Just A Theory

Ah, iPlayer. What would I do without you? I didn’t manage to catch the BBC4 broadcast of the first episode of Science and Islam last night, but thanks to the wonderful catch-up service I am able to provide you with a full review. Of course, services like the iPlayer would be impossible without the internet, which in turn could never arisen without first inventing the computer. And what makes computer software tick? Algorithms.

An algorithm is basically a set of instructions, broken down in to simple steps. A computer can follow an algorithm to do pretty much anything, which is why we find them so versatile. As presenter Jim Al-Khalili (a physicist born in Bagdad) tells us, algorithms were invented by a Persian man known as Mohammad ebne Mūsā Khwārazmī, or al-Khwārizmī. Even the word algorithm is derived from his name.

It’s not just algorithms that have been given to us by medieval Arab scholars. The words algebra and alkalis both betray their Arabic origin, but so much of science is attributed to the West. The three part series seeks to unearth the unsung heroes of Islamic science.

The rulers of the Islamic empire realised that with knowledge comes power, and as they spread their influence across the globe the sought out scientific texts from many different regions and cultures. These texts were translated into Arabic, the official language of the empire, which just so happened to be a very scientific language. Originally intended to communicate the teachings of the Koran without misinterpretation, its detailed scripts allowed a precise and unambiguous description of many scientific phenomena.

Much of our modern knowledge can be traced back to this extensive library. In one part of the programme, Al-Khalili visits a modern surgeon to get him to perform a cataract operation by following an Arabic text and using replica instruments from the time. Thankfully for the squeamish the operation is carried out on an eye that has long since been separated from its owner, and the surgeon admits that the instructions are based on sound principles. Indeed, Islamic science provides us with one of the very first anatomical diagrams, showing how the eye is controlled by surrounding muscles.

It’s easy to draw parallels between this programme and an earlier BBC4 one, namely Marcus du Sautoy’s The Story of Maths. Both adopt a sort of travelogue approach, but whilst the earlier programme consisted of nothing but all du Sautoy, all the time, Science and Islam is nicely broken up with contributions from many others. They do cover similar ground however, especially when Al-Khalili meets mathematician Ian Stewart to examine one of the early texts on al-jabr; that is, algebra.

The conclusion of this episode is that by gathering texts from many different places, Islamic scientists proved that science is a universal concept that belongs to no one religion or culture; rather, it can be appreciated by everyone. No arguments here. I will say that at an hour, the programme was perhaps overly long. I can lay the same criticism against it as I did to The Story of Maths - less of our narrator wandering through generic marketplaces please! At least there was no dodgy CGI, however.

As I said at the start, I watched the programme on iPlayer, so of course so can you. If you liked The Story Of Maths, or perhaps if you missed it but want to learn about the history of science, I suggest you give it a look.

University of Warwick mathematician Ian Stewart has provided New Scientist with a scientific guide to gift wrapping. Very festive. Professor Stewart informs us about the “sausage conjecture”, which asks what the most efficient way to wrap a group of circles or spheres is.

The tastiest way to wrap a sausage

For two and three dimensions, we have the answer: round circular objects (like mince pies) should be stacked end to end like a sausage if you have six or fewer, but for seven or more you’re better off arranging the pies in a hexagon and wrapping them that way in order to minimise the paper used. For spherical objects (Christmas puddings, of course) the split comes at 56 or fewer versus 57 or more.

So far, so simple, and good enough for anyone looking to wrap presents this Christmas. You might think we could just leave it there, but mathematicians never can. Extend to problem to four dimensions, and matters become predictably more complex. Now, you might be asking what a four-dimensional sphere looks like, and the truth is it’s impossible for the human mind to visualise. Mathematicians have no trouble with higher dimensions however - just add another number to your coordinate system. So, whilst we need two numbers to describe any point on a circle, and three numbers for a sphere, a group of four numbers will let us mathematically explore a so-called hypersphere.

How exactly do you go about wrapping a group of hyperspheres them? Well, for 50,000 or fewer you’re looking at a hyper-sausage, and for 100,000 you’re looking at something distinctly un-sausage-like - thought no-one knows exactly what. As for the specific trade off point, it isn’t as clear cut as with circles or spheres, but it definitely lies between 50,000 and 100,000 hyperspheres.

So what about the “sausage conjecture”? Unfortunately, it’s nothing to do with the trimmings at Christmas dinner, but rather states that for objects with five-dimensions or more, sausages are always best. This rather uninituive result, given the rules for two, three and four dimensions, was put forward in 1975 by Hungarian mathematician László Fejes Tóth.

