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2+1=3, or 1+2=3; which we read 1 plus 1 plus 1 equals three; 2 plus 1 equals 3, or 1 plus 2 equals 3.
b 22. Let the line
be 2 inches long, and 5
d the line
'3 inches long: then,
9 the middle of a b c being at b, and the middle of a b c d at g, half-way between 6 and c, it is plain that we can divide a b c evenly; that is, into 2 equal parts, without breaking either of its units. Also, that we cannot so divide abcd; because the unit b c will be broken at g; and each half of the line abcd will contain one inch and a half.
23. A number consisting of whole units only is called an Integral or Whole Number, (integer, Lat., whole.) A number containing broken parts is called a Broken or Fractional Number, (fractus, Lat., broken.) Hence the numbers 2 and 3 are whole numbers; but one and a half is a fractional number.
24. A number whose half is integral is called an even number. And a number whose half is fractional is called an Odd Number. Hence 2 is an even number; and 3, an odd number.
25. Beginning with the Unit, (which, though in itself, really not a number, is often, for the sake of form, called Number 1,) we progress regularly by adding a unit at a time, in forming what is called, the Scale of Natural Numbers : thus, 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine, &c. It is easy to see that, in this scale, the numbers are odd and even ad infinitum ad infinitum or ad inf., Lat., to infinity or without end.)
26. Though the Scale of Natural Numbers evidently progresses ad inf., seeing that, however great a number may be, we can still add à unit, the figures 1, 2, 3, 4, 5, 6, 7, 8, 9, are the only ones by which any number, whole or fractional, however great, is expressed; from which it is easy to infer that each of these figures is made to assume new values, increasing ad inf.
27. The next number in the scale, called ten, is expressed by the figure 1, which assumes this new value merely by a new position or place, and to show this the figure 0, Nought or Cipher, (which signifies nothing is placed on the right of it, thus : 10. Because O bus no value or signification, except merely empty place, the other figures are, by way of distinction, called Significant Figures ; sometimes Digits,
(digitus, Lat., finger.). The first idea of the method of expressing numbers having been, it is supposed, derived from the fingers.
28. In the number 10, the nought is considered as occupying the place which the digits occupy when alone ; and, therefore, this first, or right-hand place, is called the place of units. The unit being thus removed one place towards the left, has, merely from its place, assumed a new value equal to ten times its former value ; and, hence, the place where it stands, or second place, is called the place of tens.
29. The place of units being empty, we continue the regular formation of numbers, by substituting, for the nought, the nine digits in succession : thus, beginning with ten, we have the numbers 10 ten, 11 eleven, 12 twelve, 13 thirteen, 14 fourteen, 15 fifteen, 16 sixteen, 17 seventeen, 18 eighteen, 19 nineteen : the numbers 11, 12, &c. being ten and one, ten and two, ten and three, &c. to ten and nine, inclusive.
30. Continuing the formation, we add 1 to the 9 units of 19; and transferring the ten, thus formed, to the second place, by a unit, as before : we add this 1 to the 1 already there, which makes 2 tens, or 20, twenty, and the place of units again becomes vacant. Then, repeating the digits in this place, as before, we form the numbers 21 twenty-one, 22 twenty-two, &c. to 29 twenty-nine.
31. By adding 1 to the 9 units of 29, and transferring the ten, thus formed, to the second place, we have the number 3 tens or 30 thirty. Continuing thus, we form the tens in succession, which are 10 ten ; 20 twenty ; 30 thirty ; 40 forty; 50 fifty; 60 sixty; 70 seventy; 80 eighty; 90 ninety. Again repeating the digits, we have 91 ninety-one, 92 ninety-two, &c. to 99 ninety-nine, which is the greatest number expressed by two figures.
32. By adding 1 to the 9 units of 99, and carrying the ten to the second place, by the figure 1, as usual; this, together with the 9 already in this place, makes ten tens, which we express precisely in the same manner as we did ten units, namely, by the figure 1 carried to the next place towards the left, by which means the place of tens, as well as the place of units, becomes vacant, and we supply both accordingly with ciphers, thus : 100, and read one hundred. The third place is, therefore, called the place of hundreds.
33. Thus we see that the use of the cipher is to hold the place of an order of units which becomes vacant.
34. Continuing the formation of numbers in the empty places, as at first, we have 101 one hundred and one; 102 one hundred and two, &c., to 199 one hundred and ninety-nine. Again, proceeding as in Art. 32, we form the number 200 two hundred ; and in like manner, as above, successively, the numbers 300 three hundred ; 400 four hundred; 500 five hundred ; 600 six hundred ; 700 seven hundred ; 800 eight hundred; 900 nine hundred; as also the numbers 901 nine hundred and one ; 902 nine hundred, and two, &c., to 999 nine hundred and ninety-nine; which is the greatest number expressed by three figures.
