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Euclid’s Method of proving Unique Prime Factorisatioon

December 1, 2013

It is often said that Euclid (who devoted Books VII – XI of his Elements to Number Theory) recognized the importance of Unique Factorization into Primes and established it by a theorem (Proposition 14 of Book IX). This is not quite correct. Modern authors usually present UPF in the following way

THEOREM Any positive integer N can be written as a product of primes in one and only one way barring changes in order. i.e.  N = pa qb rc…..

        But what Euclid establishes by Book IX Proposition 14 — Heath, whose translation I use throughout calls ‘theorems’ ‘propositions’ — is
“If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it.”

      Now, from this one can, with the help of one or two other theorems, deduce Unique Prime Factorization (UPF), but Euclid does not actually do so. For one thing, Euclid would need to show that every (natural) number can be presented as a product of primes if Proposition 14 is to have a universal application. He goes some way to doing this in Propositions 31 and 32 of Book VII : Any composite number is measured by some prime number” and “Any number either is prime or is measured by some prime number”. But, for some reason, we lack the clinching Proposition, that all numbers can be written as a product of primes and that there is only one way of doing this barring changes in order.

Euclid’s presentation of Number Theory is so idiosyncratic, not to say perverse, that many readers, flipping through the Elements,  do not even realize that he ever dealt with numbers at all. This is because Euclid insists on presenting (whole) numbers as line segments
A ______________       B __________  and not, as one would expect, as collections of discrete elements, e.g. by such sequences as ● ● ● ● ● ● ● ●  or □ □ □ □  It is true that, by presenting numbers as lines Euclid gains generality : we can see in the above that A > B but we are not limited to specific magnitudes. Also, Euclid did not have the … facility which we have.

However, I doubt if this was the real reason. By Euclid’s time geometry had almost entirely ousted arithmetic as the dominant branch of mathematics much in the way that algebra subsequently ousted geometry. Pride of place in the Elements is given to the theory of proportion developed by Eudoxus. In the books devoted to Number Theory Euclid only deals with whole numbers (presented as line segments) and ratios between whole numbers which mimic ratios between sides of triangles and other figures. He does not mention ‘fractions’ as such though Greek housewives and practical people must have been well acquainted with them. Why is this? Partly no doubt because of the influence of Plato who, though not himself a mathematician, was well versed in the higher mathematics of his time and remains one of the most important theorists in the history of mathematics. Plato’s view that the ‘truths of mathematics’ are in some sense independent of human experience, while nonetheless underlying it, is the view held by  many pure mathematicians today. Plato considered mere calculation with numbers to be a lowly activity, the affair of craftsmen and merchants, while geometry was a discipline that ennobled the practitioner by fixing his eye on the eternal. This explains the radical ‘geometrization’ of number that we find in Euclid.

In his Books on Number Theory, it would seem that Euclid was building on a much older arithmetic tradition which not only presented numbers as discrete entities but actually used objects such as pebbles or shells in calculations and formed them into shapes — which is why we still speak of ‘triangular numbers’, ‘square numbers’ and so forth (Note 1). The material of Book VII, the basic Book dealing with Number Theory, looks as if it goes back a very long way indeed and this  is at once an advantage and a drawback.

It is an advantage because Euclid kicks off with an eminently practical procedure (rather than an abstract theorem in our sense), the so-called Euclidian Algorithm, and makes it the foundation of the entire edifice. Most of Euclid’s proofs are by contradiction and thus ‘non-constructive’  but the Euclidian Algorithm not only demonstrates that a ‘least common measure’ of two or more numbers always exists, but actually shows you how to obtain it. Remarkably, the Euclidian Algorithm works perfectly well in any base, or indeed without any base at all — and this suggests that it is a very ancient procedure. It was quite possibly  discovered before written numbers even existed : in effect, it shows you how to group or bag up two different collections of similarly sized  objects without anything being left over, using the largest  possible bag size. Proposition 1 is a special case of this : when the largest bag size possible turns out to be the unit. Such an outcome  situation must have seemed extraordinary to the people who first discovered it, and indeed mankind has ever since been fascinated by ‘prime numbers’ — they were originally called ‘line numbers’ because they could only be laid out in a line or column, never as a rectangle.

However, probably because they are based on an ancient source, Euclid’s presentation in the Books devoted to Number Theory is not  so impeccably logical as in the other Books. Euclid does not introduce any new Axioms in Book VII, the first of the four books dealing with Number Theory, though he does give twenty-two Definitions. He presumably  assumed that the general Axioms, given in Book I, sufficed. In fact, they do not. Operations with or on numbers differ from operations on geometric figures since plane figures and solids do not have ‘factors’ in the way that numbers do. As Heath notes, Euclid does not state as an Axiom that factorisation is transitive (as we would put it), i.e. “If a ∣ & B ∣ C, then a ∣ C”, nor does he prove it as a theorem though he assumes it throughout. The Euclidian Algorithm would not work without this feature and a large number of other Propositions would be defective. Indeed, as Heath specifies, we not only need the above but the Sum and Difference Factorisation Theorems which, in Euclid’s parlance, would be

If A measures B, and also measures C, then A measures the sum of B and C, also the difference of B and C when they are unequal and B is greater than C.

i.e. a ∣ B  & a ∣ C, then a ∣ (B + C), also a ∣ (B ‒ C) when  B > C 

        An even more serious admission, from our point of view, is that Euclid does not explicitly state the Well-Ordering Principle, namely that Every non-increasing sequence of natural numbers has a least member though he assumes it in various propositions. Given the strong anti-infinity bias of Greek thought, Euclid would doubtless have thought it unnecessary.

Euclid proves Proposition 14 (If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it) in the following way :

“Let N = pqrs… where  p, q, r…. are primes. Suppose a prime u different from primes p, q, r… and which divides N. Then N = u × b.
But if any prime number divides (m × n) and does not divide m, it must divide n [VII. 30].

Now, pôN and p does not divide u since u, p are primes and u ≠p Therefore, p ∣ b. And the same applies to q, r….
        Therefore, pqr… ∣ b                                                       

But this is contrary to the hypothesis, since b < N and N is the smallest number that can be divided by pqr….
        Therefore, N has no prime factors apart from p, q, r…

It should be noted that this is a Proof by Contradiction and that it applies only to the case where p, q, r… are each of them distinct primes.

What Propositions does this proof rely on?

Firstly, on VII. Proposition 30 “If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.”

This is one of the most important theorems in the whole of Number Theory and I call it the Prime Factor Theorem. What applies here is the special case when one at least of the two original numbers is prime — and a different prime from the ‘dividing number’.
But Euclid also needs to prove, or to have proved, that, N really is, in our terms, the Least Common Multiple of p, q , r…. This he has done in Book VII. Propositions 34 and 35  which detail the procedure for finding the Least Common Multiple, first of two numbers (Prop. 35), and secondly of three or more numbers (Prop. 36). As a special case, Euclid shows that the LCM of two numbers a, b that are prime to each other is ab  and that the procedure can be applied as many times as we wish so that the LCM of a,b,c…. where a, b, c are primes is abc… (He is also scrupulous enough to show (Proposition 29) that a prime and any other ‘number it does not measure’ are prime to each other, which makes any two primes ‘prime to each other’.)

Euclid does not generalize Proposition 14 to powers of these primes, i.e. to our pa qb rc…  though this extension is in effect covered by the propositions about Least Common Multiples VII. 34, 35 and 36 taken together with VII. 31 and 32.
The propositions concerning LCMs are very much what one would expect and are easily assented to. The same does not apply to the Prime Factor Theorem which is by no means ‘intuitively obvious’ nor that easy to establish.

In modern terms Euclid’s proof of the Prime Side Theorem is as follows:

“Suppose p ∣ N (= ab) where p is prime, and p does not divide a.
Then (p, a) = 1  [VII. 29]
Let ab = pm = N where m is some number.
Then p ∣ a = b ∣ m [VII. 19]
But since (p, a) = 1, p/a is in its lowest terms. Therefore m must be a multiple of a and b a multiple of p [VII. 20, 21].
So, if p ∣ ab where p is prime, then either p ∣ a or p ∣ b (or both).”

         The key proposition here is VII. 19, the Cross Ratio Theorem: “If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers will be proportional.”   