Whilst it’s no Fermat’s Last Theorem or Riemann Hypothesis, some headway has been made with the sausage conjecture. In 1998 Ulrich Betke, Martin Henk and Jörg Wills proved that it was true for 42 or more dimensions, just leaving the cases 5 to 41. Perhaps you’d like to contemplate them as you wrap your Christmas presents!

I’m sorry to have two similar stories so close together, but when I saw that the Sun had published a formula for determining if your boobline is too low, I just had to say something.

Apparently, following the dress-popping antics of Britney Spears “scientists, undies experts and mathematicians have been trying to figure out where the decency perimeter lies.” I’ll quote the “result” in full.

The equation is O=NP(20C+B)/75.

To figure out the naughtiness rating (O), you times the number of nipples exposed, from zero to two or expressed as fractions of nipple shown (N) with the percentage of exposed frontal surface area (P).

The sum in brackets is 20 multiplied by the cup size (C), where A cup is one, B is two, C is three and D or above is five.

Add that figure to B, the bust measurement in inches. Then divide your answer by 75. Any score higher than 100 is counted as obscene.

Can anyone spot the immediate problem with the equation? It’s this: if N is zero, then O will be zero, because anything multiplied by zero is zero. In other words, if no nipples are shown then the “naughtiness rating” will always be zero! Hardly scandalising, I think you’ll agree.

What’s worse is the Sun actually demonstrate this in the article, with their example calculation for Britney:

Britney’s tight fitting Roberto Cavalli dress showed off around 70 per cent of her breasts, and experts at Wonderbra think she is a 32D. Without any nipple exposure, Britney’s formula works out as 0×70x(20×5+32)/75 = 123.2.

They’ve clearly multiplied by zero, and yet got a non-zero number! What’s worse, the sub-editor who wrote the headline has substituted the O in the equation for a 0, rendering it completely meaningless. It’s a shame actually, because for once everything in this formula is quantifiable in an non-subjective manner. Don’t get me wrong, it’s a load of rubbish (why multiply the cup size by 20? Why is a score of 100 obscene), but I have to give whoever came up with this formula some small amount of credit for dealing in actual measurements.

That’s the other problem actually - who did come up with this? The Sun quote William Hartson, “who holds an MA in Maths from Cambridge University”, and is also the author of “Drunken Goldfish and Other Irrelevant Scientific Research”. Ah, I thought to myself - another book to shill - but no, Drunken Goldfish was published in 1987! I think the Sun may have just gone to Mr Hartson for an “expert” quote. A listing on another book at Amazon indicates that he writes “surreal humour” for the Daily Express. Further on in the article, a spokesperson from Wonderbra is quoted. Maybe they came up with the formula? It’s possible, but I can’t find any information indicating this to be the case.

Really, I’m over-thinking this. The article is little more than an excuse to publish pictures of scantily clad women, under the pretence of evaluating them with the formula. Sex sells papers, as is well documented on Just A Theory with what I like to call the Scarlett Johansson School of Science Reporting. Still, as you should’ve realised by now, I can’t resist a “formula for” story. Thankfully however, my reasons are the exact opposite of the mass media!

Long term readers of Just A Theory may remember that one of the very first posts here was about a pet hate of mine: junk equations. Back then it was a formula for fame, but this time it’s the bane of students with essay deadlines ever: procrastination. Thankfully I handed in my essay yesterday, so I have some free time to rip in to this nonsense.

Professor Piers Steel has, according to the Telegraph spent “more than 10 years” studying why people procrastinate. Depending on who you ask, he’s either a psychologist or a business professor at the University of Calgary (the Telegraph say the former, the Daily Mail and the Times the latter).

On to the equation itself. It’s U = EV/ID, where U stands for “utlity”, or your desire to complete a given task. E is the expectation of succeeding in your task, whilst V is the value of completing it. I is the immediacy of the task, and finally D is your personal sensitivity to delay.

Well, that’s what the Telegraph says. The Daily Mail give a different formula: U = EVTC, where T is your tendency to delay work, and C the consequence of not completing it. By simple substitution, it must be that 1/ID = TC. Now, I can see an argument for saying that T has just been re-written as 1/D (in the same way that you can write 0.5 as 1/2), as they are both about delay, but how does the immediacy of the task (I) relate to the consequence of not completing it (C)? Already I’m starting to see the cracks in this equation…

For the definitive answer I went to Prof. Steel’s website, which provided me with the following:

Yet more variables! We’ve already met U, E, V and D, but now we have G (which seems to be standing in for the Greek letter Gamma which was actually used in the equation). Confusingly, G appears to be taking the place of D in the equation described by the Telegraph, whilst D here is now I. To avoid any further confusion, I will refer to Steel’s form of the equation, U = EV/GD from now on. To reiterate: E is expectancy of successful completion, V is the value of completion, G is the sensitivity to delay, and D is the immediacy of the task.

