are both knaves.” What kind of person are Anna and Bill? Chris and Diane are also residents of the island of knights and knaves. Chris says "We are the same kind", but Diane says "We are of different kinds”. What kind of person are Chris and Diane? Solution: Anna states they are both knaves; this can’t be true, as if it were she would be a knave and therefore would be lying. So, the statement must be false, and Anna is therefore a knave. Also, since she always lies, they can’t both be knaves, so Bill must be a knight. In the second example, the two give different answers, one of which must be true, so one of them must be lying, and therefore Diane is telling the truth. This means Chris is a knave and Diane is a knight. Show/Hide

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. Interesting links: "A gentle infinity" - One of Dorothy's compositions Amelia and the Mapmaker, the project on the Poincaré conjecture Jorge Luis Borges, on Wikipedia Marcus Du Sautoy on Borges for BBC Radio 4's Great Lives Recounting the rational numbers, at The Math Less Travelled Puzzle: Which are there more of: whole numbers, or square numbers? (If you think the answer is obvious, try counting them). Solution: 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. Show/Hide

This week the topic was coordinate geometry and structural engineering. We interviewed John Read, who’s a structural engineer, about his work and how he uses mathematics to design structures and buildings. Interesting links: What do structural engineers do?, at the Institute of Structural Engineers website Structural engineering, on Wikipedia Introduction to coordinate geometry, at Maths Open Reference Coordinates revision at GCSE Bitesize Puzzle: A square has one corner is at coordinates (1,12) and the corner opposite it is at coordinates (22,15). Calculate the coordinates of the other two corners. What is the area of the square? Solution: Show/Hide

This week the topic was the most efficient way to pack shapes in 2D and 3D space. We interviewed Jacek Wychowaniec, who’s a scientist studying applications of materials science to biology. We talked about how he uses many different types of maths in his work, and how he’s been developing substances which can be used to help regrow damaged nerves. Interesting links: Allotropes of carbon on Wikipedia Packing Problems on Wikipedia Packing 3D shapes at Nrich What is graphene? at Gigaom Manchester Graphene institute website Is graphene a miracle material at BBC News Puzzle: If you pack circles onto a surface using a square arrangement (each circle is sitting in one section of a square grid and they all touch), what percentage of the surface is covered by squares? How much of the surface is covered if you pack the circles on a grid of hexagons? Solution: If circles are arranged in a square grid, they cover 78.5% of the area. If this is changed to a hexagonal arrangement, where each row of circles is offset from the one above and below, this increases to 90.7%. A detailed description of how to calculate this can be found here: http://nrich.maths.org/604&part=solution Show/Hide

This week the topic was trigonometry. We interviewed Stephanie Yardley, who’s a solar physicist. We talked about the research Stephanie does into activity on the surface of the sun, and how she uses trigonometry to analyse data from satellites and telescopes. Interesting links: Space weather information booklet Videos at the Space Weather Prediction Centre website Sine, cosine and tangent at Maths is Fun Shape of the sine and cosine graphs, at BBC Bitesize Using trigonometry to find components of vectors How to find vector components, at For Dummies Puzzle: You want to calculate the height of a tall building. You set up a device for measuring angles, on a 1m high tripod, which is 200m away from the building. The angle above horizontal, when looking at the top edge of the building, is 15 degrees. What is the height of the building in metres? Solution: The height of a triangle with base 200m and angle 15 degrees is 53.6m. This, added to the height above the ground you are measuring from, means that the height of the building is approximately 54.6m. Show/Hide

This week the topic was exponential growth, and pension investments. We interviewed Simon Perera from Lane, Clark & Peacock about his work as an actuary, what an actuary is and how it involves predicting the growth of investments. Interesting links: Actuarial science on Wikipedia Exponential growth on Wikipedia Exponential growth at Maths is Fun Be an actuary website The wheat and chessboard problem on Wikipedia Workplace pensions at Gov.uk Puzzle: James has 250 friends on Facebook. He sees a really funny photo and sends it to two of his friends. The next day, each of those friends sends it to two of James’ other friends who haven't seen it yet. If this repeats, how many days will it take (including the first day on which James originally sent the photo) until all of James friends have seen it? Solution: 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. Show/Hide