This cumbersome statement shows the importance of algebraic notation which the Greeks did not have. Remember that Euclid is speaking only of ratios between hypothetical line segments, not of ‘rational numbers’ as modern mathematicians understand them. However, bearing this in mind, Euclid’s proof may be presented thus :

“Let ac/ad  =  c/d = a/b
        But  a/b = ac/bc
So ac/ad  =  ac/bc
This can only be true if ad = bc   

Conversely, let ad = bc
Then ac/ad = ac/bc
However, ac/ad = c/d
Also ac/bc = a/b
Therefore a/b = c/d

The above itself depends on the legitimacy of ‘cancelling out’, likewise the legitimacy of multiplying and dividing numerator and denominator by the same factor. Euclid has already dealt with such issues and I will not trace the derivation any further back. He has, I think, made a proposition by no means obvious — the ‘Prime Factor Theorem’ — entirely acceptable and, if we accept the latter, we must accept Book IX Proposition 14. Apart from some tidying up and expansion, Unique Prime Factorization in the Natural Numbers has been established.
There was in fact another way to prove the Prime Factor Theorem which is more in keeping with the general plan of Book VII since the above theorem can be made to depend on the Euclidian Algorithm. The latter, for those who are not familiar with it or need a reminder, is a foolproof method for finding the lowest common factor of two numbers. Algebraically, take two different whole numbers M and N where M > N 

 M = a N + R1
N = b P + R2
P = c Q + R3
…….

         This sequence will eventually come to an end since the divisors are diminishing i.e. Q < P < N < M
        (This is an example of Euclid appealing to the Well-Ordering Principle without stating it as such.)

        Suppose, the very next line leaves no remainder.
Q = d T
For  N = b P + R= b (cQ + R3) + R2     
N = b P + R= b (cQ + R3) + R2     

 Note 1 :  “It seems clear that the oldest Pythagoreans were acquainted with the formation of triangular and square numbers by means of pebbles or dots; and we judge from the account in Speusippus’s book On the Pythagorean Numbers, which was based on the works of Philolaus, that the latter dealt with linear numbers, polygonal numbers, and plane, and solid numbers of all sorts….”   (Heath, History of Greek Mathematics p. 76)   
 

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ADDITION

September 14, 2013

ADDITION , the basic arithmetic operation, is not quite so straightforward and unambiguous as one might suppose. When we ‘add’ one thing to another, or to a collection, the originally separate items remain separate after combination : they do not fuse or merge.

Addition is a strictly numerical operation which tells us nothing about the sizes of the objects that are brought together, nor their colour, mass and so forth. The only assumption made is that the items with which arithmetic  deals are discrete and remain so. Even when dealing with liquids which do combine together imperceptibly to form a whole, we often tend to still think in terms of discrete items : when we talk of ‘adding’ a bucketful of water to a pond, the implication is that the water in the pond is made up of ‘so many bucketfuls’, i.e. could be broken down into units. If we want to take merging into account we end up with the formula 1 + 1  = 1  which, though it is a perfectly correct representation of what goes on when, say, we combine two droplets of rainwater, looks extremely peculiar. It would be quite possible, though probably not worthwhile, to develop ‘Rainwater Arithmetic’ where no matter how many items you add together the net result is 1. Many liquids have an upper limit to merging : if you carry on adding droplets of oil to an initial droplet lying on water, the sheet of oil eventually splits in two. In such a case 1 + 100,000,000 = 2 or something of that order of magnitude. Our arithmetic, then, concerns entities with an upper limit of 1 , i.e. substances which never merge at all or, if they do , immediately break apart.

There are at least three different senses to addition which we might call  ‘adding’, ‘adding on’ and ‘adding up’. In the first case we join together two groups or collections of comparable size. In the second case we tack on a smaller collection to a larger, and in the third case we do not so much join together as rearrange. In abstract mathematics there is no difference in the operations involved but in concrete terms there is all the difference in the world.

The most ancient of the three types of addition is undoubtedly ‘adding on’. If we go back to the time when objects or events were recorded by notches on a bone or knots in a cord (which I shall call the tally system) there is nothing to ‘add up’ because there is no numerical base : what is on the stick or bone is the ‘total’ and that’s all there is to it.  A new item, the birth of a child or a caribou kill, will be recorded at the end of the list which will, if there are very few items most likely be arranged in a row, or in several rows more or less underneath each other as on the Ishango bone, perhaps the earliest example of written numerals. In Wild West days, Billy the Kid “had so many notches on his gun” — one notch, one dead person. The amount grows by accretion just like an organism : the longer the list the more items, children born, midsummers, caribou kills and so on.

But when you add ciphered numbers the final amount does not grow noticeably  : 2 + 3 = 5 where 5 is no larger or longer than 2 or 3. This is not in the least a trivial observation since it is the root cause of the current very widespread misunderstanding of what numbers essentially are, namely the symbolic representation of real or imaginary collections of certain objects, collections which increase or diminish in size.

And when you add up a column of figures all you are doing is getting the separate amounts into a tidier form. What was separate (distinct) before addition remains separate after addition : the difference is that the whole lot is now grouped in a systematic fashion according to the powers of the base, usually ten. But what is there at the end was there at the beginning.

However, when you ‘add’ fresh recruits to a regiment, new employees to a firm’s workforce or money to someone’s bank account you are not just rearranging what is already there but bringing in someone or something new from outside. It is at first sight somewhat surprising that this makes no difference arithmetically speaking, the reason being that by the time the new objects are actually joined to the existing group they are no longer completely ‘new’ since they are already in existence, even if, as in the case of an ‘addition’ to a family they have only just been born.

Despite the difference between adding up and adding on, the operation of addition remains, like division, an ‘inert’ one : strictly speaking nothing is created or destroyed. A primary schoolteacher teaching addition to a child has to have the extra building blocks required concealed to one side : if there is nothing there to add, addition cannot take place. This is quite distinct from a truly ‘creative’ operation where something new is produced from within as when a mother gives birth or when unicellular organisms like bacteria reproduce by splitting in two (mitosis). I did as a matter of fact start to  construct an arithmetical system where the basic operations are splitting and merging instead of adding and subtracting.   SH  14/9/13

 

   


Bases

June 4, 2013

Bases

The first thing to realize about bases is that Nature does not bother with them. Nature does not group objects into tens, hundreds, thousands and so on, not even into twos, fours, eights. Bases in the mathematical sense are entirely a matter of human convenience — Nature only uses base one. We are so used to thinking decimally and writing numbers in columns that we tend to consider that an amount, expressed in our modern Hindu-Arabic  positional numerals, is somehow truer than if it were expressed in, say, Chinese Stick Numbers. Even, there is a tendency to think that the modern representation of a quantity  is somehow numerically truer or more real  than the actual quantity  — an example of the delusionary thinking that doing mathematics all too readily gives rise to. Asked how many stones there are in a certain pile, an unhelpful but perfectly correct answer would be to transfer all the stones into a wheelbarrow and empty them out at the enquirer’s feet.

          So why do we bother with bases? Partly for reasons of space : an amount expressed in base one requires a lot of paper or wood or whatever material we are employing and this was and is an important consideration. But the main reason is that our perception of number is so defective that we find it very hard indeed  to distinguish between amounts of similar objects or marks beyond a certain very small quantity (seven at most). So what do we do? What we need is a second fixed quantity, a second ‘unit’, in terms of which we can assess larger quantities. What do we choose?

‘Ones’ are given us by Nature and the sort of objects we shall want to assess either are collections of ‘ones’, like trees or sheep, or can be treated as if they were, e.g. mountains, villages and so on. We speak of ‘whole’ numbers thus implying that they are in some sense ‘entire’, ‘indivisible’. If our standard ‘one-object’ is a pebble it really is indivisible in a pre-industrial society and a cowrie shell, though it can be broken in two, ceases to be a proper shell if this is done. There is, in concrete number systems,  no question of dividing a ‘one’ and ‘fractions’ (‘broken numbers’), if defined at all, are represented by smaller objects or marks, not by splitting up the ‘one-object’. Whole number arithmetic was by Greek times firmly associated with the physical theory of atomism as put forward by Democritus who himself wrote works on arithmetic now lost.

A mass of ‘ones’ is perceptually unmanageable unless related to certain standard amounts we are familiar with. But Nature is extremely unhelpful in providing us with exact standard amounts. Litters of kittens and puppies are by no means standard, and apart from a slight prejudice in favour of the quantity five, the amount of petals in a flower or branches on a tree varies amazingly.
Familiar fixed amounts such as the ‘number’ of hills on the skyline could only be of strictly local relevance and at the end of the day about the only available standard amounts given to man by Nature are the fingers which, counting the thumbs, come in fives. It is thus no surprise that number systems are dominated by the amounts five, ten and twenty.

