Digit

“ 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 notation of the decimal form is such that the content of DL 10 can be passed out onto a card in the usual way and if the card is then listed the digits of the numbers α,m will then appear on the listing paper in the usual way. Or in other words only the first 10 minor cycles of DL 10 are used, and a decimal digit is represented by the minor cycle in which a pulse occurs, and its significance by the position of it within the minor cycle. ”

“ The principal rule—known as the "sluggard's rule"—given for this purpose during the last half of the sixteenth century, the half century after the time of Luther, Melanchthon, and Erasmus, was this: Subtract each digit from ten, and write down the differences; multiply the differences together and add as many tens to their product as the first digit exceeds the second difference, or the second digit the first difference. ”

“ Digit by article. Whan a digit multipliethe an article, thow most bryng the digit into þe digit, of þe whiche the article [has] 13 his name, and euery vnyte 41 shalle stonde for .10., and euery article an .100. (3) Composite by digit. Whan the digit multipliethe a nombre componede, þou most bryng the digit into aiþer part of the nombre componede, so þat digit be had into digit by the first rule, into an article by þe seconde rule ”

“ Such expressions, curious as they seem at first thought, are no more than the natural outgrowth of systems built up by the slow and tedious process which so often obtains among primitive races, where digit numerals are combined in an almost endless variety of ways, and where mere reduplication often serves in place of any independent names for higher units. ”

“ The least significant digit appears at time P1 and the most significant at P32. In the partial sums register similarly the least significant digit is to be considered to be in the units position and the most significant to have reversed sign and to appear 63 pulses later. In order to keep track of which part of the partial sum is available at any moment we have a signal ODD which is stimulated during the first minor cycle of the stimulation of S, and thereafter in alternate minor cycles so long as S is stimulated. ”

“ 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 ”

“ Since each disc can present any digit, and each circle any sign, the discs of every column may be so adjusted [6] as to express any positive or negative number whatever within the limits of the machine; which limits depend on the perpendicular extent of the mechanism, that is, on the number of discs to a column.
Each of the squares below the zeros is intended for the inscription of any general symbol or combination of symbols we please; it being understood that the number represented on the column immediately above, is the numerical value of that symbol, or combination of symbols. ”

“ Method.—The quotient consists of three equal digits; the digits of the number taken are: 1st. one of these equal digits; 2d. this digit increased by a unit; 3d. this digit diminished by a unit. The same principle holds when the digits of the number taken differ by 2, 3, or 4. It is a very pretty problem to prove that the sum is always divisible by 9 and 18.
145Take any number, divide it by 9, and name the remainder. Multiply the number by some number which I name, and divide the product by 9, and I will name the remainder. ”

“ This process is called multiplying by right- and left-hand components. It has the great advantage that no carrying is necessary to complete any line of the original multiplications. Some computing machines use this process. Built into the hardware of the machine is a multiplication table up to 9 × 9. The machine, therefore, can find automatically the right-hand digit and the left-hand digit of any partial product. ”

“ In this system, then, the only digit employed is 1. The 0 plays the same important part in it as in the decimal system. It multiplies the figure that immediately precedes it by the value of the radix. The symbol 40, in our denary scale, represents ten times four; in the octenary it would denote eight times four, and in the quinary five times four. These two symbols, 1 and 0, then, are the only ones that enter into calculations. It is evident that thought in arithmetical work is almost superseded, and that all numerical operations are reduced to the manual labor of writing. ”

“ Now if, as in this case, all the carriages were known, it would then be possible to make all the additions of digits at the same time, provided we could also record each carriage as it became due. We might then complete the addition by adding, at the same instant, each carriage in its proper place. The process would then stand thus:—
Now, whatever mechanism is contrived for adding any one digit to any other must, of course, be able to add the largest digit, nine, to that other digit. ”

“ The character of the pattern on any digit is determined by causes of whose precise nature we are ignorant; but we may rest assured that they are numerous and variable, and that their variations are in large part independent of one another. ”

“ When the multiplicand is taken from TS 3 we have to make some slight modifications to it before it is in suitable condition for adding into the partial sum. We have to convert the periodic signal with period 32 or 34 into a sequence of 64 digits of which 32 form the original content of TS 3, and the rest is a sort of padding. We may call the 32 digits the genuine digits. Those digits of padding which are less significant than the genuine digits are to be all zero, those which are more significant are to be the same as the most significant genuine digit. ”

“ The digit 1 always represents a pedestrian who moves just one step, and that means from one unit into the next; the digit 2 a horse, which moves twice as fast, that is, which moves 2 units; and the digit 3 an automobile which moves three times as fast, that is, 3 units. Moreover, the black digits stand for men, horses, and automobiles which move parallel to the track and cannot cross the track, and are therefore to be disregarded in looking out for dangers. The red digits, on the other hand, are the dangerous ones. ”

