Scales of Notation — Chambers's Encyclopaedia

Scales of Notation, in Arithmetic, have to do with the representation of numbers of any magnitude by means of a few symbols. We ordinarily express numbers in terms of the first nine digit symbols and the symbol known as the cipher—i.e. ten in all. The number ‘ten’ is then represented by 10, a combination of the ‘one’ and cipher symbols, and so on in the familiar manner. Mathematically there is no reason why ten should be chosen in preference to any other number as the radix of our common scale of notation. Its convenience arises from the way in which it suits our numeration or naming of numbers. Historically the development of decimal arithmetic—of which our denary scale is the highest phase—is intimately connected with the fact that man has ten fingers. The full significance of the denary scale will be best seen if we take some other number, seven say, as the radix. Our object is now to express all numbers in terms of the cipher and the first six digits only. The number seven itself will be represented by 10, eight by 11, twelve by 15, fourteen by 20, forty-nine, or seven times seven, by 100, and so on. In other words, 49 to radix ten is the same number as 100 to radix seven. As another example take the number of days in the year, and let it be expressed in terms of scales whose radices are twelve, ten, seven, and three. Remembering that 365 to radix ten is a concise notation for $3 \times 10^2 + 6 \times 10 + 5$ , we must, in order to express it in the duodenary scale, throw it into a similar form with twelve substituted for ten. We find easily

\begin{aligned} 365 &= 30 \times 12 + 5 \\ &= (2 \times 12 + 6) \times 12 + 5 \\ &= 2 \times 12^2 + 6 \times 12 + 5. \end{aligned}

Hence 365 to radix ten is the same number as 265 to radix twelve. In the other cases we get

\begin{aligned} 365 \text{ to radix ten} &= 265 \text{ to radix twelve} \\ &= 1 \times 7^3 + 0 \times 7^2 + 3 \times 7 + 1, \text{ or } 1031 \text{ to radix seven} \\ &= 1 \times 3^5 + 1 \times 3^4 + 1 \times 3^3 + 1 \times 3^2 + 1 \times 3 + 2, \\ &\quad \text{or } 111112 \text{ to radix three.} \end{aligned}

These examples show that the simplicity of having a few symbols is balanced by the disadvantage of having to use long expressions for large numbers. The attempt to work in other than the denary scale is moreover greatly hampered by our lifelong habit of thinking and naming our numbers according to a decimal system.

The fact that there are twelve pence to the shilling and twelve inches to the foot has often suggested the introduction of the duodenary scale. According to this scale twenty-four feet nine inches would be represented symbolically by 20·9. To use this scale we should be compelled to invent two new symbols for what we call ten and eleven. But unless we altered our numeration so as to be in accord with the symbolism, the method would be impracticable. For example, we should have to rename fourteen and twenty-six so as to bring them into line with their duodenary symbols 12 and 22. At present in all calculations involving shillings and pence or feet and inches we are compelled to work partly in the duodenary scale; but the numbers themselves are expressed both symbolically and verbally according to the denary scale and decimal nomenclature. As a matter of history the denary scale is the only one that has ever been used purely; to establish any other would necessitate a complete revolution in modes of thought and habits of language.

Very similar to the mixture of decimals and duodecimals in the examples just given is the method of sexagesimals, which still survives in the subdivision of hours and degrees. There are sixty minutes to the hour (or degree), and sixty seconds to the minute. This method is of great antiquity, and had no doubt an astronomical origin. To the early astronomers it offered special facilities for calculation and for representation of fractions. It was used extensively by Ptolemy and the Alexandrian mathematicians, who employed for symbols the usual Greek numerals as far as the symbol for sixty (see NUMERALS). At its best, however, the sexagesimal notation must have been very cumbersome, even when assisted, as it probably was, by use of the Abacus (q.v.). It is evident that it does not form a pure scale; to do so sixty distinct symbols including the zero would be required. The Alexandrians no doubt borrowed the system from the Chaldeans. In the older Babylonian inscriptions there is found a sexagesimal notation identical, in so far as it is a notation, with that used by Ptolemy and his school. The symbolism is of course quite different, all numbers being represented by appropriate combinations of two cuneiform characters. The numbers up to nine are represented each by the proper number of the simple wedge-shaped character. Ten is symbolised by the angle-shaped character, two of which give twenty, three thirty, four forty, and five fifty. Sixty, however, is represented by the same simple character as one; five times sixty, or 300, by the same character as five, and so on. The famous tables of Senkereh contain the squares and cubes of all numbers from 1 to 60, expressed in terms of this sexagesimal notation. A few examples will suffice. Thus we find A diagram showing a square of a sexagesimal number. It consists of two large brackets containing smaller symbols. The left bracket contains three 'V' symbols and the right bracket contains three 'Y' symbols. The entire structure is enclosed in a larger set of brackets. given as the square of A diagram showing a sexagesimal number. It consists of three 'V' symbols and three 'Y' symbols, enclosed in a set of brackets. —i.e. $48 \times 60 + 36 = 2916 = (54)^2$ .

We may most simply exhibit the Babylonian method by using heavy figures for the tens and light figures for the units. Thus the above example would be translated 4836 = square of 54. Others from the table of Senkereh are 12153 = cube of 17 $(1 \times 60^2 + 21 \times 60 + 53 = 4913 = 17^3)$ 73 = cube of 3 (i.e. thirty).

This last must mean $7 \times 60^2 + 30 \times 60$ , although there is nothing in the notation to show what place in the sexagesimal representation is to be occupied by the 3 (or thirty). The example is instructive as showing how far short the Accadians and Assyrians fell of our modern cipher system. It is clear, however, that they were in possession of a sexagesimal scale as true and as complete as the much later Alexandrians. It was used probably only for purposes of calculation; for in simply representing numbers the Assyrians, if not the earlier Accadians, used another scale, in which a special symbol for the hundred was introduced. In this scale, however, the sexagesimal symbolism for 60, 70, 80, and 90 was retained. In the later cuneiform inscriptions of the Persians all trace of the sexagesimal scale is obliterated.