Numerator

“ In the case of decimal fractions the psychological process is more complex still; for the pupil has given him one of the terms, the numerator, from which 20 he must mentally deduce the denominator from the position of the decimal point. This doubles the work of reading and recognizing the fractional number. But it makes addition and subtraction of fractions nearly as easy as that of simple numbers and assists also in multiplication of fractions. But division of decimals is a much more complex operation than that of common fractions. ”

“ The modern method is to deal with fractions which have 100 as denominator; such fractions are called percentages. They only apply accurately to divisions by 2, 4, 5, 10, 20, 25 or 50; but they have the convenience of fitting in with the denary scale of notation, and they can be extended to other divisions by using a mixed number as numerator. ”

“ But I know not by what reason you set that one counter for 500 betwene two lynes.
M. you shall remember this, that when so euer you nede to set downe 5, 50, or 500, or 5000, or so forth any other nomber, whose numerator 118 a. *is 5, you shall set one counter for it, in the next space aboue the lyne that it hath his denomination of, as in this example of that 500, bycause the numerator is 5, it must be set in a voyd space: and bycause the denominator is hundred, I knowe that his place is the voyde space next aboue hundredes, that is to say, aboue the thyrd lyne. ”

“ A vulgar fraction is reduced to a decimal by dividing the numerator (increased sufficiently with ciphers) by the denominator. Thus, 1⁄2 = ·5, 1⁄8 = ·125, &c.
The value of a decimal, of any denomination, is found by multiplying it by the number of parts in the next less denomination, and cutting off as many places to the right hand as there are decimals, and so on until the terms are exhausted. Thus, ·634 oz. ”

“ But is it not clear that supreme bliss and envy are only the numerator and the denominator, respectively, of the same fraction, happiness? ”

“ As men advanced in the use of numbers, they adopted a group derived from the fingers of both hands; thus ten became the base of numeration.
Notation, like numeration, began with ones, advanced to fives, then to tens, etc. Roman notation consisted of a series of signs signifying 1, 5, 10, 50, 100, 500, 1000, etc.,—a series evidently the result of counting by the five fingers and the two hands, the numbers signified being the products of continued multiplication by five and by two alternately. ”

“ It is a useful exercise to multiply and divide numbers represented in other scales of notation than the common or decimal one. The rules are in all respects the same for all systems, the number carried being always the radix of the system. Thus, in the quinary system we carry fives instead of tens. I now give an example of multiplication and division: [Pg 170]
Another way of turning a number from one scale into another is as follows: Multiply the first digit by the old radix in the new scale, and add the next digit ”

“ While considering the extent of the savage's arithmetical knowledge, of his ability to count and to grasp the meaning of number, it may not be amiss to ask ourselves the question, what is the extent of the development of our own number sense? To what limit can we absorb the idea of number, with a complete appreciation of the idea of the number of units involved in any written or spoken quantity? Our perfect system of numeration enables us to express without difficulty any desired number, no matter how great or how small it be. ”

“ From this last arrangement seems to have come the fundamental law of the decimal notation in which its superior utility consists, and upon which quite recently has been based the metric system of weights and measures. By placing any of the digits in the place of the zero to make the numbers between ten and twenty, we have the law established. The science of arithmetic, like all other sciences, was very limited and imperfect at the beginning, and the successive steps by which it has reached its present extension and perfection have been taken at long intervals, and among different nations. ”

“ Devices and illustrations of a material kind are necessary to build up in the pupil’s mind at the beginning a clear concept of a tenth, etc., etc., and then to show that one-tenth written as a decimal is only a shorthand way of writing 1/10 as a common fraction, and so on. He sees very soon that the decimal is only a shorthand common fraction, and this notion he must hold to. This is the vital point in decimals. The idea that they can be changed into common fractions and the reverse at will establishes the fact in the pupil’s mind that they are common fractions and not uncommon ones. ”

“ But now the important fact, at present ignored, and to which I draw your attention, is that it is perfectly possible to have all the facilities which a method of notation like that of decimals gives, along with all the facilities which duodecimal division gives. It needs only to introduce two additional digits for 10 and 11 to unite the advantages of both systems. The methods of calculation which now go along with the decimal system of numeration would be equally available were 12 made the basic number instead of 10. ”

“ The division of the fingers into two sets of five each is still in his mind, but it is no longer the leading idea. As the count proceeds further, the quinary base may be retained, or it may be supplanted by a decimal or a vigesimal base. How readily the one or the other may predominate is seen by a glance at the following numerals:
By the time 20 is reached the savage has probably allowed his conception of any aggregate to be so far modified that this number does not present itself to his mind as 4 fives. ”

“ Any such mode of counting, whether involving the use of the fingers or not, is to be regarded simply as an extraneous aid in the expression or comprehension of an idea which the mind cannot grasp, or cannot retain, without assistance. The German student scores his reckoning with chalk marks because he might otherwise forget; while the Andaman Islander counts on his fingers because he has no other method of counting,—or, in other words, of grasping the idea of number. A single illustration may be given which typifies all practical methods of numeration. ”

“ If a savage expresses the number 3 by the combination 2-1, it will at once be suspected that his numerals will, by the time he reaches 10 or 20, become so complex and confused that numbers as high as these will be expressed by finger pantomime rather than by words. Such is often the case; and the comment is frequently made by explorers that the tribes they have visited have no words for numbers higher than 3, 4, 5, 10, or 20, but that counting is carried beyond that point by the aid of fingers or other objects. ”