Alternatively, of course, we could start from the other end and opt for a ‘man-made’ secondary unit whose size would depend on our perceptual needs and what exactly it is we want to assess or measure. These three criteria 1.) availability of a standard amount; 2.) human perceptual  limitations and 3.) appropriateness for assessment purposes, conflict and one of the main problems of early numbering was how to reach an acceptable compromise between them.

Two is the first possibility for a ‘secondary unit’ but, although it has come into its  own in the computer era (because of the two states On and Off), it is clearly too small to be of much use for ordinary  purposes. A language spoken in the Torres Straits had a word for our ‘one’, namely urapan and a word for our ‘two’ okasa and that was about it. Their numerals went

1.       urapan                   4.       okasa okasa

2.       okasa                     5.       okasa okasa urapan

3.       okasa urapan         

Understandably, since even a number as small as 11 would require six words, the natives referred to anything above 6 as ras — ‘a lot’ (Conant, p. 105).

A few things, or rather events, are viewed in threes witness phrases like “third time lucky” and we group quite a lot of things in fours (seasons, points of the compass &c.) but  the obvious first choice for a ‘secondary unit’ is five. Beyond five we really feel the need for a ‘secondary unit’ since  collections like    l l l l l l l l and   l l l l l l re  practically indistinguishable. Also, as it happens we have the five fingers to be able to check (by pairing off) whether we are separating out the items into groups correctly.

The Old Man of the Sea in Homer ‘fives’ his seals but for most herdsmen five would still have been rather too small as a secondary unit. So where do we go next? If we remain guided by the fingers the next possibilities are ‘both hands’ and what many primitive languages referred to as ‘the whole man’ i.e. ten and twentyTwenty is in some ways a better choice since, if we keep the option open of reverting to five for trifling amounts we can cope with very sizeable collections using batches of twenty. The Yoruba used twenty cowrie shells as their principal counting amount after the unit. Some modern European  languages which have long since become decimal show traces of an earlier vigesimal (twenty-based) system which probably suited farmers better. Hence Biblical terms like ‘three score years and ten’ in English and the French soixante dix-huit (sixty-eighteen).

A secondary  unit is, unlike the unit, not actually indivisible — since it is still made up of standard ones — so how do we keep it together if we are using objects as numbers? This depends on the choice of standard object and in practice is one of the motivations for the choice of object in the first place (or second place at least). Heaps of pebbles are heavy enough not to blow away but can all too easily be disturbed by people bumping into them, while piles of flat objects unless they are paper thin readily tip over and in any case really flat objects are hard to come by in nature. This is where shells are advantageous since if of the cowrie variety they stack up neatly and, even better, can be pierced and threaded on strings to make number rosaries. Beads make good numbers but since they are manufactured items they would not have been amongst the very earliest examples of object numbers.

The clay Number Ball I have already mentioned would not be suitable for secondary (or tertiary) standard amounts precisely because the bits tend to adhere together : its use would be for assessing limited quantities in the field which, if required, could be recorded back at the Number Hut using a different system, knotting or incising.

On my island I opt for a stick of standard length as my ‘one-object’ and I instruct the natives to tie sticks together into a bundle when we reach the fixed secondary amount which tentatively I fix at our twelve. Lacking a  sign on this computer for a bundle of sticks I represent this amount by . At the back of the Number Hut I set up partitions to make alleyways for concrete numbers and I suspend from the roof an example of the bundle the alleyway is to contain and its decomposition into smaller bundles or individual sticks as a sort of Système Internationale prototype. For the moment I only propose to use the extreme right two alleyways, the first for individual sticks and the second for bundles of the specified size.  Whenever we have  ⁄ sticks in the extreme right alley, they are tied together and transferred (literally ‘carried’) to the next alleyway. The system can be used for the temporary recording of data but it is best to restrict its use to calculation, simple additions and subtractions, while using a different system for recording purposes. I can, for example, paint three vertical lines on a piece of bark to make it into a Number Board and paint in particular configurations of the sticks and bundles.  When painting I do not use short cuts, I just represent the sticks and bundles as well as I can turning stick numbers into stroke numbers.  As yet I do not proceed any further : all quantities are to be represented by sticks and/or Number Bundles of a single fixed amount and, for the time being, I do not set an upper limit to the amount of bundles. Thus the components of our number system so far are only  l  and     where

       =        l l l l l l l l l l l l  

A  feature of this still very rudimentary system is that at any moment a bundle     can be reduced to so many sticks simply by untying the cord and transferring them into the appropriate alley. Even, it is possible to have second thoughts about the ‘secondary unit’ and change it for another, since all you have to do is untie the bundles and tie the sticks up again using a different set amount. We might, for example, want to revert to five if the quantity to be assessed turns out not to be so great, or, conversely, jump ahead to twenty for a really large herd of goats or clump of trees. With systems that depend on threading objects on a string or wire, changing the secondary unit is either impossible or time-consuming and so would tend not to be done.

If the first ‘greater unit’ is set at ten and we are dealing with sticks, the problem of distinguishing different numbers less than ten  remains — we have met requirements 1.) and 3.) but not 2.) . The earliest Egyptian written numbers, perhaps based on still earlier number sticks, got round this problem, or tried to, by arranging the sticks in set patterns. But the patterns are not very distinctive or memorable. Far more striking are the excellent domino or dice dot numbers. Domino patterned numbers stop at six and the arrangements for the playing-card seven, eight and nine are not so striking — I have sometimes I caught myself having to look at the corner to distinguish between a seven and a nine.

The stratagem of arranging near identical marks in a pattern is an attempt to enlist shape in the service of number : if you recognize the shape you don’t need to count the dots. Shape recognition is distinction by type which the principle of distinction by number must displace in the cultural development of the species. For all that , even today, we feel at ease with shape and respond to it ‘naturally’ (perhaps because of the sexual instinct and childhood memories) while number is at first unappealing, it appears cold and  inhuman. The supposedly artificial distinction between the arts and the sciences is rooted in this primeval struggle between distinction by shape and distinction by number, a struggle which the latter is obviously  winning. In the pre-industrial past it was quite the reverse  : most ‘primitive’ tribes considered that distinctions of shape and thickness were so much more significant than numerical distinctions that they developed a large  and sophisticiated vocabulary to deal with the former while contenting themselves with half a dozen number words. And even in the domain of number itself shape cast its shadow: many societies used different number words  depending on the overall shape of the objects being counted. The Nootkans, for example, used special terms for counting round objects and traces of this practice persist in the ‘numerical classifiers’ of modern Japanese and Chinese1.

A drawback of numerical distinction by patterning is that every new number requires its own special arrangement which must then be committed to memory. This certainly limits the range but the same patterns could be re-used with slight differences or could be combined in various ways. One would not have thought the effort involved was that great, not that much more than is involved in learning the alphabet. Also, the idea of familiarising people with numerals by way of card and board games is delightful (though presumably not done  deliberately). For some reason this promising system never got extended beyond six (otherwise we would have in our heads patterns for higher numbers) and taken in itself constitutes something of a dead-end in the history of numbering.

Domino numbers are a curious and attractive relic of days long gone. 

Q1. If we use dominoes as numbers, what base are they in? And what is the largest amount that can be represented by a single set of dominoes ?  

(To be continued)

Bases

February 28, 2013

“He who examines things in their growth and first origins will obtain the clearest view of them” (Aristotle).

The first thing to realize about bases is that Nature does not bother with them. Nature does not group objects into tens, hundreds, thousands and so on, not even into twos, fours, eights. Bases in the mathematical sense are entirely a matter of human convenience — Nature only uses base one. We are so used to thinking decimally and writing numbers in columns that we tend to consider that an amount, expressed in our modern Hindu-Arabic  positional numerals, is somehow truer than if it were expressed in, say, Chinese Stick Numbers. Even, there is a tendency to think that the modern representation of a quantity  is somehow numerically truer or more real  than the actual quantity  — an example of the delusionary thinking that doing mathematics all too readily gives rise to. Asked how many stones there are in a certain pile, an unhelpful but perfectly correct answer would be to transfer all the stones into a wheelbarrow and empty them out at the enquirer’s feet.
          So why do we bother with bases? Partly for reasons of space : an amount expressed in base one requires a lot of paper or wood or whatever material we are employing and this was and is an important consideration. But the main reason is that our perception of number is so defective that we find it very hard indeed  to distinguish between amounts of similar objects or marks beyond a certain very small quantity (seven at most). So what do we do? What we need is a second fixed quantity, a second ‘unit’, in terms of which we can assess larger quantities. What do we choose?
‘Ones’ are given us by Nature and more often than not this is all we are given. The sort of objects we shall want to assess either are collections of ‘ones’, like trees or sheep, or can be treated as if they were, e.g. mountains, villages and so on. We speak of ‘whole’ numbers thus implying that they are in some sense ‘entire’, ‘indivisible’. If our standard ‘one-object’ is a pebble it really is indivisible in a pre-industrial society and a cowrie shell, though it can be broken in two, ceases to be a proper shell if this is done. There is, in concrete number systems,  no question of dividing a ‘one’ and ‘fractions’ (‘broken numbers’), if defined at all, are represented by smaller objects or marks, not by splitting up the ‘one-object’. Whole number arithmetic was by Greek times firmly associated with the physical theory of atomism as put forward by Democritus who himself wrote works on arithmetic now lost.
A mass of ‘ones’ is perceptually unmanageable unless related to certain standard amounts we are familiar with. But Nature is extremely unhelpful in providing us with exact standard amounts. Litters of kittens and puppies are by no means standard, and apart from a slight prejudice in favour of the quantity five, the amount of petals in a flower or branches on a tree varies amazingly. Familiar fixed amounts such as the ‘number’ of hills on the skyline could only be of strictly local relevance and at the end of the day about the only available standard amounts given to man by Nature are the fingers which, counting the thumbs, come in fives. It is thus no surprise that number systems are dominated by the amounts five, ten and twenty.

