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In order to understand the peculiar principle upon which larger numbers than nine are expressed in figures, let us suppose the case of a teacher who employs a class of boys to keep an account, with their fingers, of the number of nuts which he removes, one by one, from a heap before him-the first boy putting up a finger for every nut removed, and beginning again when his fingers are all up; the second boy putting up a finger for every occasion on which the first boy is obliged to begin again ; the third boy putting up a finger for every occasion on which the second boy-his fingers being all up—is obliged to begin again ; and so on. It is evident that, according to this arrangement, every finger held up by the first boy would represent one nut; every finger held up by the second boy, ten nuts (or as many as the first boy's ten fingers); every tinger held up by the third boy, one hundred nuts (or as many as the second boy's ten fingers) ; &c. So that—to take a particular case-four of the first boy's fingers, three of the second boy's, and two of the third boy's would represent, respectively, four nuts, thirty nuts, and two hundred nuts ; altogether, two hundred and thirty-four. The following is the way in which this number would be expressed if every finger were represented by a “stroke”-the boys being supposed to stand in a row, and in the order indicated :
(3rd boy)|(and boy)| (1st boy)
) If, however, instead of four, three, and two strokes, we wrote the digits 4, 3, and 2, respectively, the number two hundred and thirty-four would appear under this more concise form
234. It is easy to see that, upon the principle just explained, these digits, if differently combined, would express a different number. For instance: three of the third boy's fingers, four of the second boy's, and two of the first boy's would represent (altogether) three hundred and forty-two, which would be written
342; four of the third boy's fingers, two of the second boy's, and three of the first boy's would represent four hundred and twentythree, which would be written
423; &c. Comparing the three combinations 234, 342, and 423, we see that the same digit has different values in different places, and that different digits have different values in the same place. Thus, 4 represents four in the first combination, forty in the second, and four hundred in the third ; 3 represents thirty in the first combination, three hundred in the second, and three in
the third ; whilst 2 represents two hundred in the first combination, two in the second, and twenty in the third.
After the removal of five hundred and sixty nuts, five of the third boy's fingers, six of the second boy's, and none of the first boy's would be up. This number of nuts would be written
560: the cipher, whilst representing no portion of the number, being necessary in the first place, that 6 may stand in the second, and 5 in the third place; just as the first boy, although having no finger up, would be obliged to keep his place, that there may be no mistake as to the places of the other two boys.
8. When a number is represented by a combination of figures, we find what number it is by adding the individual values of the figures together.
Thus, the combination 365 represents three hundred and sixtyfive-3 representing three hundred ; 6, sixty; and 5, five. The combination 708 represents seren hundred and eight—the value of 7 being seven hundred ; (of o, nothing ;) and of 8, eight.
9. The cipher has no value in any situation: it merely serves, in certain cases (when no other figure would answer), to keep the digits in their proper places.
10. The value of a digit depends—partly upon what digit it is, and partly upon the place it occupies.
The mere name does not indicate the value of a digit, because, as we have seen, “the same digit has different values in
* In almost every treatise on Arithmetic, we are told that “a digit has two values--simple and local.” To be convinced of the absurdity of this, we have merely to take any combination of figuressay 789, and reflect whether, in this combination, 9 has any other value than nine, or 8 any other value than eighty, or 7 any other value than seven hundred. Those who employ the terms simple and local must intend “simple” to mean “non-local;" but when has a digit a nonlocal value? If it be said that the simple or non-local value of the digit 9, for example, is nine, the answer is this: 9 must occupy a particular “ place” (the units' place) to stand for nine—just as it must be in a particular place to represent ninety or nine hundred; so that nine is as “local" a value as any other. The truth is that a digit might have any number of values, because it could be written in any number of different places, in no two of which would its value be the same. So long, however, as it remains in the same place (whatever the place may be), a digit has one and only one-value.
different places.” The digit called “one (1), for example, sometimes represents ons, sometimes ten, sometimes one hundred, &c. Neither does its place indicate what a digit stands for, because " different digits have different values in the same place." In the place in which I would stand for one, 2 would stand for two, 3 for three, &c.; in the plaee in which i would stand for ten, 2 would stand for twenty, 3 for thirty, &c.; in the place where I would stand for one hundred, 2 would stand for tuo hundred, 3 for three hundred, &c.