“ In nearly all the examples given the decimal becomes the leading element in the formation of all units above 100, just as was the case in the Celtic scales already noticed.
Among the northern tribes of Siberia the numeral scales appear to be ruder and less simple than those just examined, and the counting to be more consistently vigesimal than in any scale we have thus far met with. The two following examples are exceedingly interesting, as being among the best illustrations of counting by twenties that are to be found anywhere in the Old World. ”

“ Few tests of relative stages of civilisation are more trustworthy than that of the definite conception of high numbers. The prevalence of a decimal system of numerals among widely severed nations, alike in ancient and modern times, has been universally ascribed to the simple process of counting with the aid of the fingers. ”

“ If the strong man who had six fingers on each hand, and six toes on each foot, had invented a system of numeration, it would have gone in twelves, nearly like the duodecimals which our carpenters use; unless, indeed, he had been stupid after the manner of very strong men, and not gone beyond sixes. We see how the Romans, though they inherited from their Eastern ancestors a numeration by tens up to decem, and then beginning again undecim, &c., yet when they began to write a notation could get no farther than five—I., II., III., IV., V. ”

“ Select a multiplicand digit. 2. Select a multiplier digit. 3. Refer to the multiplication table with these digits. 4. Obtain the value of their product, called a partial product. 5. Add the preceding carry. 6. Set down the right-hand digit. 7. Carry the left-hand digit. We can, however, simplify this subroutine for a machine by delaying the carrying. We collect in one place all the right-hand digits of partial products, collect in another place all the left-hand digits, and delay all addition until the end. ”

“ A most inviting field for research is, however, furnished by the primitive methods of counting and of giving visible expression to the idea of number. Our starting-point must, of course, be the sign language, which always precedes intelligible speech; and which is so convenient and so expressive a method of communication that the human family, even in its most highly developed branches, never wholly lays it aside. It may, indeed, be stated as a universal law, that some practical method of numeration has, in the childhood of every nation or tribe, preceded the formation of numeral words. ”

“ Any other origin is difficult to reconcile with observed facts, and still more difficult to reconcile with any rational theory of number system development. Dismissing from consideration the quinary scale, let us briefly examine once more the natural process of evolution through which the decimal and the vigesimal scales come into being. After the completion of one count of the fingers the savage announces his result in some form which definitely states to his mind the fact that the end of a well-marked series has been reached. ”

“ The use of visible signs to represent numbers and aid reckoning is not only older than writing, but older than the development of numerical language on the denary system; we count by tens because our ancestors counted on their fingers and named numbers accordingly. So used, the fingers are really numerals, that is, visible numerical signs; and in antiquity the practice of counting by these natural signs prevailed in all classes of society. In the later times of antiquity the finger symbols were developed into a system capable of expressing all numbers below 10,000. ”

“ That the first numbers of a numeral scale are usually derived from other sources, we have some reason to believe; but that all above 2, 3, or at most 4, are almost universally of digital origin we must admit. Exception should properly be made of higher units, say 1000 or anything greater, which could not be expected to conform to any law of derivation governing the first few units of a system. ”

“ In this scale of notation every number is expressed by means of the ten digits, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, by giving each digit a local as well as its proper or natural value. The value of every digit increases in a tenfold proportion from the right towards the left ”

“ The fingers are, however, often employed in counting numbers far above the first decade. After giving the Il-Oigob numerals up to 60, Müller adds:8 “Above 60 all numbers, indicated by the proper figure pantomime, are expressed by means of the word ipi.” We know, moreover, that many of the American Indian tribes count one ten after another on their fingers; so that, whatever number they are endeavouring to indicate, we need feel no surprise if the savage continues to use his fingers throughout the entire extent of his counts. ”

“ The uniform employment of twelve as the divisor and multiplier in tables of weight, distance, time, and space, as well as in arithmetical notation, has all the conveniences of the decimal system of France, and some others besides due to the greater convenience of twelve as a base. ”

“ Here is a subtraction sum. We cannot take 9 units from 3 units, so we get change from the next row, that gives 13 units, from which we take 9, and have 4 left. We have nothing to take from our 8 remaining tens, so we write 8. We have no hundreds, so we cannot take away 9, but we change one of our thousands into 10 hundreds, and take away 9, leaving 1; lastly we take away 2 from our 4 thousands, and get 2—altogether 2184.
Decimal fractions.Now would come in naturally the extension of this system of notation to decimal fractions, marking the unit by a full stop. ”

“ Almost all civilized languages, however, except the Teutonic, and practically all uncivilized languages, begin their direct numeral combinations as soon as they have passed their number base, whatever that may be. To give an illustration, selected quite at random from among the barbarous tribes of Africa, the Ki-Swahili numeral scale runs as follows:53
The words for 11, 12, and 13, are seen at a glance to signify ten-and-one, ten-and-two, ten-and-three, and the count proceeds, as might be inferred, in a similar manner as far as the number system extends. ”

“ The theory that man’s primitive mode of counting was palpable reckoning on his hands, and the proof that many numerals in present use are actually derived from such a state of things, is a great step towards discovering the origin of numerals in general. Can we go farther, and state broadly the mental process by which savage men, having no numerals as yet in their language, came to invent them? ”