100 Essential Things You Didn't Know You Didn't Know Page 10

  This checking system will catch a lot of simple typing and reading errors. It detects all single-digit errors and most adjacent swops (although 90 becoming 09 would be missed).

  Another check-digit bar code that we are always encountering (and totally ignoring, unless we are a supermarket check-out assistant) is the Universal Product Code, or UPC, which was first used on grocery products in 1973 but has since spread to labelling most of the items in our shops. It is a 12-digit number that is represented by a stack of lines, which a laser scanner can read easily. The UPC has four parts: below the bars, two separate strings of 5 digits are set between two single digits. For example, on the digital camera box on my desk at the moment this looks like

  0 74101 40140 0

  The first digit identifies the sort of product being labelled. The digits 0, 1, 6, 7, 9 are used for all sorts of products; digit 2 is reserved for things like cheese, fruit and vegetables that are sold by weight; digit 3 is for drugs and heath products; digit 4 is for items that are to be reduced in price or linked to store loyalty cards, and digit 5 is for items linked to any ‘money-off ’ coupons or similar offers. The next block of five digits identifies the manufacturer – in my case Fuji – and the next five are used by the manufacturer to identify the product by its size, colour and features other than price. The last digit – 0 here – is the check digit. Sometimes it isn’t printed but is just represented in the bars for the code reader so it can accept or reject the UPC. The UPC is generated by adding the digits in the odd-numbered positions (0+4+0+4+1+0 = 9), multiplying by three (3 × 9 = 27), and then adding the digits in the even-numbered positions to the result (27+7+1+1+0+4+0 = 40 = 4 × 10), and checking that it is divisible by 10 – which it clearly is.

  That just leaves the bars. The overall space inside the outermost single digits (our two zeros) is split into seven regions, which are filled with a thickness of black ink, which depends on the digit being represented, with light bands and dark bands alternating. On each end of any UPC there are two parallel ‘guard bars’ of equal thickness, which define the thickness and separation scale that is used by the lines and spaces in between. There is a similar set of four central bars, two of which sometimes drop below the others, which separate the manufacturer’s ID from the product information and carry no other information. The actual positions and thicknesses of the bars form a binary code of 0s and 1s. An odd number of binary digits is used to encode the manufacturer’s details, while an even number is used for the product information. This prevents confusion between the two and enables a scanning device to read these numbers from right to left or left to right and always know which block it is looking at. And you thought life was simple.

  43

  I’ve Got a Terrible Memory for Names

  The ‘t’ is silent, as in Harlow.

  Margot Asquith, on her name being mispronounced by Jean Harlow

  If you have ever had to make a note of someone’s name over the telephone, then you will know that it is a tricky business being sure of the spelling. Usually you respond to uncertainty by asking them to spell out their name. I recall how my doctoral research supervisor, Dennis Sciama, whose unusual surname was pronounced ‘Sharma’, could spend a significant amount of his working day spelling out his name to phone callers who did not know him.

  There are occasions when oral and written messages can’t be repeated or have been wrongly written, and you want to minimise the possibility of missing out on a person’s real identity when you look them up in your files. The oldest scheme in operation to try to ameliorate this problem is called the Soundex phonetic system, and dates from about 1918, when it was invented by two Americans, Robert Russell and Margaret Odell, although it has gone through various small modifications since then. It was originally designed to help with the integrity of census data that was gathered orally, and was then used by airlines, the police and ticket booking systems.

  The idea was to encode names so that simple spelling variants, like Smith and Smyth, or Ericson and Erickson, that sounded the same were coded as the same, so that, if you entered one of the group, then the other variants would appear as well, thus ensuring that you were not missing one in the filing system. Anyone searching for relatives or ancestors, especially immigrants with foreign names that might have been slightly modified, would find this encoding useful. It will automatically seek out many of the close variants that you would otherwise have to search out one by one and will also find variants you hadn’t even thought of. Here is how it works for names.

  Keep the first letter of the name, whatever it is.

  Elsewhere, delete all occurrences of the following letters: a, e, i, o, u, h, y, w

  Assign numbers to the other letters that remain so that:

  b, f, p, v all become 1

  c, g, j, k, q, s, x, z all become 2

  d and t both become 3 and l becomes 4

  m and n become 5 and r becomes 6

  If two or more letters with the same number were next to one another in the original full name keep only the first of them.

  Finally, record only the first four characters of what you have left. If you have fewer than four then just add zeros on the end to make the string up to a length of four.

  My name is John, and the process will change it to Jn (after steps 1 and 2), then J5 (after step 3), and the final record is J500. If your name is Jon you will get the same result. Smith and Smyth both become S530. Ericson, Erickson, Eriksen and Erikson all give the same record of E6225.

  44

  Calculus Makes You Live Longer

  As a math teacher, I understand how important it is for students to see that mathematics can connect with life. Mortuary science gives me a novel and unique way to do that. After all, what could be more universal in life than death? Once my students learn about rates of decay and embalming theory, they seem eager to return to the study of calculus with a renewed rigor.