The Secondary Unit

 What there is no doubt we do need and have done from very remote times is a secondary unit. This must be clearly distinguished from a base  since the latter is an extendable sequence of ‘unit’ sizes, a ‘geometric series’ like 1, 10, 100, 1000 and so on. Why didn’t ‘early’ societies (with some exceptions) go straight for the base system? The answer is that they didn’t need it and that it doesn’t come naturally, at any rate to practical people. For most purposes two ‘significant amounts’ or the first and second powers of the base are quite sufficient. Even one will do if it is of reasonable size because you don’t actually have to stop at the ‘square’ of the base as if it were a brick wall cutting off all access to a numerical  beyond. Fifteen hundred, apart from being more succinct,  sounds a good deal more natural than the pedantic ‘one thousand and five hundred’ which is what we ought to say by rights (Note 1). Without even defining the hundred  we could still cope perfectly well with quantities up to 999 reckoning in so many tens e.g. by speaking of 810 as eighty-one tens and a five. Generally we do not need to go anything like so far and the language is littered with sets of number words which, though they show base potential, peter out into nothingness without ever even making it to the ‘cube’ stage.
The ‘natural’ way of cultivating the wilderness is to clear an area and then when you’ve planted that, to clear another. Natural at any rate if you have to do a fair amount of the work yourself. If you are a conqueror you may, of course, have an eye on infinity and eternity from the start but this is  folie de grandeur. The first man to introduce wholesale decimalisation was the Ch’in Emperor and he is thought to have hastened his death by imbibing elixirs of immortality.
Numbers were measures before they became numbers ¾ even ‘one’ itself, the ‘father (better mother) of all numbers’, is essentially a measure, one drop, one mouthful, one foot. Our standard weights and measures are really only numbers that remain tied to particular contexts and functions. The Imperial system of liquid measures is an application of base two with four left out since the numbers involved are 1, 2 and 8.

                   1 pint     =  1 pint

                   2 pints   =   1 quart

                   4 quarts =  1 gallon  =  8 pints

Verbally, the number system stops here : although there are obviously larger quantities than the gallon we have  no special words for them, it is someone else’s job to work them out.

In our number words and pre-decimal measures we find a surface  order with an underlying picturesque confusion where all sorts of sets of numbers leave their traces. In the Avoirdupois weights we seem to have two sets of numbers, one proceeding from the ‘small’ end and one from the ‘large’ end, most likely developed by persons performing different functions. It makes sense to have the pound finely divided for sales over the counter to individuals, thus the appearance of the  large, but not too large, secondary unit 16 in sixteen ounces to a pound. From the wholesaler’s point of view we want a large quantity defined straight off, the hundredweight (which is not a hundredweight). We now quarter the hundredweight as it is always useful to divide something into four equal parts and we nearly but not quite converge with the rising 16 system. But there are not 32 pounds to a quarter but the anomalous 28. Is a systematic base system preferable even supposing we had a more suitable base than 10? Not necessarily for the people doing the work. Their principal concern is not logical  consistency but the ready availability of convenient set amounts which the chosen number system or systems should favour and promote. Moreover, once they have what they want, various landmark fixed amounts, they leave the system to its own devices.
There is certainly, within the context of a pre-industrial economy, no need for a number sequence stretching out into infinity :  on the contrary this very feature would have in many cultures provoked a certain malaise as indeed it still does to persons like myself. The idea of really large magnitudes is frightening like the idea of really large intervals of time. Although it is now unfashionable, even politically incorrect, to speak of cultures having specific traits, it is surely no accident that it was the Hindu mathematicians who gave us the first fully positional indefinitely extendable written number system. Indian thinkers, both Hindu and Buddhist, were obsessed with large numbers and vast spans of time : the kalpa for example is a period which lasts 4320 million years. Armed with the decimal base number system the Indians built ‘number towers’ reaching unimaginable heights and not only could they write down these quantities but they had names for many of them. In one legend the Buddha, challenged to list the numbers (read ‘powers’) beyond 107, answers with the names for all the powers up to the colossal tallaksana  or 1053 ¾ i.e. 1 followed by fifty-three zeroes (Note 2) . The Indian approach to large numbers is quite different from that of Archimedes who is, surprisingly in some ways, much more in line with the earlier ‘clear an area’ approach. He wrote a treatise on large numbers but showed none of the delectation and religious awe that the Hindu and Buddhist mathematicians clearly felt. Archimedes was concerned to show that the finite Greek number system could be extended upwards and outwards to deal with colossal quantities like the amount of grains of sand in the, for him finite, universe. But his aim was to tame the beyond not to lose himself in admiration of it. To the amazement of modern commentators, he did not quite hit upon the artifice of full positional notation.
The acceptance (or imposition) of a single indefinitely extendable base system has taken a very long time and is of comparatively recent date. For centuries individual and local numbering systems co-existed with the State imposed one, Roman or Napoleonic, especially in country areas. Until very recently by far the greater part of the inhabitants of Europe were illiterate and many of them used their own numbering systems and ‘peasant numerals’ like the notched sticks of Swiss cowmen. Inns could scarcely have carried on at all without the one-base slate and chalk system where the reckoning was totted up at the end of the evening or month, and in Spain it was once common for the innkeeper to toss a pebble into the hood of a traveller’s cape for each drink consumed. Today rustic numbering systems like the ‘milk sticks’ of cowmen in the upper Alps or the use of knots or pebbles  are things of the past : everyone has at long last agreed to at least write numbers in the approved standard decimal fashion. But for all that we do not think or feel in the way we write. In effect we still use the Babylonian secondary unit, sixty, for the subdivision of the hour, although we  express the quantity in a ten-base. We think ‘sixty minutes’ as a ‘chunk of time’  divisible into so many units, not as six tens of time. We do indeed have the availability of intermediate amounts, five minutes, ten minutes, quarter of an hour, but they are subordinate to the hour and the minute. A day is experienced as a unity which is in the first place decomposable into the unequal ‘halves’ of daytime and nighttime. We do not experience or think the day as two tens of time plus four units.

To the administrator, of course, the use of a consistent base-system is as necessary as the use of a ‘universal’ official language (Latin, English in the Commonwealth &c.). To him numbers have finally ceased to be tied to objects or activities, have become contextless,  in much the same way as, at a further level of abstraction, functions, to the contemporary mathematician, have ceased to be tied to numbers.

Alternatively, of course, we could start from the other end and opt for a ‘man-made’ secondary unit whose size would depend on our perceptual needs and what exactly it is we want to assess or measure. These three criteria 1.) availability of a standard amount; 2.) human perceptual  limitations and 3.) appropriateness for assessment purposes, conflict and one of the main problems of early numbering was how to reach an acceptable compromise between them.
Two is the first possibility for a ‘secondary unit’ but, although it has come into its  own in the computer era (because of the two states On and Off), it is clearly too small to be of much use for ordinary  purposes. A language spoken in the Torres Straits had a word for our ‘one’, namely urapan and a word for our ‘two’ okasa and that was about it. Their numerals went

         1.      urapan                  4.      okasa okasa
          2.      okasa                     5.      okasa okasa urapan
          3.      okasa urapan                

 

Understandably, since even a number as small as 11 would require six words, the natives referred to anything above 6 as ras — ‘a lot’ (Conant, p. 105).