Besides changing variables like they were underpants, the problem with all of these formulas is that the values in them are completely unscientific and not at all measurable. Granted, your expectation of completing a task successfully could be expressed as a probability, for example, but such a measure is very subjective. What are the odds of getting an A for an essay? They simply can’t be calculated.

The other issue is the mathematical validity of the formula. If your sensitivity to delay is very low (and thus you have a small G), your utility value will be high - but surely it should be the other way around? If you don’t like to put things off, you’re less inclined to procrastinate! So maybe G should be measured from 1 to 10, with 1 being a high sensitivity and 10 being low. All this really illustrates is that it is very easy to come up with a formula for anything - as long as you fiddle the numbers to give that answer that you want!

Actually, it appears that this formula has more than one thing in common with the fame formula from my early post. Like that example, this equation is being used by its creator to publicise an upcoming book. Of course, all of the newspapers that have picked up this story are giving him a nice little bump of free advertising.

It shouldn’t need saying again, but I’m going to any way: these formula stories are a complete waste of time. They’re the absolute dregs of scientific journalism, and you shouldn’t pay any attention to them whatsoever. So, stop reading this and get back to work!

Football statistics. The lengths of roads in Britain. Fundamental physical constants. What do all these groups of numbers have in common? The answer may surprise you, but it is this: in all three data sets, numbers that begin with 1 are far more common than those whose first digits are 2, 3 and so on up to 9.

Now, you might expect all numbers to start with 1 to 9 equally. As there are nine numbers you would think that chances of any number beginning with 1 would be 1/9, or around 11%, but actually it is more like 30%! So what makes these groups so special?

It’s a bit of a trick question actually. It turns out that for many large sets of data, numbers that start with a 1 crop up around 30% of the time. Numbers that start with a 2 occur around 18% of the time, and the probability decreases with each successive number. It’s not just in the examples I listed above that this happens, but also stock exchange data, population figures, and many more. This seemingly strange phenomenon is all thanks to Benford’s law.

It was first observed by a man named Simon Newcomb in 1881. As both a mathematicians and an astronomer, he often used logarithms in his work. The logarithm of a number in a particular base is the power to which that base must be raised in order to produce that number. It’s easy to illustrate with an example: in base 10, the logarithm of 1000 is 3, because 10³ = 1000. We write that log₁₀(1000) = 3.

Logarithms can be used to make complex calculations much simpler, as long as you know how to convert back and forth to regular numbers. Nowadays we can let a computer do all the work, but back in Newcomb’s time people were forced to rely on weighty tombs of pre-calculated logarithms. One day, Newcomb realised that the pages of the book he was using became more worn the closer you were to the front. He came up with a formula that described the probabilities, as shown by this handy graph:

Benford's law in action: lower leading digits are far more common

Why then do we call this Benford’s law, if Newcomb came up with the formula? Well, Newcomb dismissed the idea an ideal curiosity. It was forgotten until 1938, when physicist Frank Benford noticed the exact same occurrence. He decided to investigate, and gathered masses of data to see if the rule was universal. Looking at sets similar to the ones described at the start, he found that it was really true: when it comes to large amounts of data, not all numbers are created equal.

Of course, you have to be careful when applying Benford’s law. A list of secondary school pupils and their ages will not follow the law, since all pupils must be aged 11-18 the probability of a leading 1 is 100%! On the other end of a scale, a collection of dice rolls will show that each number has a 1 in 6 chance of appearing; dies are truly random. Benford’s law will only apply in cases that fall somewhere between these two extremes, but thankfully this still includes a lot of data.

An interesting fact about Benford’s law is that it applies no matter the units of measurement used. You can measure your roads in miles or kilometres, and Benford’s law will still apply. This is known as scale invariance, and can actually be used to mathematically derive Benford’s law. You can work out which distributions of first digit probability stay the same when you switch from miles to kilometres, and it turns out there is only one: the formula that Newcomb came up with.

Benford’s law is a fascinating mathematical fact, but surprisingly it also has practical applications. Get this: it solves crime. No, really. If you’re a crooked accountant who likes to cook the books, Benford’s law will catch you out if you aren’t careful. If our dodgy dealer doesn’t know about Benford’s, they will probably pick numbers at random, or tend to stay in the “middle” (4, 5, 6, 7). Either approach will result in a data set that looks perfectly reasonable at a casual glance, but an analysis with Benford’s law will reveal that not enough numbers start with 1.

