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figure on the left will be tens, with regard to that value, and the next hundreds. Wherefore, in each period, going from right to left, we say units, tens, hundreds : that is to say, with respect to the digits, any figure on the right is a number of units; in the middle, a number of tens, and on the left, a number of hundreds. Hence, we read the number, under any period, as if it stood alone, except merely at last pronouncing the name of the period. Thus, the number 243 is read two hundred and forty-three, under whatever period it may stand. For example, in the following number:
243 243 243 Beginning at the left, we say, two hundred and forty-three Millions; two hundred and forty-three Thousand; two hundred and forty-three.
47. Hence also, by its place, we easily determine the value of each figure in a number, without reference to the others. Thus, in the above, beginning at the left, the figure 2 is read, first, two hundred Millions; then, two hundred thousand; and, lastly, two hundred. The figure 4 is, first, forty Millions; then, forty Thousand ; and, lastly, forty. The figure 3 is, first, three Millions ; then, three s'housand; and, lastly, three.
48. Again : from the same universal law, it is evident that we can, in a large number, read a part, consisting of a number of consecutive figures, taken at pleasure, (the left-hand figure of such part being significant,) as a number of units of the order signified by the place of the right-hand figure of the part taken. Thus, in the number
128 029 625 343 One hundred and twenty-eight Billions; twenty-nine Millions; six hundred and twenty-five Thousand, three hundred and forty-three: taking the figures two and two in order, we read, twelve tens of Billions ; eighty hundreds of Millions; twenty-nine Millions ; sixty-two tens of Thousands; fifty-three hundreds and forty-three : or, four and four, we read, one thousand two hundred and eighty hundreds of Millions; two thousand nine hundred and sixty-two tens of Thousands ; five thousand three hundred and forty-three: or, taking the two middle figures, we read, ninety-six hundreds of thousands: or, taking eight figures, beginning with the figure 8, we read
eighty Millions, two hundred and ninety-six Thousand, two hundred and fifty-three hundreds, or any other part in the same manner, at pleasure.
49. From the above, we easily perceive that any number, having a significant figure in the place of units, may be considered as a binomial; (bis, twice, and nomen, name ;) that is, as consisting of two names or parts, one of which is the number of tens expressed by all the figures on the left of its unit figure; and the other the number of units expressed by that figure. Thus, the above number, considered as a binomial, consists of twelve Billions, eight hundred and two Millions, nine hundred and sixty-two Thousand, five hundred and thirty-four tens and three units. This general binomial property will be found important in some of our future details.
50. When some of the intermediate orders or periods of a number contain ciphers only, we omit such orders or periods in reading, as also the name Units : because, all numbers being composed of units, that name is applicable to any order or period. Thus, the following number,
1 000 009 022 is read One Billion; nine Thousand and twenty-two.
Let the following numbers be expressed in words: 1. 101 4. 1001 7. 10010 10. 900090 2. 110 5. 1010 8. 10002 11. 300100 3. 111 6. 1100 9. 10120 12. 506011 13. 5006030
16. 694032580795251 14. 40206017
17. 875875875875875 15. 95000070004 18. 12000679800979040
20. 83006500010110099580000428 51. When familiar with the Table, and the names derived from it, we dispense with actually pointing the periods and writing their names. Thus, in the number
65128075000095008 beginning at the right hand and taking the figures three at a time, we count the number of periods, saying, 1, 2, 3, 4, 5, 6, and (42) subtracting 2, we have 4 for remainder, by the Table quatuor, and hence, quadrillions, for the name of the highest period. Wherefore, having in this the figures 65, we read sixty-five Quadrillions; four hundred and twenty-eight Trillions; seventy-five Billions; ninety-five Thousand and eight.
To read this as a binomial, we begin with the tens in counting the periods; and finding 6 only in the highest, we say, sic Quadrillions; five hundred and forty-two Trillions; eight hundred and seven Billions ; five hundred Millions ; nine thousand five hundred TENS, and eight UNITS.
In like manner, let each of the following numbers be read as a Natural and as a Binominal Number :
4. 6850685006850006850000685 Let the following number be read in parts, taking the figures consecutively two and two, four and four, and five and five, as in Art. 48:
610261026 10261026402 52. To express, in figures, a large number given in words, we first write the names of the periods, beginning with Units and ending with the highest or first name mentioned in the reading. Then, beginning with the left-hand period, we write its number underneath, as if it were to stand alone. As we progress from left to right, the order of the places is, under each period, hundreds, tens, units; consequently, in writing its number, we first consider whether it contains hundreds, and write a cipher or digit in the left-hand place accordingly. We next consider whether it contains tens; and write, accordingly, a cipher, or digit in the middle place. Lastly, in like manner we supply the place of units.
If, in the reading, any period is wanting-that is, not mentioned we must, of course, write 000 under it.