A few things, or rather events, are viewed in threes witness phrases like “third time lucky” and we group quite a lot of things in fours (seasons, points of the compass &c.) but  the obvious first choice for a ‘secondary unit’ is five. Beyond five we really feel the need for a ‘secondary unit’ since  collections like  ½½½½½½ and ½½½½½½½ are  practically indistinguishable. Also, as it happens we have the five fingers to be able to check (by pairing off) whether we are separating out the items into groups correctly.

The Old Man of the Sea in Homer ‘fives’ his seals but for most herdsmen five would still have been rather too small as a secondary unit. So where do we go next? If we remain guided by the fingers the next possibilities are ‘both hands’ and what many primitive languages referred to as ‘the whole man’‘ i.e. ten and twentyTwenty is in some ways a better choice since, if we keep the option open of reverting to five for trifling amounts we can cope with very sizeable collections using batches of twenty. The Yoruba used twenty cowrie shells as their principal counting amount after the unit. Some modern European  languages which have long since become decimal show traces of an earlier vigesimal (twenty-based) system which probably suited farmers better. Hence Biblical terms like ‘three score years and ten’ in English and the French soixante dix-huit (sixty-eighteen).

A secondary  unit is, unlike the unit, not actually indivisible — since it is still made up of standard ones — so how do we keep it together if we are using objects as numbers? This depends on the choice of standard object and in practice is one of the motivations for the choice of object in the first place (or second place at least). Heaps of pebbles are heavy enough not to blow away but can all too easily be disturbed by people bumping into them, while piles of flat objects unless they are paper thin readily tip over and in any case really flat objects are hard to come by in nature. This is where shells are advantageous since if of the cowrie variety they stack up neatly and, even better, can be pierced and threaded on strings to make number rosaries. Beads make good numbers but since they are manufactured items they would not have been amongst the very earliest examples of object numbers.

However, on reflection I decide that introducing such a system would be premature. Within the bounds of a self-sufficient fishing, hunting or agricultural economy there would neither be any need for an indefinitely extendable number system nor would it have any special appeal. In the first place an inhabitant of such a society would not anticipate needing to assess really big quantities. Although a peasant needs more numbers than a hunter, in the past he probably rarely if ever needed numbers extending beyond about 400 , supposedly the upper numerical limit of a typical 19th century Russian peasant. Large amounts of fruit, potatoes and so on would, of course, not be counted but be assessed by weight just like coins that we hand in to the bank. (Banknotes are still counted but in most banks the work is now done by a machine.)

The practical man, craftsmen, herdsman or farmer does not deal in ‘numbers’, he deals in fixed amounts that are significant in terms of his or her  daily work and/or perceptual apparatus. And such quantities do not usually  correspond to the transition points of an extendable base system like our own. To judge by the traces they have left on our language the two most popular ‘significant amounts’ beyond the unit in English speaking countries were, and to a certain extent still are, the dozen and the score. Although twelve as such is beyond our perceptual capacity, the image of two boxes each containing half a dozen eggs must by now have penetrated to the collective  unconscious, at any rate the English speaking one. Twelve, like ten, seems about the right size for making bundles or piles, but is better than ten in many ways because it can be halved, quartered and chopped into three equal portions. For dealing with larger amounts, the score which was originally a ‘score’ or notch a farmer made on a piece of wood as twenty animals passed through a gate, is about all you need so long as you know at least  twenty number words off by heart. The publisher of the red book travel guides to Europe is said to have counted the steps leading up to Milan Cathedral by transferring a pebble from one pocket to the other each time he mounted twenty steps.

A brief list of ‘significant quantities’ on a world-wide scale with reasons for their significance would perhaps be:

5                         hand, fingers,  right size perceptually

6                         half of dozen

10                        both hands, right size for base

12                        right size for base, many factors

 20                        hands and feet, multiple of 5 and 10

60                        many factors, multiple of all previous

100                    square of 10, many factors

Some of the above numbers are significant perceptually, notably 5 since this is around the stage when we cease to be able to assess objects numerically without counting them individually. Thus 5 combines significance because of its use in one-one correspondences (by way of the fingers) with significance as a perceptual ‘unit’. But it has the serious drawback that it cannot be divided up at all (has no factors). It is thus significant but not convenient as a ‘secondary amount’.
Having many factors is really more a matter of convenience than  significance as such but since previous significant amounts are amongst the factors of 60 and 100 these numbers acquire significance acquire significance by proxy. 60 is particularly rich in factors  and has the remarkable property of being a multiple of all previous significant amounts.
100 means nothing to us perceptually though it undoubtedly did to a Roman centurion who would have had in his mind’s eye the terrain covered by his infantry when lined up ready to give battle. 100 is around the ‘acquaintance’ mark, i.e. near the maximum number of persons one is able to relate to personally ¾ I believe I have read somewhere (Desmond Morris?) that 128 is about the limit and that this generally corresponds to the maximum number of persons one has in one’s address book.
But of course 60 and 100 are above all significant because of their divisibility ¾ the main use of 100 is in percentages though it still has the defect that one cannot divide it into three properly.  It must be stressed that a number’s divisibility is not just a matter of interest to modern number theorists : wholesalers or state suppliers receive commodities in bulk which they must subsequently sell or distribute to individuals and it is important that standard quantities should be easy to divide up. This is the reason why so many of the old Imperial measures are built around 20 or the powers of 2 ¾ as it is one of the main reasons why there is such hostility to metric weights and measures.
   One of the troubles with the transition points of a base-system, the ‘powers’ of the base, is that they are significant and convenient not in a practical but purely mathematical sense. Technically speaking, the unit, the base and its powers are the successive terms of a geometric sequence  1, b, b2, b3, b4 ……. with common ratio b. My choice of twelve for the bundles that are to go into the second alleyway means setting b at 12. We would thus have 1, 12, 144, 1728… (since 144 = 122, 1728 = 123). Now 144 and 1728 apart from being too large are not meaningful amounts in our day to day experience. The same goes for smaller choices of base.  5 is perhaps the most ‘significant amount’ of all in real-life terms but 52 = 25 is nothing special and 53 = 5 ´ 5 ´ 5 = 125  even less. 6 certainly has some valid claims on our attention as a significant quantity but 36 has none.
One suspects that the success of a hundred as a ‘significant amount’ is due to its being a multiple of the significant amounts 5 and 20 ¾ it is actually 5 ´ 20 ¾ rather than it being the square of ten. A thousand is just a word meaning ‘large quantity’ and the ambiguous meaning of billion (a million millions or a thousand millions?) shows how vague such large quantities are. A million only has meaning with respect to wealth — and even this sense has been eroded by devaluation so that we find it necessary to replace the word millionaire by multi-millionaire which is even vaguer.

 Non-base extendable systems

Most people assume that once you have defined your ‘secondary unit’ you are somehow obliged to turn it into a true base (and I tended to think along these lines myself before writing this book). But of course you aren’t. The ‘tertiary unit’ or next standard amount we choose to define can be anything at all in principle. One of the few numerically advanced peoples that still used object numbers, the Yoruba,  took a pile of twenty cowrie shells as their ‘secondary unit’. They then combined five such piles of cowrie shells to make 100 in our reckoning, and combined two such piles to define their second most important amount after the unit, 200 in modern numbers. If they had operated a true base system the next halting point would have been 202 or 400 which presumably they considered too large. (Algebraically the Yoruba sequence goes 1, b, 10b and not 1, b, b2 ). For a somewhat different, but nonetheless pragmatic reason, the Mayans, who also took 20 as their ‘secondary unit’, then moved on to 360 (instead of 400) for the next transition point in order to get close to the number of days in a year — or so it has been conjectured.

It does not in practice matter too much for addition and subtraction if the transition points are not in proper sequence (‘proper’ as we see it today) though it is desirable that they should be multiples of the first ‘significant amount’. Thus, anticipating a further fixed amount in my stick system I have already decided to opt for 60 since it is a multiple of 12 while 20 or 100 are not. Odd though it sounds, there is much to be said for defining a large ‘secondary unit’ and then defining  ‘units’ rather smaller instead of larger than it. This is in effect what the Babylonians did by taking 60 as principal amount after the unit which they noted as    . Such an enormous secondary unit makes it absolutely essential to have one or more halting points, or sub-bases,  in the intermediary territory which the Babylonians provided at the five and ten points. They defined the five transition by grouping the one-symbols and introduced a special mark for ten      but otherwise they used only the ‘one-symbol’ right up to 60 itself. For quantities > 60  the Babylonians proceeded by using 60 as a  true base, i.e. the next halt was at 602 = 3,600. In their case they seemed to have no misgivings about using such a huge  amount as tertiary unit which seems to contradict what I said earlier. But the Babylonian scribes who developed and used the sexagesimal number system were not hunters or herdsmen but officials helping to run a vast empire. They needed large numbers and spent their lives dealing in them as did the Egyptian scribes.