There you have it. Benford’s law: a mathematical oddity that just happens to have it’s uses. For fun, I thought I’d see how Just A Theory page views stack up against Benford’s law. Wordpress can track how many times each page has been visited, so I grabbed the first digit from each of these numbers. The results:

The Benford's law prediction (blue) closely matches the page views (red)

It’s not perfect, but Just A Theory doesn’t have that many pages (yet!). As with anything statistical, a larger data set will get you closer to a mathematically predicted distribution. Still, it’s pretty impressive to see Benford’s law in action. Maybe I’ll try again in a year’s time, when I have more data!

Cash for codebreakers

Bletchley Park, home to the Allied codebreakers of World War II, has secured a grant of £330,000 to restore the roof of the historic site. The Grade II-listed mansion is at risk due to previous neglect.

Codebreakers who were at Bletchley include Alan Turing, arguably the founder of computer science. The need to crack the German Enigma machine lead to great developments in cryptoanalysis and other sciences. It’s a fascinating place that I’d love to visit one day, so hopefully this new money will help preserve the site.

China plans their own Moon buggy

The Chinese media has reported the nation’s ambitions to put an unmanned buggy on the moon by 2012 as a step along the road to a full-on manned mission.

The news follows on from China’s previous space efforts at the end of September, in which they broadcast footage of a first space-walk back to those watching on Earth. It could also be seen as an answer to the American’s testing their latest moon buggy prototype.

China says that its lunar mission will include three steps of “orbiting, landing and returning”, but has not yet set any dates for manned moon mission yet.

Not lead into gold, but tequila into diamonds

Mexican scientists have discovered a way to turn tequila into diamonds. It turns out that the chemical makeup of the drink has a ratio of hydrogen, oxygen, and carbon atoms which places it within the “diamond growth region.”

The scientists turned to tequila not for its intoxicating quality, but because previous efforts to create diamonds from organic solutions such as acetone, ethanol, and methanol had proved unsuccessful. They then realised that their ideal compound of 40% ethanol and 60% water was remarkably close to tequila.

Luis Miguel Apátiga was one of the researches from the National Autonomous University of Mexico:

“To dissipate any doubts, one morning on the way to the lab I bought a pocket-size bottle of cheap white tequila and we did some tests,” Apátiga said. “We were in doubt over whether the great amount of chemicals present in tequila, other than water and ethanol, would contaminate or obstruct the process, it turned out to be not so. The results were amazing, same as with the ethanol and water compound, we obtained almost spherical shaped diamonds of nanometric size. There is no doubt; tequila has the exact proportion of carbon, hydrogen and oxygen atoms necessary to form diamonds.”

The diamonds were made by heating tequila to transform it into a gas, and then heating this gas further to break down the molecular structure. The result: solid diamond crystals, about 100-400 nanometres in size. They could be used to coat cutting tools, or as high-power semiconductors, radiation detectors and optical-electronic devices.

Well, not quite, but close. Notices of the American Mathematical Society have published details of computer programs that can provide rock-solid mathematical proofs.

This is extremely important, because in maths, proof is king. You could count prime numbers (which can only be divided by one or themselves) until the proverbial cows come home, and by the time you get to one squillion - not actually a real number, but let’s say it’s pretty big - you might be satisfied to say there were an infinite number of primes. Not so the mathematician, who will only be convinced by a logical proof.

The trouble is, even in the most basic proof you have to make some assumptions of previous results in the field. It doesn’t really matter because a sufficiently advanced reader will be able skip over these leaps of logic, but some theorems become some long and complex that even without dotting all the mathematical “i”s the proof can reach hundreds of pages long.

Checking such a proof would be incredibly arduous, but for the mathematician it must be done. This is where computers come in. A computer can develop a “formal proof”, in which every single statement is checked all the way back to first principles.

We’re not even talking 1 + 1 = 2 here. The Principia Mathematica, a seminal work on the foundations of mathematics published nearly a century ago, does not reach a proof of 1 + 1 = 2 until page 379. And mathematicians use pretty small fonts.

This demonstrates how ridiculous it would be to create such a formal proof by hand. It would be like providing a full dictionary definition of each word in this blog post along with the text - madness. Yet, for mathematicians to be truly, truly sure, a formal proof is the only way to go.

The computers aren’t quite there yet. Mathematicians still have to break the proofs down before they are fed into the program, so there we’re not quite at the level where your PC can leaf through the latest mathematical journals. It is possible, however, to let the computers “explore” the mathematics on their own - and perhaps even come up with some points the humans may have missed.

Ultimately, mathematicians would like to have formal proofs of at least the most important theorems. Thomas Hales, one of the authors writing in the Notices, says that such a collection of proofs would be akin to “the sequencing of the mathematical genome”. Impressive stuff.

