“Don’t interrupt,” Bruno said as we came in. “I’m counting the Pigs in the field!”
“How many are there?” I enquired.
“About a thousand and four,” said Bruno.
“You mean ‘about a thousand,’” Sylvie corrected him. “There’s no good saying ‘and four’: you can’t be sure about the four!”
“And you’re as wrong as ever!” Bruno exclaimed triumphantly. “It’s just the four I can be sure about; ‘cause they’re here, grubbing under the window! It is the thousand I isn’t pruffickly sure aboutLewis Carroll (Sylvie and Bruno Concluded)
quoted in Dehaene 2011, p. 95
According to Dehaene in The Number Sense: How The Mind Creates Mathematics (Dehaene, 2011), numbers are either round or sharp. Round numbers represent approximate quantity whilst all other numbers are sharp and have a precise meaning. The conversation above sounds strange because Bruno is using a sharp number, a thousand and four, as if it were a round number like a thousand. Sylvie as ever is correct.
If you have to describe a population of 5,424,000 individuals and you are unsure about the exact number or do not wish to be precise, you might say that the population is 5 million, implying that it is 5 million plus or minus a million. If you are a little more sure about the number, you might say that the population is 5 million 400 thousand, implying that your estimate is accurate to the nearest hundred thousand. Obvious problems arise if the actual number is the same as what might otherwise be regarded as a round number. This can be overcome by using a locution such as, the population is exactly 5,424,000.
All the languages of the world seem to have selected a set of round numbers. Why this universality? Probably because all humans share the same mental apparatus and are, therefore, confronted with the difficulty of conceptualizing large quantities. The larger a number, the less accurate is our mental representation of it. Language, if it wants to be a faithful vehicle for thought, must incorporate devices that express this increasing uncertainty.Dehaene 2011, p.96
Round numbers are therefore a device that lets language express inaccuracy and uncertainty much as it is presented by the Approximate Number System.
Approximate Number Pairs
All languages also have a large vocabulary of words for expressing various degrees of numerical uncertainty. For instance, the English words, about, almost, approximately, around, barely, circa, close to, just about, more or less, nearly, roughly etc. Some of these words such as around, close to and nearly are expressly spatial.
In many languages one can also indicate an approximate quantity by saying a pair of numbers, in English for example, one might say “four or five toys”. Unfortunately, in English this can have a disjunctive as well as an approximate meaning. It can mean either four or five which just means one or the other, as well as approximately four or five which might mean three four, five or six etc.
In approximate expressions like these, not all combinations of numbers are possible or acceptable, Following work by Channell and Sigurd, Pollmann and Jansen in The Language User as an Arithmetician suggest that acceptable pairings follow a number of simple rules.(Pollmann and Jansen, 1996)
Pollmann and Jansen 1996, p.223
- The Roundness Rule. At least one of the pair of numbers needs to be round, that is in one of the following sets of numbers: the integers 1-20 and multiples of 5, 10, 20, 50 or 100.
- The Ordering Rule. The smallest number of an approximate pair expression comes first, so one can say two or three toys but not three or two toys. An exception is twelve or one, probably because one-o-clock comes immediately after twelve-o-clock.
- The Difference Rule. Both numbers must be of the same order, so one can say twenty or thirty people but not in this sense ten or one thousand dollars.
- The Sequence Rule. The numbers in a two number approximate expression have to be part of the same arithmetic sequence, and have to follow each other in that sequence.
Pollmann and Jansen then expand the sequence rule as follows
Pollmann and Jansen 1996, p.224
- The starting point of any sequence has to be equal to the difference between the numbers in the sequence; so when the difference is 1, the starting point also has to be 1 and the sequence is 1, 2, 3, 4, 5, etc. When the difference and starting point is 2, the sequence is 2, 4, 6, 8, 10, 12, etc. And when the difference and starting point is 10, the sequence is 10, 20, 30, 40, 50, etc. Expressed like this, the sequence rule correctly predicts that expressions like 3 or 4, 4 or 5 and 20 or 30 are acceptable, while expressions like 1 or 3 and 2 or 5 are not.