Practically speaking we require very different fixed standard amounts depending on the context. To divide a pound weight into sixty ounces would appear slightly crazy but we find it most convenient to divide up a fairly short interval of time, the hour, into sixty minutes, while we divide up a somewhat larger interval, the day, much less finely. The numbers 60, 12 and 24 are not imposed on us in the way the number of days in the year is : we could divide up the day into 60 or 72  or any (even) number of ‘hours’ and divide each ‘hour’ into 12 or for that matter 17 ‘minutes’. The unsystematic way in which we divide up the day seems right : there is, as far as I know, no SI project to decimalise time (though the ancient Egyptians did just this) and the very idea fills me with horror.

___________________________

Note 1

“Yet it was the Indians who reckoned the age of the Earth as 4.3 billion years, when even in the 19th century many scientists were convinced it was at most 100,000 years old ( the current estimate being 4.6 billion).” The apparent source for this is: Pingree, David, Astronomy in India, in Astronomy Before the Telescope, p.123-42.  Quoted Chasing the Sun by Richard Cohen, p. 132

1  I recently came across an interesting example of how restricting the idea of always keeping to a base is. I noticed, or had read somewhere, that the Binomial Coefficients were powers of 11 and this made sense since they can be defined by starting with 1 and getting the next term by shifting what you’ve got across one column and adding. Thus

          1                             1  =   110

                                      1        0 

                                      1        1                            11  =  111

                             1        1        0  

                             1        2        1                           121  =  112

                   1        2        1        0

                   1        3        3        1                         1331  =  113

          1        3        3        1        0

          1        4        6        4        1                       14641  =  114

         

 However, what happens now?  The next line of Pascal’s Triangle is supposed to be

                   1        5        10      10      5        1 

 This isn’t a power of 11 surely.  But who says you can’t overstep the base if you want to?

 (1 x 105) + (5 x 104) +  (10  x 103) +  (10 x 102) +  (5 x 10) +  1 

  =   100,000 + 50,000 + 10,000 + 1,000 + 51   = 161,051   = 115

   

Classifiers      In other cultures different bases were used depending on the different objects being counted. Flat objects like cloths were counted by the Aztecs in twenties, while round objects like oranges were counted in tens.  The use of classifiers obviously marks an intermediary stage between the era when numbers were completely tied to objects and the era when they became contextless as now. We still retain words like ‘twin’ and ‘duet’ to emphasize special cases of ‘twoness’, note also ‘sextet’, ‘octet’ &c.     The complete dissociation of verbal and written numerals from shape and substance is today universally seen as ‘a good thing’ especially by mathematicians. But classifiers were doubtless once extremely useful because they emphasized what people at the time felt to be important about certain everyday objects and activities, and they remain both a picturesque reminder of the origins of mathematics in the world of objects and our sense-perceptions. The removal of all such features from mathematics proper seems to be a necessary evil but at least let us recognize that it is in part an evil : the banning of contextual meaning from mathematics, the language of science and administration, is typical of the depoeticization of the modern world.

Base sixty
It is not known why the Babylonians chose 60 as their most significant amount after the unit. The fact that 602 = 360  is close to the number of days in the year may have something to do with it. Certainly, 60 would not have been the original choice. It has been suggested that 60 evolved as a compromise solution to the separate claims of 5 and 12  which were already well established as ‘secondary units’ within the territories conquered by the Babylonians. Since 60 has as factors all the main bases and significant amounts smaller than it, including 10 and 20, it had something for everyone : it was a numerical Pax Romana.

 

Gnomon :The World’s First Scientific and Mathematical Instrument ?

February 1, 2013

GnomonA gnomon was originally a sort of set-square that could be stood on its edge and was used to measure the lengths of shadows — present-day sundials have a ‘gnomon’ on the top though the shape is more complicated (Note 1). Thales is supposed to have used a gnomon to estimate the height of the Great Pyramid by employing properties of similar triangles and it was data amassed by similar methods that enabled Eratosthenes to estimate the circumference of the Earth by comparing noonday shadows cast at different localities (Note 2). The gnomon thus provided a precious link between three different disciplines : geometry, astronomy and, as we shall see, arithmetic : it was perhaps the first precision instrument of physical science.

Sets of gnomons put together — or drawings of them — became a surprisingly useful early calculator and enabled the early Greek mathematicians to investigate spatial properties of numbers.

        
   
  
   

Each coloured inverted L shape border in the above represents an odd number with the unit in the top left hand corner. The early Greek mathematicians deduced the important property that Any square number can be represented by successive odd numbers commencing with unity. As we would put it:  n2  =  1 +  3 + 5 + ……(2n + 1)   For example, 41 +  3 + 5 + 7   And this can be extended to the observation that Every difference between two squares can be represented by a sum of successive odd numbers. Thus, 5–  2=  5 + 7 + 9    (It was in fact this relation which struck me as being quite astounding that instigated my interest in mathematics which up to then I had despised.)
But much more can be got out of the simple diagram above. The Egyptians were certainly aware of certain cases of the the property forever associated with Pythagoras, namely that The Square on the hypotenuse is equal to the sum of the squares on the other two sides of a right-angled triangle since they used stretched ropes with lengths in the ratio 3, 4, 5 to lay out an accurate square corner. However, they may not have realized that this property applied to all right angled triangles. The question provided a fruitful contact between geometry, the science of shape, and arithmetic, the science of exact quantity and the gnomon most likely played an important role here. Greek mathematicians were interested in sets of numbers that were ‘Pythagorean triples’, i.e. numbers a, b, c where   a2 = b +  c2 .
Now,  adding on a gnomon “preserves the square form”  and, more significantly for the present discussion, that the difference of two successive squares is an odd  number.

        +               =           
                                       
                                       
                          

Some sharp sighted mathematician, perhaps Pythagoras himself or one of his disciples, realized that if the gnomon is itself a square we have a Pythagorean triple. (This follows from the observation that adding on the relevant gnomon leads from one square to the next.) So, if we select an odd square number, we can make it the gnomon and thus give an example of a Pythagorean triple. The first odd square is 

          (our 9) and to make it into a gnomon we stretch it out into three parts , two equal and the third a unit
                   This provides the outer framework for the two squares :

              The inner square has side 4 and the outer side 5. This gives the simplest
              Pythagorean triple  52 = 42 + 32 .
      
      

      

However, any odd square will do and, since 49 = 72 we can construct a Pythagorean triple involving it. The gnomon is 24 + 1 + 24  giving 24 for the side of the larger square and 23 for the smaller one. This gives the triple  242 = 23  + 72 .  The series of Pythagorean triples using this procedure is endless : it suffices to find an odd square number.
This procedure can be generalised if we allow a gnomon to be made up of more than one row + column. For example, we might allow the gnomon to have three rows + three columns.

………………        

………………       

………………       

…          
…          
…          

If r is the side of the inner square, the outer square is (r + 3) instead of (r + 1) and the little square in the bottom right hand corner will be 3 x 3 = 9 instead of 1. The gnomon is made up of two rectangles (r x 3)  the little square giving  (6r + 9) = 3(r + 3)   We must thus find a square which is equal to the gnomon or solve  3(2r + 3) =  m2   for some m.   Since  m2  is divisible by 3 this makes m a multiple of 3 as well. We must also have m large enough so that r is at least 1.  The first possibility is m = 21 = 7 x 3  so that  3(2r + 3) =  212    This makes r = 72  This will be the side of the inner square while the outer one will be 72 + 3 = 75.  So, if this reasoning is correct, we should find that 752 = 722 + 21 which is indeed the case (check this). So a rather more spaced out but still unending set of Pythagorean triples can be manufactured where the difference between the sides of the squares is 3 rather than 1.  It is left to the interested reader to concoct other sets.
As a matter of fact we have reason to believe that the early Pythaogoreans knew of such sets of triples and it is plausible that they hit upon them using some such method which has its basis in the manipulation of sets of wooden gnomons and/or actual counters on actual boards. Interestingly enough, the Babylonians a thousand years earlier were aware of Pythagorean triples and seem to have had some method of concocting them (Note 3)The basic formula for all Pythagorean Triples is given in Euclid — or rather can be deduced from the argument given in Euclid which is mainly verbal since the Greeks did not have our algebraic notation. I shall not give it here — you can get it from Wikipedia or some other site by the click of a key — as I am more interested in seeing how such formulae arose in the first place and indeed in (re)discovering them for myself, something that I encourage you to do as well. In the next post I will examine the slightly more complicated problem of an isosceles right-angled triangle, i.e. one where the two smaller sides are equal. This provoked a trmendous rumpus at the time because it raised the issue of so-called ‘incommensurables”. If the short side is set at unity, the square on the hypotenuse comes out at  of  12 + 12 = 2 so the side itself is the square root of 2. But was there such a number? In the ideal world of Platonic forms (not yet elaborated) certainly there was, but in the Pythagorean world of number where number meant ratio between two integers there was apparently no such quantity and thus no such length.     SH   1/1/13