Last night BBC 4 broadcast the first episode of a new four part series entitled The Story of Maths. It’s presented by Marcus du Sautoy, Oxford professor and pop-sci mathematician extraordinaire, who takes a look at the history of maths and why it is so important. This initial outing focuses on the three ancient civilizations who were the founders of maths: the Egyptians, the Babylonians, and the Greeks.

The Egyptians were practical problem solvers, and their need for bureaucracy and land management lead to the development of a counting system. Common problems, such as how to split nine loaves of bread between 10 people, were worked out in detail, but the Egyptians never realised the power of a generalised proof, forcing them instead to work out the same problem multiple times, but with different numbers. As he walks around a modern Egyptian market, and marvels at the Pyramids, du Sautoy demonstrates some of their ancient methods. (For those still wondering, each person receives one half, one third, and one fifteenth of a loaf.)

The Babylonians used maths to solve every day problems as well, but they also taught more generalised solutions in schools. Most of the mathematical records we have from those times are actually preserved clay tablets that record the workings of school children. They knew about quadratic equations like x² + 3x + 2 = 0, and du Sautoy blames the “recipes” used to solve such problems for poor maths teaching in modern classrooms.

Finally, we get to the Greeks, who in du Sautoy’s opinion are the true founders of maths - they were the inventors of proof, which opened up “a gulf between the other sciences” and are as true today as they were 2,000 years ago (a point he feels the need to make twice).

It’s a good primer to early maths, and I imagine it will be the most accessible programme of the series, since mathematics is a field that builds on its past and becomes increasingly complex. As one of the talking heads points out, Greek mathematics is still taught in schools today - because more modern concepts are completely inaccessible. Even at undergraduate level I spent most of my time learning about the 17th and 18th centuries; the 1970s were about the upper limit. This does make me wonder whether the series will remain engaging to the average viewer as it reaches more modern times.

I only have one criticism and it’s nothing to do with du Sautoy, who was excellent as always. It might be a small quibble, but the computer graphics used to illustrate his narrations were absolutely terrible. As du Sautoy was sent flying around on slices of Pyramid and hot air balloons, I found it increasingly difficult to concentrate on what he was saying, as all I could think about was how cheap and cheesy looking the animations were. Seriously, they would not have looked out of place a decade ago. It seems silly to knock the programme for this reason, but production values are an important part of getting your message across, and doing it badly just doesn’t help.

Next week, du Sautoy heads east. I expect we will be hearing about Chinese and Arabic mathematicians, along with algorithms and the number zero. It should be interesting, and I do recommend you watch this first episode, despite the dodgy CGI.

Playing a video game for 20 minutes a day can increase your mathematical potential, a study by Learning and Teaching Scotland has found. Apparently a daily dose of Brain Training on the Nintendo DS helped Scottish school children gain higher scores on their maths tests.

For the uninitiated, Brain Training is a fairly simple game that challenges players with short tests such as mental arithmetic or counting. The idea is to play the game daily with a view to improving your “Brain Age”, a fairly unscientific measure of how “young” your brain is. It’s pretty popular - even Nicole Kidman is at it - but can it really improve your thinking power?

To find out, over 600 pupils in 32 primary schools were given a maths test at the beginning of the study. For the next nine weeks, those in the control group received their normal teaching, whilst the other group were given 20 minutes of Brain Training at the start of each day. At the end of the study period the pupils were tested again, and the two scores compared. The control group showed some improvement, but those training their brains saw a further increase of 50%, from an average of 78 to 83 out of 100. They were also able to solve problems faster, dropping five minutes from an average 18.5 to 13.5 off their total test time.

Interestingly, children who were less competent at maths found the game more beneficial than their more able classmates, showing a larger increase in test scores overall. It could be that they find this non-traditional method of teaching more engaging than their standard lessons. The research also showed that both girls and boys benefited equally from using the game.

All positive results then, but will we be seeing Brain Training in classrooms an time soon? Unfortunately, I think the cost of equipment might prove to be prohibitive. The researchers who carried out the study make it clear they did not receive any financial aid from Nintendo, so presumably they forked out for the game and console themselves. Brain Training sells for around £15, whilst a Nintendo DS is close to £100. For a typical primary class of about 28 pupils, that works out at about £3200.

It makes me wonder if this would be the most cost-effective method of improving pupils mathematical ability, and perhaps more research is needed to find the teaching method with best “pound per percentage-point” ratio. Still, if you’d like to have fun and improve your mind at the same time, it could be that Brain Training is just the game for you. Personally, I think I’ll stick to Super Mario.

