100 Essential Things You Didn't Know You Didn't Know (14 page)

Garfield had originally intended to become a mathematics teacher after he graduated from Williams College in 1856. He taught classics for a while and made an unsuccessful attempt to
become
a school headmaster, but his patriotism and strong views led him to run for public office, and he was elected to the Ohio State Senate just three years later and qualified as a barrister in 1860. He left the Senate and joined the army in 1861 and rose swiftly up the ranks to become a major-general, before moving sideways into the House of Representatives two years later. There he remained for 17 years until he became the Republican Presidential candidate in 1880, winning a narrow victory in the popular vote over the Democratic party candidate, Winfield Hancock, on a ticket that promised to improve education for all Americans. Garfield is still the only person ever to be elected President directly from the House of Representatives.

Garfield’s most interesting contribution was nothing to do with politics at all. In 1876, while serving in the House, he liked to join with fellow members of Congress to talk about subjects of a mind-broadening sort. Garfield presented for his colleagues’ amusement a new proof of Pythagoras’s theorem for right-angled triangles. Later, he published it in the
New England Journal of Education
, where he remarks that ‘we think it something on which the members of both houses can unite without distinction of party’.

Mathematicians had been teaching this theorem to students for more than 2,000 years, and they usually stayed close to the proof given by Euclid in his famous
Elements
, which he wrote in Alexandria in about 300
BC
. This proof was by no means the first. Both the Babylonians and the Chinese had good proofs, and the ancient Egyptians were well acquainted with the theorem and were able to use it in their adventurous building projects. Of all the proofs of Pythagoras’s theorem that have been found over the centuries, Garfield’s is one of the simplest and the easiest to understand.

Take a right-angled triangle and label its three sides a, b and c. Make a copy of it and lay the two copies down so they form a V-shaped crevice with a flat base. Now join up the two highest points of the two triangles to make a lop-sided rectangle – a shape that we call a trapezium (or, in America, a trapezoid). It is a four-sided figure, which has two of its sides parallel, shown above.

Garfield’s figure consists of three triangles – the two copies of the one we started with and the third one that was created when we drew the line between their two highest points. He now simply asks us to work out the area of the trapezium in two ways. First, as a whole it is the height, a+b, times the average width ½ (a+b); so the area of the trapezium is ½ (a+b)
²
. To convince yourself of this you could change the shape of the trapezium, turning it into a rectangle by making the two widths equal. The new width would be ½(a+b).

Now, work out the area another way. It is just the sum of the areas of the three right-angled triangles that make it up. The area of a right-angled triangle is just one half of the area of the square you would get by joining two copies of it together along the diagonal, and so it is one-half of its base length times its height. The total area of the three triangles is therefore ½ba + ½c
²
+ ½ba = ba + ½c
²
, as shown.

Since these two calculations of the total area must give the same total, we have:

½(a+b)
²
= ba + ½c
²
.

That is,

½(a
²
+ b
²
+2ab) = ba + ½ c
²

And so, multiplying by two:

a
²
+ b
²
= c
²

just as Pythagoras said it should.

All prospective candidates in the American elections should be asked to give this proof during the televised Presidential debates.

42

Secret Codes in Your Pocket

No one can buy or sell who does not have the mark.

Book of Revelation

Codes mean spies secret formulae and countries at war – right? Wrong: codes are all around us, on credit cards, cheques, bank notes and even on the cover of this book. Sometimes codes play their traditional role of encrypting messages so that snoopers cannot easily read them or preventing third parties from raiding your online bank account, but there are other uses as well. Databases need to be kept free of innocent corruption as well as malicious invasion. If someone types your credit card number into their machine but gets one digit wrong (typically swopping adjacent digits, like 43 for 34, or getting pairs of the same digit wrong, so 899 becomes 889) then someone else could end up being charged for your purchase. Enter a tax identification number, an airline ticket code or a passport number incorrectly and the error could spread through part of the electronic world, creating mounting confusion.

The commercial world has tried to counter this problem by creating means by which these important numbers can be self-checking to tell their computers whether or not the number being entered qualifies as a valid air ticket or bank note serial number. There is a variety of similar schemes in operation to check the validity of credit card numbers. Most companies use a system
introduced
by IBM for credit-card numbers of 12 or 16 digits. The process is a bit laborious to carry out by hand, but can be checked in a split second by a machine, which will reject the card number input if the digits fail the test because of error or a naively faked card.

Take an imaginary
Visa
card number

4000 1234 5678 9314

First, we take every other digit in the number, going from left to right, starting with the first (i.e the odd-numbered positions), and double it to give the numbers 8, 0, 2, 6, 10, 14, 18, 2. Where there are double digits in the resulting numbers (like 10, 14, 18) we add the two digits together to get (1, 5, 9), which has the same effect as subtracting 9. The list of doubled numbers now reads 8, 0, 2, 6, 1, 5, 9, 2. Now we add them together and then add all the numbers in between (i.e. the even-numbered positions) that we missed out the first time (0, 0, 2, 4, 6, 8, 3, 4). This gives the sum, in order, as

8+0+0+0+2+2+6+4+1+6+5+8+9+3+2+4 = 60

In order for the card number to be valid, this number must be exactly divisible by 10, which in this case it is. Whereas, if the card number was 4000 1234 5678 9010, then the same calculation would have generated the number 53 (because the card number only differs in the last and third-last digits) and this is not exactly divisible by 10. This same procedure works to verify most credit cards.

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.

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’.