13. As the place in which I represents one UNIT is termed the units' place of the UNITS' periodso, 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. : There are places to the right, as well as to the left, of the units' place; so that the right-hand figure of a combination is not always the units' figure.* Returning to * 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. - hundreds' place units' place the combination at page 7, 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 It thus appears 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; but more than this appears: on.* 14. In the place immediately to the right of the units' place, I would be ten times less in value thanor would represent the tenth part of a unit; 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; in the next place on the right, I would be a thousand times less in value than—or would represent the thousandth part of—a unit; &c. 15. As the values of the first three figures to the left of the units' period are expressed in thousands; of the next three, in millions; of the next three, in billions; &c.-so, the values of the first three figures * 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. - one UNIT -one UNIT to the right of the units' period are expressed in thousandths; of the next three, in millionths; of the next three, in billionths; &c. Let us re-write the last combination, omitting the names which we intend to dispense with, and observing where I represents, respectively, one thousandth, one millionth, and one billionth :— Now, if we remember that I becomes ten times greater in value when removed one place, and a hundred times greater when removed two places, to the left, it will be evident that the number of THOUSANDTHS which the digit represents in the place marked (b) is ten, and in the place marked (a) one hundred—the number represented in the place marked (c) being one; that the number of MILLIONTHS represented in the place marked (e) is ten, and in the place marked (d) one hundred-the number represented in the place marked (ƒ) being one; and that the number of BILLIONTHS represented in the place marked (h) is ten, and in the place marked (g) one hundred—the number represented in the place marked (k) being one:— - one billionth -one UNIT To the right, therefore, as well as to the left, of the units' place, I is everywhere read "one," or "ten," or 66 one hundred": thus-one thousandth, one millionth, one billionth, &c.; ten thousandths, ten millionths, ten billionths, &c.; one hundred thousandths, one hundred millionths, one hundred billionths, &c. 16. The place in which I represents one THOUSANDTH is called the units' place of the THOUSANDTHS' period; the place in which I represents one MILLIONTH, the units' place of the MILLIONTHS' period; the place in which I represents one BILLIONTH, the units' place of the BILLIONTHS' period; &c. The place in which I represents ten THOUSANDTHS is called the tens' place of the THOUSANDTHS' period; the place in which I represents ten MILLIONTHS, the tens' place of the MILLIONTHS' period; the place in which I represents ten BILLIONTHS, the tens' place of the BILLIONTHS' period; &c. The place in which I stands for one hundred THOUSANDTHS is termed the hundreds' place of the THOUSANDTHS' period; the place in which I stands for one hundred MILLIONTHS, the hundreds' place of the MILLIONTHS' period; the place in which I stands for one hundred BILLIONTHS, the hundreds' place of the BILLIONTHS' period; &c. : We see, then, that every other period, as well as the UNITS' period, has a units', a tens' and a hundreds' place; - hundreds' place - units' place that the places of every other period occupy the same relative positions as those of the UNITS' period-the units' place being invariably on the right, the hundreds' place on the left, and the tens' place in the middle; and that, immediately to the left of the hundreds' place of one period, is found the units' place of the next higher period-in other words, that, immediately to the right of the units' place of one period, is found the hundreds' place of the next lower period. In every situation, therefore, a figure occupies the units', or the tens', or the hundreds' place of some period 17. The units' place (of the UNITS' period) has a point, called the DECIMAL POINT, immediately to the right of it. When, however, no figure occurs farther to the right, the point is left unwritten. Accordingly, in every case in which the decimal point is written, the figure immediately to the left of it is the units' figure; and in every case in which the point is not written, the right-hand one is the units' figure. Thus, the units' place is occupied by 4 in the first of the following combinations, by 3 in the second, and by 2 in the third; whilst, in the fourth combination, the units' place is unoccupied, being the place immediately to the left of that in which the digit 2 occurs * (a) 234 (b) 23'4 (c) 2.34 (d) 234 *It will be well to remember that the decimal point does not occupy a "place." |