11. The place in which the digit i would represent a unit is termed the “units' place.” The next place on the left, where I would represent ten (units), is called the “tens' place.” The next place on the left, where I would stand for one hundred (units), is called the “hundreds place.” [Thus, in the combination 365, the units' place is occupied by 5, the tens' place by 6, and the hundreds' place by 3; because the digit i would represent one if substituted for 5, ten if substituted for 6, and one hundred if substituted for 3.] In the places farther to the left, I would represent, respectively, one thousand, ten thousands, one hundred thousands; one million, ten millions, one hundred millions ; one billion, ten billions, one hundred billions; &c. :
12. The first three places in which “units” occur) constitute what is called the UNITS' “period;" the next three (in which “thousands” occur), the THOUSANDS' period; the next three (in which millions” occur), the MILLIONS' period ; &c.
Farther to the left would come TRILLIONS, QUADRILLIONS, QUINTILLIONS, SEXTILLIONS, SEPTILLIONS, OCTILLIONS, NONIL- *
LIONS, &c., respectively : a trillion being a thousand billions ; a quadrillion, a thousand trillions ; a quintillion, a thousand quadrillions ; &c.* It will be seen that i is everywhere read “one,” or
ten," one hundred”: thus—one unit, one thousand, one million, &c.; ten units, ten thousands, ten millions, &c. ; une hundred units, one hundred thousands, one hundred millions, &c.
13. As the place in which i represents one UNIT is termed the units' place of the Units' period— so, the place in which i represents one THOUSAND is termed the units' place of the THOUSANDS' period; the place in which i represents one MILLION, the units' place of the MILLIONS' period; &c. Again : as the place in which i represents ten UNITS is termed the tens' place of the UNITS' period—so, the place in which i represents ten THOUSANDS is termed the tens' place of the THOUSANDS' period ; the place in which i represents ten MILLIONS, the tens' place of the MILLIONS' period ; &c. In like manner, as the place in which i stands for one hundred UNITS is called the hundreds' place of the UNITS' period— so, the place in which I stands for one hundred THOUSANDS is called the hundreds place of the THOUSANDS' period; the place in which I stands for one hundred MILLIONS, the hundreds' place of the MILLIONS' period ; &c. :
- hundreds' place
- hundreds' place
- hundreds' place
* In the English system of notation, a combination of figures would be divided into periods of six each-a billion meaning a million of millions; a trillion, a million of billions; a quadrillion, a million of trillions; &c. This system may be said to have been entirely superseded by the one described above, and which is sometimes spoken of, for the sake of distinction, as the French or Continental system of notation.
14. There are places to the right, as well as to the left, of the units' place; so that the most righthand figure of a combination is not always the units' figure.* Looking at the combination before us, we observe that, as we pass from left to right, the digit i becomes continually smaller in value, until it represents only a unit. We observe, moreover, that the digit becomes ten times less in value when removed one place to the right, one hundred times less when removed two places to the right, one thousand times less when removed three places to the right, and so on.t Bearing this in mind, we are prepared to be told that, in any of the places to the right of the units' place, I would be less in value than-or would represent only part of—a unit; and further, that, in the place immediately to the right of the units' place, I would be ten times less in value than-or would represent the tenth part of-a unit; that, in the next place on the right, I would be a hundred times less in value than—or would represent the hundredth part of--a unit; that, in the next place on the right, I would be a thousand times less in value than-or would represent the thousandth part of- unit; &c.
* When we speak of the “units' place,” or the “units' figure,” without mentioning any period in particular, the UNITS' period is understood to be the one referred to.'
† This is merely another way of saying that the digit becomes ten times greater in value when removed one place to the left, one hundred times greater when removed two places to the left, one thousand times greater when removed three places to the left, &c.