The cipher is of no use, when placed on the left of all the significant figures; because, in this situation, it does not show their position towards the left.
To express, in figures, the number Forty Quintillions ; ten Quadrillions; two Billions; ten Millions; eleven Thousand, four hundred and twelve; we first write, as above directed, the names of the periods, Quintillions Quadrillions Trillions Billions Millions Thousands Units.
40 010 000 002 010 011 412 Then, as the number first dictates Forty Quintillions, we write 40 under Quintillions, as if it were to stand alone. Next, for the ten Quadrillions, as this number contains no hundreds, we first write 0 in the place of hundreds; then, as ten is expressed by a unit in the place of tens, we write 1 in that place. Lastly, as there no units, we write 0 in the place of units : thus, we have 010 (nought, one, nought) under quadrillions. Next, as there are no Trillions mentioned in the number, we write 000 (nought, nought, nought) under Trillions. Then, for the two Billions, as this number contains neither hundreds nor tens, we first write 0 in the place of bundreds; then, 0 in the place of tens; and, lastly, 2 in the place of units: thus we have 002 (nought, nought, two) under Billions. We next write ten Millions, as we did ten Quadrillions, namely, 010, (nought, one, nought,) under Millions. Then, for eleven Thousand, as this is ten and one, we first write 0 in the place of hundreds; then 1 in the place of tens; and, lastly, 1' in the place of units: thus, we have 011 (nought, one, one) under thousands. Lastly, we write four hundred and twelve, 412, (four, one, two,) under Units, as we should under any other period, namely, as if it were to stand alone.
Let the following Numbers be expressed in Figures : 1. Eighty Thousand, six hundred and twenty. 2. Eleven Millions ; four hundred Thousand and four. 3. Fourteen Millions, nineteen Thousand and eleven. 4. Five hundred and four Millions; one hundred Thousand
5. Nine hundred and nine Billions; thirty Thousand and ten.
6. Two Trillions; three hundred Billions; forty Millions ; four Thousand.
7. One hundred and ninety Quadrillions; twelve Billions; one Thousand.
8. Fourteen Nonillions; sixteen Sextillions; three hundred and twenty-six Quintillions; fourteen Millions and six.
9. Five hundred Decillions; eight Octillions; twenty Septillions; one hundred and one Quadrillions; one hundred and eleven Trillions; nine Billions ; five hundred Thousand and forty-two.
10. Ten Duodecillions; one hundred and nineteen Nonillions; fifteen Octillions; thirty Septillions; four Quintillions ; five Trillions; fifty Billions; ninety-one Millions; one hundred and ten Thousand and thirteen.
53. The Unit, which is repeated or multiplied in forming a number, we shall call the Genéric Unit, (genus, Lat., kind, sort,) because it shows of what kind the number is. Hence the abstract unit, represented by the figure 1, is the generic unit of all natural whole numbers. Also, in general, the
unit from which a number derives its name is the generic unit of that number.
54. We have seen (49) that a natural number, having its unit figure significant, may be considered binomial. This we shall call a Natural Binomial, (Nat. Bin.) But a number of consecutive figures, taken at pleasure, is (48 and 53) read as a natural number, having, for its generic unit, a unit of the order of its right-hand figure: consequently, if this figure be significant, the number expressed by those figures may also be considered binomial, having for its units the number expressed by its right-hand figure, and, for its tens, the number expressed by the figures contained in it on the left of that figure. This we shall call a Factitious Binomial, (Fac. Bin.)
Wherefore, in the same number; though we can have but one Natural Binomial, we may constitute many Factitious
55. If, on the right of a Nat. Bin., we place a cipher, it ceases to be binomial, the whole being now (48) a number of tens.* It may, however, be considered a Fac. Bin., which may be read as before, with this condition, that the generic unit of each of its parts is ten times as great as before: hence, the number is multiplied by ten.
Again : if another cipher be placed on the right, it is (48) read as a whole, a number of hundreds : and as a Fac. Bin. the generic unit of each part is 100 times as great as at first. Hence the number is multiplied by 100.
Thus we see, that to multiply a number by 10, 100, 1000, &c., we have only to place 1, 2, 3, &c. ciphers on the right.
56. Again : in a number having 0 on the right, the part on the left of this 0, being a number of tens, is the quotient found in dividing the number by 10; because it shows how many times 10 can be subtracted. If there be two ciphers, the part on the left, being hundreds, shows how many times 100 can be subtracted : and, hence, it is the quotient in dividing by 100.
Wherefore, to cut off 1, 2, 3, &c., ciphers on the right of a number, is to divide that number by 10, 100, 1000, &c.
57. When a unit or number is repeated two or more times, the sum found by adding or combining the whole together is said to be the product of two numbers multiplied
* A number having one name is Monomial, (monos, Greek, one.)