Something doesn’t sound quite right

The type of music you like could be linked to your personality, suggests a study carried out by Professor Adrian North of Heriot-Watt University. Apparently fans of country and western are “hardworking, outgoing” whilst indie lovers are “low self-esteem, creative, not hard working, not gentle”. Sounds like a bunch of nonsense to me - what if you like both country and indie? I haven’t been able to find a published paper on the research, which might validate it a little more, but I’m not holding my breath.

Because I say so

In the latest of a series on statistics in the media, Michael Blastland talks about the pitfalls of causation and correlation. Just because event A occurred before event B, it does not mean that A caused B - and yet so many stories in the media report just that. One you should always watch out for, so have a read.

Fruit for thought

Finally, some amazing photos of fruit taken using a scanning electron microscope. The colours may be false, but its all still very pretty.

You hopefully now understand the concepts that make up Euler’s equation, so let’s move on to how the equation arises. You may have already realised that for e^iπ + 1 = 0 to be true, it must also follow that e^iπ = -1, simply by rearranging the equation.

Think about that for a second. You’ve got a combination of three extraordinary numbers - e, i and π - numbers that seem to have no relation what so ever, and they combine to make -1. How can this be? Why don’t these numbers create a horrible decimal like 8.23487 or similar? Two of the numbers, e and π, cannot even be written down in full because they are infinity long, and yet shove i into the mix and we get the simple, beautiful result: -1.

It all stems from the exponential function, e^x. A function is basically a rule for turning one number into another. You take your independent variable, x, and plug it into your function to get the dependant variable. A simple function might be f, where f(x) = x². If you stick x = 2 into f, out pops f(2) = 2² = 4. The exponential function works the same way, just replace x with whatever number you are interested in. For Euler’s equation, x = iπ gives us the desired result.

Some functions can be expressed as lots of other functions - an infinite number of functions, in fact. This representation is often of a type known as a Taylor series, and for e^x it looks like this (I’ve borrowed a graphic from Wikipedia to make it clearer):

I’ve mentioned factorials like 2! and 3! before, but a quick reminder: n! just means “multiply together all the numbers from 1 to n. The ellipsis at the end of the equation means that the pattern goes on forever. “Well how does that help?” you ask. “Now I’ve got an infinite amount of things to deal with, and we don’t seem to be getting any closer to -1!” Fear not.

You see, e^x is actually hiding two other functions you may remember from school - the trigonometric duo, sinx and cosx. These two help you work out the length of a triangle’s sides from its angles, and they crop up everywhere in mathematics. When you stick x = iy (y is just another variable, like x) into our exponential function, it turns out that the Taylor series above becomes equal to two series added together - the series for cosy and isiny.

This means that e^iy = cosy + isiny. Like so many things in mathematics, this equation has an alternate, geometric representation - in this case, e^iy traces out a circle in the complex plane, a way of representing both real and imaginary numbers. Once more, Wikipedia has an excellent representation. They have used the Greek letter phi rather than the y I use here, but it means the same.

We’re nearly there now. We just need to set y = π, which represents a turn half way around the circle. This also happens to place us on the point where the circle intersects the real numbers at -1; in other words, e^iπ = -1. This is because cosπ = -1 and isinπ = 0. This what Euler realised, and although it is thought that he never actually wrote the expression e^iπ + 1 = 0, it follows directly from his result.

Euler’s equation summarises addition, multiplication, and exponentiation using five of the most important numbers in mathematics; e, i, π, 1, 0. It’s a truth universal in any language and to any culture. There is quite journey to get the equation, but one I believe is well worth travelling. I hope my explanation here leads you to agree.

In the very first post on Just a Theory I mentioned Euler’s equation, considered by many to be the most beautiful equation in the whole of mathematics. I decided to share with you everything I know about this wonderful equation, e^iπ + 1 = 0. Let’s start with some of the more unfamiliar elements.

π: You’ve probably met π, written Pi and pronounced “pie”, before and perhaps you remember that it is the number you get when dividing the circumference of a circle by its diameter. This neat little Wikipedia animation demonstrates the principle. Pi has an infinite number of digits because it is an irrational number; this means that it cannot be represented by a simple fraction such as 1/2 or 365/789. I’m hopeless at remembering digits of Pi and normally stick to 3.14.

i: What is the square root of 4? In other words, what number must you multiply by itself to get 4? The answer is of course 2. What about the square root of -4? It can’t be -2, because -2 is also the square root of 4 - since multiplying two negative numbers results in a positive number. The answer is 2i, because i is defined as the square root of -1. i is an imaginary number - but that doesn’t mean it’s just made up. For example, many problem in electrical engineering can only be expressed using i.

e: Another irrational number like π, e is the base of the natural logarithm. What that means isn’t really important in this context - just think of it as another important mathematical constant like π. The numerical value of e is approximately 2.718.

