ELEMENTS OF ARITHMETIC.

ARITHMETIC is the science of numbers; it treats of their properties, and of the methods of making computations by means of them.

A unit, or, as it is sometimes called, unity, is one of anything; as, one foot, one man, one table, &c.

Number signifies either a unit, as, one man; or a collection of units of the same kind, as, two men, five books, seven houses, a thousand feet, &c.; two, five, seven, thousand, &c., are called numbers.

NOTATION AND NUMERATION.

NOTATION is the method of expressing numbers by means of certain signs or figures; thus—1, 2, 3.

NUMERATION is the art of numbering, that is, of reading or expressing numbers in words; thus-1, one; 2, two.

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THE FIGURES used to express numbers are the following: one, two, three, four, five, 1 2 3

4

six, seven, eight, nine, nothing or cipher. 5 6 7 8 9 0

The first nine of these figures, when standing separately or singly, thus-1, 2, 3, 9, represent the numbers from one to nine. The last, 0, called a nothing or cipher, expresses no number, and has no value in itself; but it is used to affect the value of the other figures when annexed to them, as will presently be explained.

It is by means of these ten figures and their combinations that all numbers are expressed.

Each of these figures, besides the simple value which it has when standing alone, as, 1, one, 2, two, &c., has also, when standing in connection with other figures, a local value-that is, a value depending on the place it occupies in the number.

Thus, 5 by itself means simply five; but if it become the second figure from the right, by a nothing or any other figure being placed after it-thus, 50-the 5 has ten times its former value, and means 5 tens, or fifty; and if it become the third from the right, by another figure being annexed-thus, 500-the 5 is again increased tenfold in value, and means 5 hundreds; and so on, the addition of each new figure increasing tenfold the value of those before it.

It is in this way that the nothing or cipher, though it has no value in itself, affects the value of the other figures, by altering their place or rank. Each of the other figures has a similar effect on the rest when annexed to them.

It is to be observed, however, that though the value of a figure depends on the place it occupies, the meaning of the figure always remains the same. Thus, 5 always expresses five of something, either five single things, or five groups of ten, or five groups of a hundred, &c.

NUMBERS Ten to ninety-nine are expressed by combining two figures; thus-10, ten; 20, twenty; 85, eighty-five; 99, ninetynine.

NUMBERS One hundred to nine hundred and ninety-nine, are expressed by combining three figures; thus-100, one hundred; 500, five hundred; 867, eight hundred and sixty-seven; 999, nine hundred and ninety-nine.

Thousands are expressed by four figures; thus-1000, one thousand; 7320, seven thousand three hundred and twenty.

Tens of thousands are expressed by five figures; hundreds of thousands, by six figures; millions, by seven figures; and so on. By using more figures, we are enabled to express any number, however great.

It will thus be seen that in Notation, the rank or place of a figure in any number is what determines the value it bears.

The first figure at the right in a line of figures, counting from right to left, has only its simple value; a figure in the second place has ten times its simple value; a figure in the third place has ten times the value it would have in the second place, or a hundred times its simple value, and so on; and, generally, a figure in any place has ten times the value it would have in the next lower place.

The number 10, according to which the local values of the other figures are determined, is called the base of the system; and the system of notation itself, the decimal scale of notation.

The first or lowest place in a line of figures is called the units' place; the second, the tens' place; the third, hundreds'; the fourth, thousands'; and so on, as shewn in the following NUMERATION TABLE. It is read from right to left, thus-units, tens, hundreds, &c.; the rank or position in which these stand in regard to each other should be carefully studied and committed to memory.

In expressing large numbers in figures, it is usual, for the sake of distinctness, to point off the figures as far as possible into sets of three, called periods, by means of commas, beginning at the right hand, and counting towards, the left. Thus-87,463,292. Each Period of three is named as marked in the Table.

The above number, 321,987,654,321,987,654,321, is read three hundred and twenty-one quintillions, nine hundred and eightyseven quadrillions, six hundred and fifty-four trillions, three hundred and twenty-one billions, nine hundred and eighty-seven millions, six hundred and fifty-four thousand, three hundred and twenty-one.

The periods succeeding those in the Table are sextillions, septillions, octillions, &c., but these names are seldom employed. In ordinary affairs, we rarely hear of any sum beyond hundreds of millions.

I. TO EXPRESS NUMBERS IN Figures.

Begin at the left hand, and put down the required figures one after the other in a line, taking care to put each figure in the place or rank necessary to express the number, according to the Numeration Table—that is, millions must be put in the

*This Table is given in the improved form used by the French and Italians. It has been recently introduced into this country, as being much more simple and convenient than the old form used by ourselves. The figures are grouped into periods of three, and are named and read accordingly. In the old form, which is given below, the figures are

millions' place, or seventh from the right hand; thousands in the thousands' place, or fourth from the right; and so on.

In doing this, nothings must be put in all those places of which none are mentioned in the given number. Thus, if thousands are mentioned, but no hundreds, a nothing must be put in the hundreds' place, to keep the other figures in their proper rank,

After the figures are written down, point them off, as far as possible, into periods of three, beginning at the right.

It may be useful, at first, for the pupil, in writing small numbers, to mark down as many of the places (such as units, tens, hundreds, &c.) in the Numeration Table as are required to express the given number, and then to write the respective figures below the names that express them: thus, write in figures, thirty-six thousand and seventy-three.

In writing down larger numbers, such as millions, proceed as follows: Draw three lines m t " to represent the three periods, into which the figures forming a million, or millions, are divided; below that marked m, write the millions of the given number; below that marked t, the thousands; and below that marked u, the hundreds, tens, and units.

Thus, suppose the number to be expressed is one hundred and six millions, five thousand and thirty; below the m, write 106; below the t, 005; and below the u, 030; thus

106 005 030; or without the lines, 106,005,030.

It will be observed that there being in the MILLIONS' period no tens of millions, in the THOUSANDS' period no hundreds or tens of thousands, and in the UNITS' period no hundreds or units, nothings have been inserted in all these places to keep the other figures in their right position.

grouped into periods of six, which is an arrangement very inconvenient in practice. It will be observed that the two Tables are the same up to hundreds of millions.