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, a b c d 33330, 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 b and a. 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, a b c d 33336, 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 5th Period. 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 hundreds 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: Hundreds of thousands Thousands - Tens of Trillions. Units ~Hundreds of thousands Thousands 4 7 4 2 1 3, 7 2 5 6 3 8, 1 2 3 4 5 6, 7 8 9 1 2 3, Hundreds of thousands Tens of thousands Thousands → Tens Units ~ Hundreds of thousands Tens of thousands Thousands Tens co Units 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 A unit of the second order by 1 and 0; thus 1 10 100 1000 A unit of the third order by 1 and two Os; thus A unit of the fourth order by 1 and three Os; thus 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 the second? The first in the third? The second in the fifth? &c. 8. Point off and write in words the following figures: 72358, 700, 1245, 604267, 8956238, 284563002, 123456, 7924502, 3824507266. 9. Increase the first two of the above nine numbers tenfold; the second two a hundred fold; the third two a thousand fold; and the remaining three ten thousand fold. In other words, multiply them by 10, by 100, 1000, 10,000. [The above exercises should be repeated and varied by the teacher till the subject becomes perfectly familiar to the class.] But the right-hand figure does not always represent units. Sometimes it becomes necessary to use or speak of a number less than one. Thus, with respect to the money of the United States, the dollar is considered the unit. But a sum less than a dollar frequently enters into a calculation, cents, for instance, which are hundredths of a dollar; or dimes, which are tenths. These parts of a unit of any kind are called fractions, a word signifying broken into parts. Or, let us suppose an apple to be cut into ten equal parts. One or more of these fractions or tenths of an apple may enter into a calculation. When this is the case, these tenths, as the smallest part of the number, would occupy the rank on the right. But some mark would then be necessary to show which rank was occupied by the units. The character used for this purpose is a reversed comma, called a separatrix,* placed to the right of the rank of units. When a separatrix is used, any number of figures_or ciphers can be added to the right of a number without changing the value of the other figures. Thus, if we take the four 3s again, marked with letters as before [see p. 113], and put a separatrix after the 3 marked d, to show that it stands in the place of units, we can add as many figures as we choose on either hand without changing the value of any of the 3s, for this simple reason, that their distance from the rank of units is unchanged by that addition. For example: * Some writers use a dot, others a comma, for a separatrix. Both are wrong. For, as both these characters are used with figures for other purposes, they thus give rise to much uncertainty and perplexity. For instance, if the dot be used as separatrix, there is no means of ascertain |