Note 1   “The word gnomon ….literally means an “indicator”, or “one who knows”. Specifically, it was the name of the sundial first brought to Greece from Babylonia by Anaximander, who was probably one of Pythagoras’s teachers. The word also serves to indicate any vertical object like an obelisk which serves to indicate time by means of a shadow.”  Valens, The Number of Things 

gnomon : Stationary arm that projects a shadow on a sundial” (Collins)

Note 2  Actually, it seems that Eratosthenes’ data did not depend on gnomons as such but it did rely on the measurement of noonday shadows. Reputedly, Eratosthenes based his remarkably good estimate of the circumference of the Earth on the information. presumably relayed by a traveller, that the sun at noon at midsummer’s day at Syene was directly overhead because it was reflected at the bottom of a deep well. Eratosthenes, as Librarian at Alexandria some 500 miles or so due north of Syene, knew that the shadow of a pillar cast by the sun at the same moment in time was a little more than 7 and a half degrees off the vertical . This enabled him to come up with an estimate of 4,000 miles in our reckoning for the Earth’s radius using geometrical techniques. Current estimates put the Earth’s mean radius at about 3960 miles.

Note 3     “The Babylonian tablet called Plimpton 322 (dating from between 1900 and 1600 B.C.) shows that the Babylonians had studied this problem [of Pythagorean triples] much earlier. The tablet merely lists a series of Pythagorean triples but the order in which they are listed makes us believe that the Babylonians had a general and systematic solution for the problem of finding Pythagorean triples.”

Bunt, Jones & Benient, The Historical Roots of Elementary Mathematics

Cosines and Sines

December 31, 2012

“He who sees things in their growth and first origins will obtain the clearest view of them” (Aristotle)

Our most basic ‘mathematical’ impressions are neither numerical nor geometrical. We situate an object (desired or feared) relative to certain familiar objects whose position we know or think we know. We position the new object by means of words such as ‘near’, ‘under’, ‘on top of’, ‘alongside’ and so forth. Babies and toddlers are not alone in working this way : even adults (including mathematicians and scientists) fall back on this method when confronted by a real life situation as opposed to a situation in a laboratory). If I ask where the nearest church or Post Office is, no one turns on their mobile phone complete with SATNAP (?) and tells me the latitude and longitude or even  the exact distance and orientation with respect to the present location. He or she would say, “Go right, then left, it’s the other side of the Church”, where the church is an easily recognizable landmark. If I ask where my pen or pencil is, I won’t be given the co-ordinates relative to the bottom left hand corner of the room; I will be told “It’s under the table”.
Now these relations are neither numerical or geometrical, even less algebraic : they are, as Bohm pertinently pointed out in an interview ‘topological’. Topology is that branch of geometry that deals with proximity and connectivity to the exclusion of metrical concepts. Perhaps we ought to make topology the first ‘science’ learned at school, though it probably does not need to be taught at this stage. Regrettably, topology is an exceedingly abstruse (and largely useless) branch of mathematics that only specialists study.
After ‘topological’ concepts, we have numerical ones, ‘How many of this?’ ‘How many of that?’  So-called primitive tribes get on perfectly well in their environment without numbers and arithmetic operations but the latter become necessary when we have a commercial and bureaucratic society such as developed in the Middle East in Assyria, Babylon and so on. Geometry was a somewhat later development, becoming necessary only when it was vital to assess accurately the areas of plots of land (for taxation purposes and also when State buildings became so large and complex that the coup d’oeil and rule of thumb of the craftsman/builder was no longer adequate. You can raise a pretty good straight wall without any calculation or mathematics but if you want to build a pyramid orientated in a particular way with regard to the stars, you require both a relatively advanced arithmetic and geometry.
After geometry came algebra (unknown to the Greeks) and algebra has so swamped mathematics that even Euclidian geometry, itself thoroughly idealized and remote from the natural world, seems homely and ‘concrete’ today since you can at least draw lines and, if you wish, even make spheres out of Blu-tack and play around with them.
A problem that was posed in the mathematical magazine M500 prompted some reflections on the gulf between geometry and Calculus and the even greater gulf between both of these and physical reality. The problem was about proving the basic limit     lim   (sin θ)/θ  = 1
θ → 0 . As a mathematical fundamentalist, I view sines and cosines as essentially ratios between line segments rather than infinite series.

lim sin θ)/θ  = 1  as  θ → 0

 Cosine M500 diagram

If  angle PÔD = θ  in radians, PE, the arc subtended by the angle θ, is and  > PD = r sin θ. Also, QE = r tan θ > rθ> PD

So   r sin θ  <  rθ  < r tan θ 

This inequality holds for any circle with r > 0 and all angles θ for which sin θ, cos θ and tan θ are defined. We take θ as positive (clockwise from the x axis) and, since we are only concerned with small angles,  0 < θ < π/2.

Dividing by r sin θ which is positive and non-zero we have

         1  <  θ/(sin θ)   <  1/cos θ
        1 has limit 1 since it is never anything else.
If we can show that the limit of 1/cos θ is 1 as θ → 0  the expression θ/(sin θ)   will be squeezed between two limits.
What can at once be deduced from the diagram is :

1.  r cos θ must be smaller than the radius and so, for unit radius,  0 < cos θ < 1

2. As θ decreases, cos θ increases, or “If f < θ, cos f > cos θ

1/cos θ is thus monotonic decreasing and has a lower limit of 1 which is sufficient to establish convergence. If one wants to apply the canonical test, we have to find a δ  such that, for any ε  > 0  whenever 0 < θ <  δ

1– 1/cos θ)  < e
With δ < cos–1 ((1/(1+ ε)) we should be home and, applying the ‘sandwich principle’ for limits, we have

lim        θ/(sin θ)  = 1
                 θ → 0

Turning this on its head, we finally obtain
lim   (sin θ)/θ  = 1
θ → 0

Note, however, that θ is the independent variable — sin θ depends on θ and not the reverse.  

        From here we can find the  derivative of sin θ  in a straightforward manner by using nothing more than the definition of the derivative and the ‘Double Angle’  formula sin (A + B) = sin A cos B + cos A sin B which can be easily proved geometrically for all angles A, B  where
0 < A < π/2 and 0 < B <π/2  (Note 1).

However, what’s all this got to do with the well-known power series?

sin θ = θ – θ3/3! + θ5/5!  – θ7/7! + …… 

Define a convergent power series f(x) with the convenient property that d2 f(x)/dx2 = – f(x)

        Setting A0 = 0, A1 = 1 and equating coefficients we eventually end up with 

     f(x) = x –x3/3! +  x5/5! – x7/7!  ……+ (–1)n x2n+1 /(2n–1)!…

But I’m none too happy about identifying the above series with sin x (and its derivative with cos x). For, if sin x and cos x are  geometric relations between line segments, when there is no triangle, there can be no sine or cosine. For me, geometric sin x is undefined at x = 0 (and likewise at x = π/2 &c.) although the limit of sin x as x → 0 is certainly 0. (It is distressing how often it seems necessary to point out, even to mathematicians, that the existence of a limit does not in any way guarantee that this limit is actually attained.)
All in all, I would feel a lot easier if the ‘sin x power series’ were derived (or defined) recursively term by term along with a demonstration that the difference between f(x) and sin x is always decreasing as we add more terms with limit zero. For sin x is, in  my eyes, itself the limit of a power series as n increases without bound, i.e. If  0 < x < π/2

lim  f(x) = x –x3/3! +  x5/5! …+ (–1)n x2n+1/(2n–1)!   =  sin x
n →                           

Realirty is Discrete

A more general point needs to be made. Practically all proofs in analysis and Calculus depend on the assumption that the independent variable (in this case the angle θ) can be made arbitrarily small. This is quite legitimate if we restrict ourselves to pure mathematics. But Calculus was invented by Newton and Leibnitz to elucidate problems in physics. Translated into physical terms, the basic assumption of Calculus, is equivalent to the presumption that space and time are ‘infinitely divisible’. But I do not believe they are for both logical and observational reasons. There is a growing (but still minority) view amongst theoretical physicists that Space/Time is ‘grainy’, i.e. that there are minimal distances and minimal intervals of time just as there are minimal transfers of energy (quanta). If this proves to be the case, Calculus and a lot else besides constitutes a very misleading model of a reality that is essentially discrete. The great majority of differential equations are, in any case, unsolvable analytically and increasingly the trend is to slog things out iteratively with high-speed computers taking things to the level of precision required by the conditions of the problem and then stopping. Dreadful to say so, but it looks like Calculus’s reign, like that of the dinosaurs, is drawing to a close and that the future will go to algorithmic methods, genetic or otherwise.