- Only sequences where the starting point and difference is 10n, 5 x 10n, 2 x 10n or 2.5 x 10n can be used as sources for approximate expressions. Therefore, expression pairs like 3 or 6, 40 or 80 and 600 or 1200 that would form part of sequences with differences equaling 3 x 10n, 4 x 10n or 6 x 10n are not acceptable.
- The sequence involved can only be used up to its 20th number at most. 12 or 13 and 90 or 95 are therefore acceptable, but 32 or 33 and 150 or 155 are not. If the starting point and difference is 2 or 20 etc., or 2.5 or 25 etc. the sequence might be even shorter than 20. Meaning that large numbers with small differences are not acceptable members of an estimation pair.
With regard to round numbers, Pollmann and Jansen suggest that estimations made using the Approximate Number System can be thought of as being mapped onto a set of favourite numbers. So that with any particular base n there is a set of favourite numbers that consists of:
- any integer power of the base
Pollmann and Jansen 1996, p.225
- half, double, and half of half of any integer power of the base
Giving a set of favourite numbers for our base 10 number system of something like:
n-powers of base 10 0.01, 0.1, 1, 10, 100, 1000 etc.
doubles 0.2, 2, 20, 200, 2000 etc.
halves 0.5, 5, 50, 500 etc.
halves of halves 0.25, 2.5, 25, 250 etc.
Pollmann and Jansen then wanted to see if favourite numbers like these occurred in everyday usage and examined the denominations used for different currencies. They examined the denominations used by 84 countries with a total of over 1000 separate denominations and found that they all used base 10 favourite numbers as defined above, with just 13 exceptions.
Sterling 1p, 2p, 5p, 10p, 20p, 50p, £1, £2, £5, £10, £20, £50
Euro 1, 2, 5, 10, 20, 50 cents €1, €2, €5, €10, €20, €50, €100, €200, €500
Dollar 1, 5, 10, 25, 50 cents $1, $2, $5, $10, $50, $100
The few exceptions mainly included countries that have a 3-unit coin or banknote. Before metrication in 1971, Great Britain also had threepence (3d) and sixpence (6d) coins although these would have been members of a base 12 favourite number list, there being 12d in a shilling and 6d would half of that and 3d a half of a half of a shilling.
When thinking about satisfying a particular value, for instance when proffering money or change, a so-called ‘greedy algorithm’ is sufficient. That is first proffering the largest denomination less than the required value, then subtracting that denomination from the value to create a new value and continuing the process until the value becomes zero.
If you are asked for 75p you would, if possible, first proffer 50p leaving 75 – 50 = 25p, then proffer 20p leaving 25 – 20 = 5p and finally proffer 5p leaving 5 – 5 = 0p. Greedy algorithms like this are not necessarily globally optimal but with the currencies and denominations listed above they are. This means that these denominations are good models for sets of coordinated modular sizes.
If favourite numbers are used for component sizes a greedy algorithm guarantees that any overall size can be fulfilled with the minimum number of components.
Figure 2: Favourite Numbers and Greedy Algorithm
Saliency of Favourite Numbers
Round numbers are a device that lets language express the inaccuracy and uncertainty of our perception of larger numbers as presented by the Approximate Number System. These numbers and especially their Favourite Number subset have great saliency and are the numbers that we prefer to see when measuring and prefer to use when designing.
Are there musical equivalents of these rules
In rounded numbers, the lower number needs to be a multiple of the difference. It’s fine to say “there were about 200 – 250 people there”; “there were 150 – 200 people there”; but “there were about 150 – 250 people there” feels strange.
200 is multiple of 250 – 200 = 50 OK
150 is multiple of 200 – 150 = 50 OK
150 is not multiple of 250 – 150 = 100 NOT OK
Corresponding to the rules?
Yes, that’s what’s I’m suggesting – I didn’t see a requirement for the round numbers to be a multiple of the difference. But it’s a suggestion -perhaps there are counter-examples?
Note that “there are about 150 – 250 people” is directly equivalent to “there are roughly 200 people, give or take 50 or so”, which *is* acceptable
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