35. In adding a unit to 999, to form the next number, we have first ten units, which we express by 1 in the second place; then ten tens, which we express by 1 in the third place; and, lastly, ten hundreds, which we express by 1 in the fourth place ; thus, 1000, (the three vacant places being supplied with ciphers,) and read one thousand. It is easy to see that, proceeding as above, we should succe
should successively form the numbers, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000; which are read two thousand, three thousand, &c. to nine thousand; as well as the numbers 9001 nine thousand and one; 9002 nine thousand and two, &c., to 9999 nine thousand nine hundred and ninety-nine.
36. Hence we see that the formation of new orders towards the left is without end, depending upon this universal law, namely—the value of ten units of any order is expressed by one unit of the next order towards the left-according to which, the digits assume new values, increasing towards the left ad inf. ten fold for each remove, and are, by this means, capable of expressing any number whatever.
37. We have seen that the value of a unit in the second place is equal to 9+1; in the third, to 99 +1; in the fourth, to 999 + 1, &c., according to a general law : wherefore, the value of a unit, in any place towards the left, is equal to the number expressed by a horizontal row of nines equal, in number, to the number of places on the right of that unit, plus 1: and, hence, a unit in any place towards the left expresses a number greater than that expressed by all the figures which can stand on the right of it.
NUMERATION. 38. Numeration is the general method of reading and writing numbers.
39. As there is no end to the formation of new orders on the left, and as it would, even in ordinary numbers, be inconvenient to give a new name to each new order, the figures of large numbers are separated into periods of three figures each. Thus, beginning at the right hund, and proceeding towards the left, we separate, by a commo, the first three figures, which constitute the first period. Then, proceeding towards the left, we separate three more, and so on, regularly, till the whole of the figures are thus separated. As the number may consist of any number of figures, it is evident that the lefthand period will often contain only one or two figures.
40. The first, or right-hand period, is called Units; the next Thousands; the next Millions ; and so, in succession, Billions, Trillions, Quadrillions, Quintillions, Sextillions, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, Tredecillions, Quatuordecillions, Quindecil. lions, Sexdecillions, Septendecillions, Octodecillions, Novendecillions, Vigintillions, Viginti-unillions, Viginti-billions, &c.
41. To facilitate the remembrance of the names of the periods, to show their import and relative position in the scale of numbers, as well as to extend them as far as required by any number within the scope of human thought or calculation, the following Table of Latin Numerals, from which they are nearly all derived, will be found very useful.
From this Table we also derive the names of some months; and there is this singularly coincident irregularity, viz., that as the Romans began their year in March, the month September was, as its name denotes, in their Calendar, the Seventh ; whereas, in ours,—to which the name has been transferred, and which begins two months earlier,-it is the Ninth : so, also, in the scale of the periods, as we have two names, Units and Thousands, which do not belong to the regular nomenclature, in which we may suppose that the name Millions has been substituted for Unillions; the period Billions, which, from the import of the word, should be the second, is the fourth, and, consequently, Septillions, which should be the seventh, is the ninth period.
42. To correct this discrepance, therefore, we must, in applying the Table, add 2 to the number corresponding to the name of the period, (or month,) which will give its order; or subtract 2 from the number showing its order, which will give the number corresponding to its name. Thus, for exam. ple, if I would know the order of the period Octillions or the month October ; as octo is eight, I add 2, which makes 10; I therefore say, that Octillions is the tenth period, and October, the tenth month.
43. Again : if I wish to know the name of the twelfth period, I subtract 2 from 12, and I have 10; which, in Latin, is decem : I therefore say, that the name of the twelfth period is Decillions.
Table. 1. Unus.
11. undecim. 21. viginti-unus. 2. duo, (bis,twice.) 12. duodecim.
30. triginta 3. tres, or tria. 13. tredecim. 40. quadraginta. 4. quatuor.
14. quatuordecim. 50. quinquaginta. 5. quinque.
15. quindecim. 60 sexaginta.
16. sedecim. 70. septuaginta. 7. septem.
17. septendecim 80. octoginta. 8. octo.
18. octodecim. 90. nonaginta.
19. novendecim. 99. nonaginta-novem. 10. decem. 20. viginti. 1100. centum.
44. By means of the table we may continue the nomenclature of periods, thus: viginti-trillions, &c. to viginti nonillions; trigintillions; trigintu-unillions, triginta-billions, &c.; quadragentillions ; quadraginta-unillions, &c.; quinquagentillions ; sexagentillions; septuagentillions; octogentillions; nonagentillions; centillions.
45. The above Table, as well as the names of the periods in the Seale of Numbers, we could easily extend, but such extension could, for our present purpose, be of no utility; seeing that the number 1 centillion is far above the scope of human affairs or conception. For, if we suppose a hollow globe, ten millions of millions (10 trillions) of miles in diameter, to be filled with dust, so fine that there should be one thousand millions (1 billion) of particles in each cubic inch; the whole number of particles in this mighty mass would be an inconceivably small part of a. centillion.
46. It is plain (see Art. 36) that, whatever may be the value of the units composing any figure in the scale, the next