  Professor Sweeney Todman, Mathemortician

  The difference between an amateur and a professional is that as an amateur one is at liberty to study only those things one likes, but as a professional one must also study what one doesn’t like. Consequently, there are parts of a mathematical education that will seem laborious to a student, just as all those hours of winter running in the cold and rain will be unattractive, but essential, to the aspiring Olympic athlete. If students asked why they needed to learn some of the more intricate and unexciting parts of calculus, I used to tell them this story, one that the Russian physicist George Gamow tells us in his quirky autobiography, My World Line. It is about the remarkable experience of one of Gamow’s friends, a young physicist from Vladivostok called Igor Tamm, who went on to share the Nobel prize for physics in 1958 for his part in discovering and understanding what is now known as the ‘Cerenkov Effect’.

  In the Russian revolutionary period, Tamm was a young professor teaching physics at the University of Odessa in the Ukraine. Food was in short supply in the city, and so he made a trip to a nearby village, which was under the apparent control of the communists, in an attempt to trade some silver spoons for something more edible, like chickens. Suddenly, the village was captured by an anti-communist bandit leader and his militia, armed with rifles and explosives. The bandits were suspicious of Tamm, who was dressed in city clothes, and took him to their leader, who demanded to know who he was and what he did. Tamm tried to explain that he was merely a university professor looking for food.

  ‘What kind of professor?’ the bandit leader asked.

  ‘I teach mathematics,’ Tamm replied.

  ‘Mathematics?’ said the bandit. ‘All right! Then give me an estimate of the error one makes by cutting off Maclaurin’s series at the nth term.11 Do this and you will go free. Fail, and you will be shot!’

  Tamm was not a little astonished. At gunpoint, somewhat nervously, he managed to work out the answer to the problem – a tricky piece of mathematics that students are taught in their first course of calculus in a university degree course
of mathematics. He showed it to the bandit leader, who perused it and declared ‘Correct! Go home!’

  Tamm never discovered who that strange bandit leader was. He probably ended up in charge of university quality assurance somewhere.

  45

  Getting in a Flap

  In ancient days two aviators procured to themselves wings. Daedalus flew safely through the middle air and was duly honoured on his landing. Icarus soared upwards to the sun till the wax melted which bound his wings and his flight ended in fiasco. The classical authorities tell us, of course, that he was only ‘doing a stunt’; but I prefer to think of him as the man who brought to light a serious constructional defect in the flying-machines of his day.

  Arthur S. Eddington

  Lots of things get around by flapping about: birds and butterflies with their wings, whales and sharks with their tails, fish with their fins. In all these situations there are three important factors at work that determine the ease and efficiency of movement. First, there is size – larger creatures are stronger and can have larger wings and fins, which act on larger volumes of air or water. Next, there is speed – the speed at which they can fly or swim tells us how rapidly they are engaging with the medium in which they are moving and the drag force that it exerts to slow them down. Third, there is the rate at which they can flap their wings or fins. Is there a common factor that would allow us to consider all the different movements of birds and fish at one swoop?

  As you have probably guessed, there is such a factor. When scientists or mathematicians are faced with a diversity of examples of a phenomenon, like flight or swimming, that differ in the detail but retain a basic similarity, they often try to classify the different examples by evaluating a quantity that is a pure number. By this I mean that it doesn’t have any units, in the way that a mass or a speed (length per unit time) does. This ensures that it stays the same if the units used to measure the quantities are changed. So, whereas the numerical value of a distance travelled will change from 10,000 to 6¼ if you switch units from metres to miles, the ratio of two distances – like the distance travelled divided by the length of your stride – will not change if you measure the distance and your stride length in the same units, because it is just the number of strides that you need to take to cover the distance.

  In our case there is one way to combine the three critical factors – the flapping rate per unit of time, f, the size of the flapping strokes, L, and the speed of travel, V – so as to get a quantity that is a pure number.fn1 This combination is just fL/V and it is called the ‘Strouhal number’, after Vincenc Strouhal (1850–1922), a Czech physicist from Charles University in Prague.

  In 2003 Graham Taylor, Robert Nudds and Adrian Thomas, at Oxford University, showed that if we evaluate the value of this Strouhal number St = fL/V for many varieties of swimming and flying animals at their cruising speeds (rather than in brief bursts when pursuing prey or escaping from attack), then they fall into a fairly narrow range of values that could be said to characterise the results of the very different evolutionary histories that led to these animals. They considered a very large number of different animals, but let’s just pick on a few different ones to see something of this unity in the face of superficial diversity.

  For a flying bird, f will be the frequency of wing flaps per second, L will be the overall span of the two flapping wings, and V will be the forward flying speed. A typical kestrel has an f of about 5.6 flaps per second, a flap extent of about 0.34 metres and a forward speed of about 8 metres per second, giving St(kestrel) = (5.6 × 0.34)/8 = 0.24. A common bat has V = 6 metres per second, a wingspan of 0.26 metres, and a flapping rate of 8 times per second, so it has a Strouhal number of St(bat) = (8 × 0.26)/6 = 0.35. Doing the same calculation for forty-two different birds, bats and flying insects always gave a value of St in the range 0.2–0.4. They found just the same for the marine species they studied as well. More extensive studies were then carried out by Jim Rohr and Frank Fish (!) at San Diego and West Chester, Pennsylvania, to investigate this quantity for fish, sharks, dolphins and whales. Most (44 per cent) were found to lie in the range 0.23 to 0.28, but the overall range spanned 0.2 to 0.4, just like the range of values for the flying animals.

