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character occupies as connected with other figures. The word local means pertaining to a place. For example: 1 means a single unit, or one, when it stands by itself, or when it stands in the first rank at the right hand. But when it is placed in the second rank from the right, it is ten times greater ; that is, it stands for one ten. Thus, the figure 1 in 10 or in 16 stands for ten, because it occupies the second rank. But as the figure in the first rank in 10 signifies nothing, being only used to place the 1 in the second rank, the two figures together stand for ten. In 16, as the figure in the first rank stands for 6, the the two figures together stand for sixteen. The same principle holds with all figures. Thus, 24 stands for twenty-four, because the figure 2, being in the second rank, does not stand simply for two, but for two tens, or twenty; and 40 stands for forty, because the 4 occupies the second rank. 44 is forty-four, because the first 4, being in the second rank, is forty ; the second 4, being in the first rank, is simply four. Any number, then, as far as ninety-nine, can evidently be expressed with the ten characters. The next higher number is ten tens, or one hundred. This is expressed by placing the figure 1 one place further to the left; that is, in the third rank from the right. Thus, in 100 and 124 each of the ones stands for 100, because it is in the third rank. The 2 in the second number counts for twenty, because it is in the second rank, which is the place of tens. Thus, the three figures together, 124, read one hundred and twenty-four. To express thousands, a figure must stand one place still further to the left, because ten hundred make one thousand. Thus, in the number given below,
there are four 3s, but each has a different value. The first 3 on the right, marked d, stands for three ones. The figure in the second rank, marked c, is ten times greater than the first; that is, it stands for three tens, or thirty. The third, marked b, is ten times greater than the second, or a hundred times (ten times ten times) greater than the first ; that is, it stands for three hundred. Lastly, the fourth figure, marked a, is ten times greater than the third; a hundred times (ten times ten times) greater than the second; a thousand times (ten times ten times ten times) greater than the first; that is, it stands
for three thousand. The whole number, then, reads three thousand, three hundred and thirty-three.
The first principle of decimal arithmetic, then, is derived from the tenfold increase of value of the ranks, or places, of figures. It may be expressed as follows:
I. When figures are placed horizontally, or side by side, every
figure is ten times greater than the same figure immediately on its right, and ten times less than the same figure immediately on its left.
If a cipher were placed to the right of the above four 3's, as below,
the 3 marked d would no longer stand for three units, or ones. It would now be three tens, or thirty, because it occupied the second rank from the right, which is the place of tens. The 3 marked c has also changed its place. It, also, has become ten times greater. It was formerly three tens; it is now three hundred. The same remark applies to the figures marked 6
Each is moved one place further to the left, and thus has become tenfold greater. In a word, the whole number has been increased tenfold by having a cipher placed at its right. Again, by removing the cipher, each of the other figures is changed to one rank further to the right, and thus each figure, and consequently the whole number, is decreased tenfold. The object, then, of the cipher is to enable us to place significant figures in their proper rank, and thus show their true local value.
But any other figure, by changing the rank of these 3s, would have changed their value just as effectually as the cipher. If 6 is put in place of the cipher, as below,
each three has its value increased as before, by having its rank changed one place towards the left. The only difference between the two numbers is, that six has been added to it in the one, besides the tenfold increase; whereas nothing has been
added to the other. Again, by removing the 6, we not only decrease the value of the other figures tenfold by changing their rank, but also diminish the number by six. It is evident that the same observations will hold good if any other significant figure is added or taken away.
The second principle of decimal arithmetic, then, may be expressed as follows: II. Every figure becomes tenfold greater by being removed one
rank, or place, to the left, and tenfold less by being removed one rank, or place, to the right.
The following Numeration Table, which teaches us to read the names of those figures that stand for integers, will now be readily understood :
From this table, it appears that each figure, besides its simple name of one, two, three, &c., has two other names. For instance, the first figure on the left of the table is three hiindreds of trillions, or more simply three hundred trillions; the second is six tens of trillions, or sixty trillions (the final syllable ty signifying tens), and so forth. The term units is
always omitted. Hence the third figure is not read five units of trillions, but simply five trillions; and the figure on the right of the table is not read seven units of units, but simply seven. The whole series of figures is read thus : three hundred and sixty-five trillions, four hundred and twenty-seven billions, nine hundred and eighty-four millions, two hundred and eightythree thousand, two hundred and forty-seven. Higher numbers than these are rarely required. It may be proper to mention, however, that the same principles of nomenclature can be continued to infinity, the classes or periods being named quadrillions, quintillions, sextillions, septillions, &c., to each of which, as before, are assigned three ranks or places, namely, units, tens, hundreds.
It will be observed that the figures in the table are divided by commas into periods, or classes, of three orders of figures each, commencing at the right. This should always be done when a series of figures exceeds four in number, for otherwise they cannot so easily be read. These periods, it may be noticed, are named Units, Thousands, Millions, Billions, Trillions, &c. The orders, or ranks, are the same in every period, namely, Units, Tens, Hundreds.*
* This is the French mode of separating numbers into periods. Its simplicity has led to its universal use in this country. By the English mode, formerly used here, each period has six figures, and is read as follows :
8 9 1 2 3,
It ought here to be carefully noted that the word unit, besides forming the name of the first period, and of the first order or rank of each period, may be applied to figures of any order whatever. Thus a single unit of the first order is expressed by
1 A unit of the second order by 1 and 0; thus .
10 A unit of the third order by 1 and two Os ; thus 100
A unit of the fourth order by 1 and three Os ; thus 1000 and so on for the units of higher orders. But, when units are named simply, without expressing any particular order, units of the first order are always meant.
As it is evident from the above table and explanation that there cannot be more than nine different numbers of any one denomination, since an addition of one more to the nine enlarges the number to ten, and thus carries it into the next higher rank, we thence have the third principle of decimal arithmetic ; namely:
III. Ten units of any one rank make one unit of the next
rank to the left; and one unit of any one rank makes ten units of the next rank to the right.
Exercises for the Black-board or Slate. 1. Divide 44444444 into periods of threes by commas, commencing at the right.
2. What is the general name of the first period on the right? Of the second ? Of the third ?
3. Repeat the name of the orders in the first period ? Ans. Units, tens, hundreds. Repeat those of the third ; of the second ; of the fourth. Are they the same in every period ?
4. What name is never expressed ?
5. What is the first figure on the left called ? Ans. Tens of millions. How many millions does that figure stand for? What is the second figure on the right called ? The fourth on the right ? The third on the left?
6. How many times is the second figure on the right greater than the first ? [Point to the figures on the black-board.] The third than the first? The fourth than the second ? The fourth than the first? The fourth than the third ? &c.
7. How many times is the first on the right contained in