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. Interesting links: Large Hadron Collider on Wikipedia CERN website Standard deviation at Maths is Fun What is the Higgs Boson? at HowStuffWorks Accuracy versus precision by Matt Parker Puzzle: The heights of a group of people are measured, and the resulting data has mean 1.35m, and standard deviation 0.13m. Someone in the group is 180.5cm tall. How many standard deviations away from the mean are they? Solution: Their height is 180.5cm, which is 45.5cm away from the mean. This is 3.5 times the standard deviation of 13cm. Show/Hide

This week the topic was quadratic equations and their applications. We interviewed Colin Wright, who works on radar systems for coordinating and tracking ships and boats. Interesting links: Quadratic equations in the real world Marine Radar on Wikipedia How to use the quadratic formula Path of a projectile (interactive GeoGebra page) Puzzle: A boat is going to sail 20km upstream along a river, then 20km back to where it started. Due to the speed of water flowing in the river, its speed is reduced by 2kph on the way upstream and increased by 2kph on the way downstream. 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? Solution: Since speed = distance / time we can rearrange that to get time = distance / speed. Then our speed is (x-2) on the way out and (x+2) on the way back, so 20/(x-2) + 20/(x+2) = 3.5 Multiply through by (x+2) and (x-2): 20(x+2) + 20(x-2) = 3.5(x+2)(x-2) Now expand the brackets to get a quadratic equation: 3.5x^2 - 40x - 14 = 0. We can use the quadratic formula to get two solutions (one positive and one negative), but we know that x is positive, as it’s a speed so x=11.77kph. Show/Hide

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. Cellular automata: Stephen Hawking’s introduction to the Game of Life Clips from lots of Game of Life configurations, cut together like an epic sci-fi movie trailer! Game of Life simulator on Wikipedia Explanation of, and uses for, cellular automata, at GCSE Bitesize Geography An application that uses Brian’s Brain to trigger musical notes. Graph Theory: Graphs, at Wolfram Mathworld Seven Bridges of Konigsberg: a puzzle in graph theory Puzzle: I have a 5 × 5 grid, in which the squares can either be empty (white) or infected (black). The four ‘neighbours’ of each square are the ones directly next to it: up, down, left and right. A square will become infected if two or more of its neighbours are infected. Can you find a set of squares to colour black (‘infect’) which will eventually spread the infection to the whole grid? What’s the smallest number of squares you need to do this? Solution: 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. Show/Hide

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. Some interesting links: Alison's Blog Alison’s research page and PhD abstract The Black Death on Wikipedia Population structure, at GCSE Bitesize Geography Sampling, at Revision World Stride length calculations, from the Open University Puzzle: A field is 6 metres square. Your sampling methodology dictates you should excavate 30% of the field. What would be the side length of a square trench which covers 30% of the area? Solution: The area of a field 6 metres square would be 6 x 6 = 36 square metres. 30% of this area would be 36 x 0.3 = 10.8 square metres, which would be covered by a square trench whose side length is the square root of 10.8. This is approximately 3.29 metres (to the nearest centimetre). Show/Hide

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. Bayesian probability:Introduction to Bayesian Statistics, by computer scientist Kevin Boone. A nice simple introduction to conditional probability, from Maths Is Fun. Math Goodies, introduction and questions on conditional probability. Puzzle: I have a bag containing 47 black marble and 53 white marbles. I draw out two marbles. What’s the probability that the second marble I draw out is white, given that the first marble I draw is black? Solution The bag contains 47 black marbles and 53 white marbles. This means the probability of drawing a black marble first time is 47/100, but then having done so there are 46 black marbles and 53 white marbles. The probability of drawing a white marble the second time is then 53/99, instead of 53/100, which would be the probability of drawing a white marble the first time. Show/Hide