  You can try this on humans too. A good male club standard swimmer will swim 100 metres in 60 seconds, so V = 5/3 metres per second, and uses about 54 complete stroke cycles of each arm (so the stroke frequency is 0.9 per second), with an arm-reach in the water of about 0.7 metres. This give St(human swimmer) = (0.9 × 2/3)/5/3 = 0.36, which places us rather closer to the birds and the fishes than we might have suspected. However, arguably the world’s most impressive swimmer has been the Australian long-distance star Shelley Taylor-Smith, who has won the world marathon swimming championships seven times. She completed a 70 km swim in open sea water inside 20 hours with an average stroke rate of 88 strokes per minute. With an effective stroke reach of 1 metre that gives her the remarkable Strouhal number of 1.5, way up there with the mermaids.

  fn1 The unit of frequency f is 1/time, of size L is length, and of speed V is length/time so the combination fL/V has no units: it is a pure dimensionless number.

  46

  Your Number’s Up

  Enter your postcode to view practical joke shops near you.

  Practical Jokeshop UK

  Life seems to be defined by numbers at every turn. We need to remember PIN numbers, account numbers, pass codes and countless reference numbers for every institution and government department under the Sun, and for several that never see it. Sometimes, I wonder whether we are going to run out of numbers. One of the most familiar numbers that labels us geographically (approximately) is the postcode. Mine is CB3 9LN and, together with my house number, it is sufficient to get all my mail delivered accurately, although we persist with adding road names and towns as back-up information or perhaps because it just sounds more human. My postcode follows an almost universal pattern in the United Kingdom: using four letters and two numbers. The positions of the letters and numbers don’t really matter, although in practice they do because the letters also designate regional sorting and distribution centres (CB is Cambridge). But let’s not worry about this detail – the postal service certainly wouldn’t if it found itself running out of 6-symbol postcodes – and simply ask how many different postcodes of this form could there be? You have 26 choices from A to Z for each of the four slots for letters and 10 choices from 0 to 9 for each of the numerals. If these are each chosen independently then the total number of different postcodes following the current pattern is equal to 26×26×10×10×26×26, which equals 45,697,600 or nearly 46 million. Currently, the number of households in the United Kingdom is estimated to be about 26,222,000, or just over 26 million, and is projected to increase to about 28.5 million by the year 2020. So, even our relatively short postcodes have more than enough capacity to deal with the number of households and give each of them a unique identifier if required.

  If we want to label individuals uniquely, then the postcode formula is not good enough. In 2006 the population of the United Kingdom was estimated at 60,587,000, about 60.5 million, and far greater than the number of postcodes. The closest thing that we have to an identity number is our National Insurance number, which is used by several agencies to identify us – the possibility of all these agencies coordinating their data by using this number is the step that alarms many civil liberty groups the most. A National Insurance number has a pattern of the form NA 123456 Z, which contains six numerals and 3 letters. As before, we can easily work out how many different National Insurance numbers this recipe permits. It’s

  26 × 26 × 10 × 10 × 10 × 10 × 10 × 10 × 26

  This is a big number – 17,576,000,000 – seventeen billion, five hundred and seventy-six million, and vastly bigger than the population of the United Kingdom (and even than its projected value of 75 million by 2050). In fact, the population of the whole world is currently only about 6.65 billion and projected to reach 9 billion by the ye
ar 2050. So there are plenty of numbers – and letters – to go round.

  47

  Double Your Money

  The value of your investments can go down as well as up.

  UK consumer financial advice

  Recently you will have discovered that the value of your investments can plummet as well as go down. So, suppose you want to play safe and place cash in a straightforward savings account with a fixed, or slowly, changing rate of interest. How long will it take to double your money? Although nothing in this world is so certain as death and taxes (and the version of the latter that goes with the former), let’s forget about them both and work out a handy rule of thumb for the time needed to double your money.

  Start out by putting an amount A in a savings account with an annual fractional rate of interest r (so 5% interest corresponds to r = 0.05), then it will have grown to A × (1+r) after one year, to A × (1+r)2 after two years, to A × (1+r)3 after three years and so on. After, say, n years your savings will have become an amount equal to A × (1+r)n. This will be equal to twice your original investment, that is 2A, when (1+r)n = 2. If we take natural logarithms of this formula, and note that ln(2) = 0.69 approximately, and ln(1+r) is approximately equal to r when r is much less than 1 (which it always is – typically r is about 0.05 to 0.06 at present in the UK), then the number of years needed for your investment to double is given by the simple formula n = 0.69/r. Let’s round 0.69 off to 0.7 and think of r as R per cent, so R = 100r, then we have the handy rule that12

  n = 70/R

  This shows, for example, that when the rate R is 7% we need about ten years to double our money, but if interest rates fall to 3.5% we will need twenty.