Exponentiation: You should be familiar with the concept of raising one number to the power of another, such as 2³ = 2 x 2 x 2 = 8. This is exponentiation. What does the strange beast e^iπ represent then? If you are trying to multiply e by itself iπ times you might be left scratching your head, but all will become clear as Euler’s equation is explained.

What with it being a bank holiday, I’ll give you a break and leave it there for today. Hopefully you’ve understood these building blocks, and your ready to tackle the full equation tomorrow. If not, leave a message in the comments and let me know if you need further clarification.

If I remember anything from my days of learning foreign languages, it’s how to count. Not very impressive I grant you, but I can still knock out an “un, duex, trois” or an “ein, zwei, drei” when required. Counting is such a basic and universal skill that it is hard to imagine life without it, but certain aboriginal communities do not have words or gestures to represent numbers. A study by University College London and the University of Melbourne of children from two such communities has found the lack of words is not a hindrance to counting.

The study looked at children aged four to seven from two aboriginal groups, one speaking a langage called Warlpiri whilst the other used Anindilyakwa. Both have words for one, two, few and many, and Anindilyakwa uses numbers up to 20 in rituals but children are not taught these. As a control group the team also worked with an English-speaking indigenous community.

Professor Brian Butterworth of the UCL Institute of Cognitive Neuroscience was lead author of the study, and details the difficulty in designing questions that the children could answer:

“In our tasks we couldn’t, for example, ask questions such as “How many?” or “Do these two sets have the same number of objects?” We therefore had to develop special tasks. For example, children were asked to put out counters that matched the number of sounds made by banging two sticks together. Thus, the children had to mentally link numerosities in two different modalities, sounds and actions, which meant they could not rely on visual or auditory patterns alone. They had to use an abstract representation of, for example, the fiveness of the bangs and the fiveness of the counters. We found that Warlpiri and Anindilyakwa children performed as well as or better than the English-speaking children on a range of tasks, and on numerosities up to nine, even though they lacked number words.

It appears being able to count is an innate skill. This could explain why children with dyscalculia, a form of dyslexia relating to mathematics, find arithmetic so difficult to learn. Even with our counting system of “one, two, three” to aid them, a lack of this innate skill causes sufferers to struggle. Professor Butterworth is conducting another study in order to find the differences in brains of people with the disorder.

Andrew Hodges’ inspiration for the title One to Nine was Sudoku, the immensely popular number puzzle. Hodges comments that newspapers insist the puzzles require no mathematical knowledge, in order to not scare away an often maths-phobic British public - indeed, Sudoku does not even require numbers, since substituting nine letters or symbols into a puzzle would leave the logic required to solve it unchanged.

Hodges describes logic as one of the most fascinating elements of ‘adult mathematics’, wholly different to the ‘school maths’ that newspapers try to distance themselves from. The book aims to provide an insight into this for those who may have been turned off the subject at school.

Unsurprisingly, the book is split into nine chapters, One through Nine. Each begins with a characterisation of the number; seven ‘needs sifting and sorting out’, whereas three ‘doesn’t just talk’, but ‘thinks big’. The chapter titles are a bit of a gimmick at times. Six is the first perfect number, so-called because 6 = 1 + 2 + 3 = 1 x 2 x 3, and this leads to a discussion of factorials. Six is 3! (pronounced ‘three factorial’) because 3! = 1 x 2 x 3., and the factorial of a number n is simply the product of all numbers from 1 to n. The chapter continues with probabilities, the Enigma machine, and Euler’s equation – all very interesting topics with links to factorials, but do they really relate particularly to six, more so than any other number?

Gimmicks aside, One to Nine is a whistle-stop tour of pop-sci mathematics, with sections ranging from black holes to game theory to musical harmony. Each topic is well described and often accompanied by many useful diagrams, although some appear to have been lifted straight from a .jpg file, complete with ugly compression artefacts – a bit more care could have been take in order to provide high quality images.

Numerous equations may discourage the casual reader, but they are always accompanied by a thorough explanation in the text. Stephen Hawking was told when writing A Brief History of Time that ‘each equation in the book would halve the sales’; I hope this is not the case else I will have already lost 75% of my readership! For those who really will not abide equations, relax – they can for the most part be skipped.

Sprinkled throughout the text are problems rated on a Sudoku-like scale, from GENTLE to DEADLY. I found these to be a welcome addition, but normally skipped over any that I was unable to solve in a minute or two, so as not to slow down the pace of the book. Placing these at the end of each chapter would have made me more inclined to give them a go.

