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