PRACTICAL ARITHMETIC.

NUMERATION.

Teaches to express in words any number written in figures; and before the learner enters on the study of Arithmetic, it is necessary that he should be thoroughly conversant with reading, at a glance, any number however large which he may see written.

NOTATION

Teaches us to express in figures any number given i words, and is therefore the reverse of Numeration.

And for these purposes there are the ten following forms or figures in use amongst us :

one, two, three, four, five, six, seven, eight, nine, nought. 1 2 3 4 5 6 7 8 9 0 Thus you see the highest figure is not more than nine, the last figure, nought, being derived from the word nothing, and nothing does it express; yet with these ten figures we can express numbers so high that the mind can scarcely conceive their amount.

The manner in which this is accomplished is by writing down the figures, for any number you wish to express, in a line; you then count from the right-hand figure, which represents units, or just as many ones as the figure, if standing alone, would amount to; the

B

next figure to the left represents tens, or as many tens as it would units if standing alone, or at the right of a series; the third figure represents hundreds; and so on, every figure to the left increasing in tenfold proportion.

Let us take the first figure in the above line, 1. When written alone it but expresses one; but if the next figure, 2, is written after it thus, 12, it then would be increased to ten, and the number would express twelve, or ten and two: if to these we add another figure, or increase the line by writing the next figure, 3, on the right of the 2, thus, 123; instead of one, two, three, we have a hundred and twenty-three: the 2 and 1 being advanced one step each, the 2 represents two tens, or twenty, and the 1 represents one hundred; the 3 being in the place of units, represents but three ones, that being its own value if written singly.

Having now shewn you that the value of figures increases from their place in a series or line of figures, a glance at the following table will enable you to name the place and value of each figure in a series.

The preceding sum when written in words, or read, is one hundred and twenty-three thousand, four hundred and fifty-six billions; seven hundred and eighty-nine thousand, one hundred and twenty-three millions; four hundred and fifty-six thousand, seven hundred and eighty-nine; a sum whose magnitude we can scarcely conceive, and produced by merely writing the first line of figures from one to nine twice over.

The names of the next periods are, Trillions, Quadrillions, Quintillions, Sextillions, Septillions, Octillions, and Nonillions; each period consisting, as above, of six figures.

Before setting down lines of figures for practice in reading them, I must draw the pupil's attention to the way in which the above line is separated. Beginning with the right figure or place of units, you see a comma written between the third and fourth figure, separating the hundreds from the thousands; and between the sixth and seventh figures a semicolon is written, which divides the place of thousands from that of millions, and it also distinguishes the finishing of the first period of figures, called the period of thousands. You see also the same rule adopted at the finishing of the periods of millions and of billions; and by adopting a like rule, and writing the name of each period over it, you will be able to read with ease any of the following lines of figures.

The last line, Ex. (12), is written to point out the importance of the cipher or nought, and expresses one hundred thousand millions. Although the cipher expresses nothing of itself, without it we could not express ten, twenty, thirty, or indeed any number in which a certain number of tens is contained evenly.

PRACTICES IN NOTATION, OR WRITING FIGURES.

Write on your slate in figures the following numhers:

(1.) Five hundred and six.

(2.) Seventy-nine.

(3.) Seven thousand nine hundred and seventy-six.

(4.) Nine hundred and three.

(5.) Six thousand seven hundred and ninety-one. (6.) Eighteen thousand.

(7.) One hundred and eighty-three thousand, seven hundred and sixty.

(8.) One hundred and ninety thousand and eighteen. (9.) Six millions eight hundred thousand seven hundred and sixty-three.

(10.) Thirty-three millions one hundred and seven thousand six hundred and eighty.

(11.) One hundred and one millions, eleven thousand, one hundred and ten.

(12.) Seven billions, one hundred and nineteen thousand millions, seven hundred thousand, eight hundred and seventeen.