Helpfully, all of the solutions are provided on the website for the book, along with further notes and comments. Unfortunately the book does not feature a bibliography or recommended reading list, so if you do become engrossed in a particular topic you will have to hunt out more information by yourself, but the website does go some way to assisting with this.

If you would like to learn how mathematics is used in a variety of scientific fields and are not too afraid of a few equations, One to Nine is a good place to start. In fact, Hodges’ appropriation of Sudoku is quite apt. If you enjoy the use of logic in a Sudoku puzzle, but have dreaded school memories of multiplication tables, perhaps One to Nine can show you the world through the filter of ‘adult mathematics’.

One of my science communication pet hates is stories about scientists discovering “the formula for x”, where x is a successful sitcom, why people vote, or happiness, just to name a few.

The latest culprit is PR agent Mark Borkowski who claims to have found a “scientific formula” for fame. The formula itself is given as follows:

F(T) = B+P(1/10T+1/2T2)

where:

F is the level of fame;

T is time, measured in three-monthly intervals. So T=1 is after three months, T=2 is after six months, etc. Fame is at its peak when T=0. (Putting T=0 into the equation gives an infinite fame peak, not mathematically accurate, perhaps, but the concept of the level of fame being off the radar is apposite.);

B is a base level of fame that we identified and quantified by analysing the average level of fame in the year before peak. For George Clooney, B would be a large number, but for a fabulous nobody, like a new Big Brother contestant, B is zero;

P is the increment of fame above the base level, that establishes the individual firmly at the front of public consciousness.

Not that it really matters, but this is terribly unclear. A more correct way to write it would be F(T) = B+(1/(10T)+1/(2T^2))*P, eliminating any ambiguity as to what each symbol means, but as with all of these stories scientific accuracy is not high on the agenda. Borkowski has made the same two mistakes that always crop up in these formulas - unmeasurable variables and confirmation bias.

The unmeasurable variables in this case are F, B, and P. T is time, where the units of T are periods of 3 months - not exactly orthodox, but still completely measurable. F, B and P however are measures of fame, for which I know of no scientific units. Perhaps fame is measured in the units of star power - solar luminosity.

Yes, I’m being facetious, but it is an important point. One of the greatest tools available to a scientist are the standard units of measurement known as SI units. I’ll talk about them in more depth another day, but they include metres, kilograms, and seconds - quantities we are all familiar with. This common set of units allow scientists to communicate their findings in a meaningful way, and the results of not confirming your units can be disastrous, as NASA discovered when they mixed up feet and metres, causing an unmanned spaceship to crash.

The other problem, confirmation bias, is an interesting one. It basically amounts to “people believe what they want to believe”, and it’s definitely in action here. Borkowski wanted to match Andy Warhol’s 15 minutes of fame with his own 15 months of fame:

I started to wonder if Andy Warhol - an artist by calling but a master of the stunt and the soundbite - was right; does everyone get 15 minutes of fame? It occurred to me that it should be possible to look at fame statistically, to analyse the evidence we have all witnessed in the media, to see if fame’s decline can be quantified. The answer, I discovered, is that it can be, and that Warhol was partially right - but the first spike of fame will last 15 months, not 15 minutes.

In looking at fame “statistically”, it turns out that 15 months is exactly right! Well done, Borkowski.

This formula fits the data remarkably well, giving a precise numerical value to the 15-month theory: if I put in T=5 (corresponding to 15 months after the peak), it gives F=B+P(1/50+1/50), which works out at F=B+.04P. In other words, up to 96% of the fame-boost achieved at the peak of public attention has been frittered away, and the client or product is almost back to base level.

Of course, if you put in T = 6 (i.e. 18 months) you get F = B + 0.03P (rounding off the decimal point). Three months later, it appears our Big Brother contestant hasn’t really got much less famous than they were after 15 months. What about after two years, when T = 8? In that case, F = B + 0.02P - fame doesn’t really appear to be dropping off very quickly, does it? The claim that ‘the study showed pretty conclusively that any specific boost to fame is sustained for approximately 15 months…’ isn’t remotely conclusive - in fact, I’ve just show that you can reach an entirely different conclusion by choosing different values of T.

The reason I hate these formula stories with a passion is that they damage the public perception of science. The ideas they offer are meaningless, suggesting that all scientist do is sit around dunking biscuits in a quest for perfection. That story is nearly a decade old, so the junk equation is clearly not a new concept, and I don’t think we will be rid of it any time soon. So, the next time you see an article proclaiming that science has once more advanced, and we now know how to calculate the cuteness of puppies or the magic of rainbows, please do the only sensible thing - ignore it.