NUMERATION AND NOTATION.

NUMBERS are usually expressed either by words or by signs or characters. The whole subject of the expressing of numbers is properly called NUMERATION, and the part of it which relates to the expression of numbers by characters is called NOTATION.*

In modern arithmetic all numbers are expressed by means of ten characters, or figures, called the ten digits, which are shown here following: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The first of these, called the cipher, or zero, or nought, indicates a place kept void from any other figure, and has important significance through the effects which it is capable of producing on the values or importances of other figures placed in connexion with it, as will soon be explained. The remaining nine denote respectively the numbers one, two, three, four, five. six, seven, eight, nine. These nine are commonly called the significant figures; but this designation ought to be abandoned as conveying essentially an erroneous idea, since the figure 0, in expressing precisely the absence of the thing or things (or of the number applicable to things in general) that would be expressed by any other figure put instead of it, has quite as definite a signification as any of the other nine. When a distinctive appellation is wanted, the figures 1, 2, 3, 4, 5, 6, 7, 8, 9, may be called the value figures, and nought, 0, the void figure.

When any of the nine "significant" or value figures stands by itself, or when it is followed by no other figure, it expresses merely its number of ones, or of single things counted; but when it is followed by one figure it expresses its number of tens of the things counted; when by two, it expresses its number of hundreds of them; when by three, it expresses its number of thousands of them; and so on. Thus in 3333 each 3 stands for three of something; the first 3 at the right means three of the things counted; the second signifies 3 groups of ten each; the third, 3 groups of a hundred each; and the fourth 3 groups of a thousand each.

Thus in the expressions 5, 25, and 365 the figure 5 denotes simply five of the things counted; while in 52 and 256 it means five tens of the things counted, or fifty of them; and in 524 it denotes five hundreds of them. Thus the expression 576 means five hundreds, seven tens, and six units; or, as it is briefly read, five hundred and seventy-six.

When a number expressed by several figures is to be read off in words, the three next the right hand, taken together, are read as

*The word numeration is frequently used, in a restricted sense, as meaning merely the method of expressing in words the values of numbers already ex pressed by characters; while, in conjunction with that restricted signification, the word notation is applied to the reverse process of expressing by characters numbers already given in words, or else to the process of expressing them by characters when they are given in any way; as, for instance, when a group of objects is given to be counted, and the number is required to be noted in characters. It is better to apply the name numeration to the whole subject than to restrict it to one part.

The name digit is derived from the Latin word digitus, finger. The ten figures used in arithmetic are by some called the zero and nine digits, and by others they are called in preference the ten digits.

ing to the stoned the nest the

quintillions, sextillions, sepnames might be formed for er, it is seldom necessary to

amber, is at once discovered the right hand, expresses six sty millions will be expressed

s into periods, and of the French and Italians. It and it has been adopted ks, however, the periods have been designated by pt thousands, for which as far as hundreds of mil

As the old method is a the pupil that he pssion of large numbers al method is shown in the gn according to both and is the same as what is millions. It is scarcely the bat will be applicable gures each, instead of

being a number of units; the three next them, taken together, are read as being a number of thousands; the next three are read as being a number of millions; and so on, according to the subjoined table. When a line of figures is thus divided, the three next the right hand are called the first period, the next three the second period; and so on, as in the table.

9, 8, 7 6, 5, 4 3, 2,

&c.

Hundreds of quadrillions
Tens of quadrillions
Quadrillions

Hundreds of trillions

Tens of trillions

Trillions

Hundreds of billions
Tens of billions

Billions

Hundreds of millions
Tens of millions

Millions

Hundreds of thousands

II. Thousands ... Tens of thousands

I. Units

Thousands

Hundreds
Tens

-Units

The periods after quadrillions may be called quintillions, sextillions, septillions, octillions, and nonillions; and analogical names might be formed for the still higher periods. In actual practice, however, it is seldom necessary to name numbers exceeding millions.

The local value of any figure used in expressing a number, is at once discovered from this table. Thus, 6 in the eighth place from the right hand, expresses six tens of millions, or sixty millions; and, conversely, sixty millions will be expressed by the figure 6 in the eighth place.*

The method given above of dividing lines of figures into periods, and of naming those periods, is that which is employed by the French and Italians. It is strongly recommended by its simplicity and elegance; and it has been adopted in some works in this country. In most English works, however, the periods have been made to consist of six figures each; and they have been designated by the same names as those in the table given above, except thousands, for which there is not a distinct period. The two methods agree as far as hundreds of millions, and it is rarely necessary to name larger numbers. As the old method is still taught in many English books, and in order to warn the pupil that he must be careful to understand the words used for the expression of large numbers in the sense intended by the speaker or writer, the old method is shown in the subjoined table; and the answers of the exercises are given according to both methods at the end of the book. In France a milliard is the same as what is called a billion in the foregoing table; it is a thousand millions. It is scarcely necessary to say, that the rules and directions given in the text will be applicable in this method, if the periods be made to consist of six figures each, instead of

The cipher, or zero, expressing absence of value, is used in combinations of figures, to fill places where no value is to be expressed, and thus to make the other figures occupy those places in which they will express the intended values.

Thus the figures 365, combined in this manner, denote three hundred and sixty-five; but the expression 306050, which contains the same value figures, means three hundred thousand, no tens of thousands, six thousand, no hundreds, five tens, and no units; or simply three hundred and six thousand, and fifty.

From the principles above explained, we have the following rules for the two chief operations in numeration which are generally requisite in practice :

RULE I. To express in words the numbers denoted by lines of figures: (1.) Commencing at the right-hand side, divide the given figures into periods of three figures each, till not more than three remain. (2.) Then, commencing at the left side, annex to the value expressed by the figures of each period, except that of the units, the name of the period, according to the numeration table.

Thus, the expression, 37053907, becomes by division into periods 37,053,907, and is read thirty-seven millions, fifty-three thousand, nine hundred and seven, the term units, or ones, at the last being omitted.

three, and if the second period be called millions, the third billions, &c., as in the following table :

Units