“ Familiarity with the Number Code given in the book on Memory, will aid the child in keeping the result of a problem. The numbers of the answer can quickly take the form of an object which can be translated again into the correct numbers. Many children will not be able to hold the visual picture [Pg 71] of the digits for any length of time. There is considerable difference in the ability to hold the visual picture of the digit 127. ”

“ I must content myself with showing one little principle involved in this puzzle. The sum of the digits in the total is always governed by the digit omitted. 9/9 - 7/10 - 5/11 - 3/12 - 1/13 - 8/14 - 6/15 - 4/16 - 2/17 - 0/18. ”

“ For any cipher whatever in which the substitutes are two-digit numbers, a frequency count taken in chart form is usually far more convenient than one made by listing the numbers in advance. With only the ten digits, the 100 cells can be made larger than the 676 cells needed for letters, and the chart still be small and compact. The pairs of digits would be counted in exactly the same way as so many digrams. With numbers, it is sometimes possible to take the subsequent digram count, also, on a chart. Solution, in many cases, involves pure guess-work. ”

“ What if it be a digit. ¶ Here is þe thryde case, þe quych is þis: yf hit happe þat of þi multiplicaciouɳ come a digit, þou schalt write þe digit ouer þe hede of þe neþer figure, by the quych þou multipliest þe hiere figure, for þis nedes no Ensampul. ”

“ Arrange these figures so that the odd ones and the even ones add up alike. Complex and improper fractions and recurring decimals are not allowed.
79—THE LOCKERS PUZZLE.
A man had in his office three cupboards, each containing nine lockers, as shown in the diagram. He told his clerk to place a different one-figure number on each locker of cupboard A, and to do the same in the case of B, and of C. As we are here allowed to call nought a digit, and he was not prohibited from using nought as a number, he clearly had the option of omitting any one of ten digits from each cupboard. ”

“ How to write a number, ¶ here he telles how þou schalt wyrch whan þou schalt write a nombur. Expone versum sic, & fac iuxta exponentis sentenciam; whan þou hast a nombur to write, loke fyrst what maner nombur it ys þat þou schalt write, whether it be a digit or a composit or an Articul. if it is a digit; ¶ If he be a digit, write a digit, as yf it be seuen, write seuen & write þat digit in þe first place toward þe ryght side. if it is a composite. If it be a composyt, write þe digit of þe composit in þe first place & write þe articul of þat digit in þe secunde place next toward þe lyft side. ”

“ 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. ”

“ It is thus evident, that which-ever key is pressed down, a given number of teeth in the wheel B, will be taken and secured by the catch E; and, afterwards, the chain be again stretched by the spring F. It may be remarked, that, in the figure, all the keys are supposed pressed down: so as to turn the wheel B, a number of teeth equal to the sum of the digits 1, 2, 3—to 9. But this is merely supposed to shew the increasing deflexion of the chain, as the digits increase: for the fact can hardly ever occur. ”

“ 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. ”

“ The tendencies of digits to resemble one another will now be considered in their various combinations. They will be taken two at a time, in order to learn the frequency with which both members of the various couplets are affected by the same A. L. W. class of pattern. Every combination will be discussed, except those into which the little finger enters. These are omitted, because the overwhelming frequency of loops in the little fingers would [Pg 120] make the results of comparatively little interest, while their insertion would greatly increase the size of the table. ”

“ Then find the second digit. Than me most fynde a digit vndre the next figure bifore the triplat, the whiche with his vnder-trebille had into a trebille, afterwarde other vnder [trebille] 26 had in his produccioun, puttethe a-way alle that is ouer it in regarde of27 [the triplat. Then lade in hymself puttithe away that þat is over his hede as in respect of hym, other as nyghe as thou maist:] Multiply the first triplate and the second digit, twice by this digit. ”

“ But germinal variation may affect the rudimentary as well as the functional digits; if through such variation the supposed rudiment of a thumb develops into a digit with two phalanges, germinal variation and reversion are one and the same thing.
To attempt to reconcile the conflicting statements of various investigators the writer has made a comparative study of polydactylism in mammals normally possessing less than five toes. It was found that in the majority of cases the extra digits are developed from the so-called digital rudiments. ”

“ It was this pentadactyl peculiarity of the frog, inherited by men and women through the reptiles and mammals, that gave rise to the decimal system of numbers and other unhandy facts in human life. The decimal system arose out of the practice of early men performing their calculations on their fingers. This method of calculating is still used by primitive peoples all over the world. The sum of the digits of the two hands came, in the course of arithmetical evolution, to be used as a unit, and from this simple beginning grew up the complicated system of tens found among civilised peoples. ”