Sebastian Hayes

Note 1   Theorem  d(sin x)   =  cos x
                                          dx

Proof     By the definition of the derivative we have

d(sin x)   =  lim       sin (x + h) – sin x
            dx          h → 0                    h

=  lim       (sin x) (cos h) + (cos x)(sin h) – sin x
            h → 0                               h 

=  lim    (sin x) (cos h – 1)   +     lim   (cos x)(sin h)
            h → 0                  h                   h → 0       h

It is not obvious that (cos h – 1)/h  has a limit but (after consulting a Calculus textbook) I substitute
– 2 sin2 (h/2)  for  (cos h – 1)  and, employing the limit we have already found ((sin θ)/θ = 1) we have

lim    (sin x) (cos h – 1)   =  sin x  lim  {– (sin h/2)(sin h/2)}
        h → 0                 h                             h → 0                            h/2

                        =   sin x  lim  (– (sin h/2)) (1)    =  sin x × 0 = 0
                                      h → 0                        

        So all we have left is       lim   (cos x)(sin h)
                                                 h → 0                 h                

Employing once again the ubiquitous lim sin θ/θ = 1 as θ → 1   I end up with the desired
d(sin x)   =  cos x  lim   (sin h)/h    =   cos x
            dx                    h → 0                    

Our dreadful mathematical terminology

December 11, 2012

Open just about any book on numbers (in the English language) and you will come across the boastful claim that we have the best number system there ever has been, so good that, according to one author, it is inconceivable that it could be improved upon in any significant manner. Granted, this claim has some justification : we do indeed have a remarkably supple system of notation since we can cope with quantities as inconceivably large as the American deficit or quantities as inconceivably small as the diameter of a proton. However, “you don’t get owt for nowt” and the flexibility of this system of notation — which we Westerners did not invent but owe to the medieval Hindu and Arabic mathematicians — comes at a cost. As I pointed out in my article on Egyptian numerals, a child at the time of the Pharaohs could  see at a glance that the quantity we note as 100000 was larger than the quantity we record as 10000 since different picture signs were used for hundreds, thousands and so on. More serious still, no one stranger to our language and notation could possibly tell whether the quantity we call seven and record as 7 was larger or smaller than the quantity we call nine and record as 9. Indeed, a visitor from another world might deduce that seven was ‘larger’ than nine since it has more letters.
Indeed, when you examine the language of basic arithmetic as it is still taught in Britain and America, you wonder how anyone ever manages to become numerate at all! The thoughtful child or adult — not quite the same as the intelligent one — is immediately repulsed by the illogicality of our far-famed system (as I was). Apart from cyphered numerals such as 7 and 9 which are perhaps a necessary evil, there is the complicated and incoherent way we form our number words beyond ten.  Instead of ten-one’  we have eleven’ which has nothing to do with either ten or one. Naturally, it takes a non-mathematician to see this and point it out to the world :

“In English we say fourteen, sixteen, seventeen, eighteen, and ineteen, so one might expect that we wouild say oneteen, twoteen, threeteen, and fiveteen. But we don’t. (…) We have forty and sixty, which sound like the words they are related to (four and six). But we also say fifty and tirty and twenty, which sort of sound like five and three and two, but not really. And, for that matter, for numbers above twenty, we put the “decade” first and the unit number second (twenty-one, twenty-two), whereas for the teens, we do it the other way around (fourteen, seventeen, eighteen). The number system in English is highlym irregular.” (Malcolm Gladwell, Outliers p. 119).

“Ask an English-speaking seven year-old to add thirty-seven plus twenty-two in her head, and she has to convert the words to numbers (37 + 22). Only then can she do the math: 2 plus 7 is 9 and 30 plus 20 is 50, which makes 59. Ask an Asian child to add three-tens-seven and two-tens-two, and then the necessary equation is right there. embedded in the sentence. No number translation is necessary: it’s five-tens-nine.”
“For fractions, we say three-fifths. The Chinese is literally ‘out of five parts, take three’. That’s telling you conceptually what a fraction is. It’s differeentiating the denominator and the numerator” (Karen Fuson, quoted Gladwell) (Note 1).

Division      Let us go further. What about this nonsense about “division by” in phrases like “ten divided by five ?  Who on earth is doing the dividing? ‘Five’? This is what we do when we carry out the operation but numbers can’t ‘divide’ other numbers. In reality we asre modelling a situation where we have       ▄ ▄ ▄ ▄ ▄     objects and we sort them into groups each containing  ▄ ▄ ▄ ▄ ▄   no more, no less.
▄ ▄ ▄ ▄ ▄

How many groups do we have?  □ □  using a different symbol. If you envisage division as the sorting out a mass of similar objects into bundles or bags, an activity that still consumes a lot of time and energy in the world today, division at once makes sense. We are a magpie species and seem to have an obsessive interest in collecting objects and storing them in containers : hence the importance of division in our arithmetic system, indeed I consider it more fundamental than adding.
The rule that you are not allowed to divide by zero, which is supposed to be so bizarre and/or profound, is imposed on us by the world we live in like all the rules of arithmetic. It is simply impossible to divide up a mass of objects into so many bundles that have strictly nothing in them. Division ‘by’ zero is not allowed, not because the mathematical establishment have decreed this to be so, but because it actually is the case that you can’t divide a quantity into bundles with strictly nothing in each bundle. What you can, of course, do is divide up a massive composite object into smaller and smaller equal groups (the ‘equality’ being tested by pairing off the groups member to member) and stopping when you get to a certain point. We might decide to call it a day when we reach, for example, the size of a bean, or, more likely in modern times, the size of a molecule.

Infinite Series   In a different website someone queried my claim that infinity is “everywhere present in mathematics and everywhere absent in the real world”. It is true that infinity is not directly involved in the construction of the natural numbers themselves 1, 2, 3…. but even here we are confronted with a series that can be ‘indefinitely extended’. And every time you carry out a division of 1 into 3 and, using a claculator, get 0.33333333333,,,,,,,,  you are in reality being confronted with a sum which goes on for ever, literally 3/10 + 3/100 + 3/1000 +  3/10000 +  …..and so on. We have a cake or anything you like that can be divided up and we make ‘three’ roughly equal portions or bundles. Nothing mysterious about that. Not only is it impossible for the most sophisticated machines to divide up an object into say a hundred billion bits, but this monster 0.33333333333,,,,,,,,  does not even ‘equal’ 1/3 exactly since the series never terminates whereas 1/3 does.  (The subject of ‘equality’ in mathematics will be dealt with in a suibsequent post.)
The child is quite right to reject the absurd adult rule that a wretched stream of figures that never ends represents the simple operation of ‘dividing an object into three roughly equal bits’. It is lamentable that in this technological era, most people actually believe that 0.33333333….. is somehow ‘truer’ than the banal and homely 1/3 because this is what you get when you feed in the numbers plus the division sign into a calculator. Entirely the reverse is true : a non-terminating decimal fraction like 0.33333333333……   does not correspond to any actual state of affairs or operation in the real world that ever has or ever will exist but division into three does correspond to actual operations with actual objects. We do not in our daily life use non-terminating decimal fractions and even quite rarely do we use proper decimals since 10 is such a wretched because it can only be divided into fives and twos (as opposed to 12 where we have quarters and thirds as well as halves).  In day to day activities we use an appropriate temporary base when the quantity to be divided is small, or use the very convenient base ‘hundred’ for this is what a percentage is, i.e. so many out of a hundred. As I said, the wonder is not that there are so few people who take to mathematics with enthusiasm in the West but that there are any at all given the linguistic and conceptual muddle of our number system and its operations.                                             SH 11 December 2012

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Note 1      These excerpts are taken from the extremely interesting book Outliers written by a non-academic, Malcolm Gladwell, a book which I thoroughly recommend along with his other insightful books, Blink and The Tipping Point. I trust the author if he ever hears of this  site will, because of the nice things I say about his books, forgive me for not obtaining official permission to quote him which would be time-wasting if not impossible. SH