Numbers ten to ninety-nine, are expressed by the joining of two figures; thus-10, ten; 20, twenty; 85, eighty-five; 99, ninety-nine.

Numbers one hundred to nine hundred and ninety-nine, are expressed by the joining of 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-2000, two thousand; 7320, seven thousand three hundred and twenty.

Tens of thousands are expressed by five figures; hundreds of thousands, by six figures; and so on, the numbers increasing at a tenfold rate for every additional figure.

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 hand in a row of figures, always means units; the second from the right, tens; the third, hundreds; the fourth, thousands; and so on, as shewn in the following Numeration Table. Whenever a new figure is annexed at the right of a number, each of the others obtain, as it were, a promotion, and is made to express ten times its former value. Thus, 89 means 8 tens and 9 units, or eighty-nine; but if 3 be annexed, making 893, 8 means 8 hundreds, 9 means 9 tens, and 3, 3 units; or eight hundred and ninety-three. The annexing of a nothing (0) multiplies the other numbers in a similar way; thus-46, forty-six, when a nothing is annexed, becomes 460, four hundred and sixty.

The following Numeration Table shews how numbers progressively increase from units up to billions. 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 :

The above number, 1,987,654,321, is read one billion, nine hundred and eighty-seven millions, six hundred and fifty-four thousand, three hundred and twenty-one. By annexing another figure, we should have tens of billions; another figure, hundreds of billions; and

another figure again, trillions. But notation seldom goes to such an extent; in ordinary affairs, we rarely hear of any sum beyond billions.

In expressing large numbers in figures, it is usual, for the sake of distinctness, to point off the figures into sets of three, by means of commas, beginning at the right hand, and counting towards the left. Thus-87,463,292.

I. TO WRITE OR NOTE DOWN IN FIGURES ANY GIVEN NUMBER.

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 millions' place, or seventh from the right hand, and in no other; thousands in the thousands' place, or fourth from the right hand; and so on. In doing this, nothings must be put in all those places of which none are named in the given number. Thus, if no thousands are mentioned, a nothing must be put in the thousands' place, in order to keep the other figures in their proper rank. After the figures are written down, point them off into sets of three when necessary, beginning at the right hand.

It may be useful for the pupil first to write 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, to write in figures, thirty-six thousand and seventy-three, first write down in a row all the places, from units up to tens of thousands, this last being the highest name in the given number; and then write the figure representing tens of thousands below that title, thousands below thousands, and so on, as follows:

Here there are 3 tens of thousands, 6 thousands, 0 hundreds, 7 tens, and 3 units, which, when read, are expressed as thirty-six thousand and seventy-three. It will be observed that there being no hundreds mentioned in the number, a 0 is placed in the hundreds' place to express this.

Examples.

Three hundred and forty-seven,

347

Twenty-six thousand four hundred and fifty-one, 26,451
Five thousand and twenty,* .

5,020

*Here, as no hundreds or units are named, nothings are put down in the places of hundreds and units, in order to keep the 5 in the thousands' place, and the 2 in the tens' place. If this were not done, the number would read as 52.

Exercises.

Note down in figures the following numbers, dividing them, when necessary, by commas into sets of three:—

Seventeen, sixty-three, eighty-nine, ninety-eight, one hundred and two, one hundred and ten, one hundred and seventeen, one hundred and twenty-seven, one hundred and ninety-nine.

Two hundred, two hundred and eleven, two hundred and forty, two hundred and fifty-five, two hundred and ninety, three hundred, three hundred and eighty-eight, four hundred, four hundred and four, four hundred and seventy-six.

Five hundred, five hundred and one, five hundred and ninetynine, six hundred, six hundred and nineteen, six hundred and thirty-seven, seven hundred, seven hundred and six, eight hundred, eight hundred and thirteen, nine hundred, nine hundred and seven, nine hundred and seventy.

One thousand, one thousand two hundred and fifty, one thousand three hundred, two thousand and forty, three thousand and four, four thousand and twenty-one, five thousand one hundred, six thousand three hundred and eleven, seven thousand and eighty-one, eight thousand nine hundred and fifteen, nine thousand nine hundred and ninety-nine.

Ten thousand, ten thousand and ten, ten thousand and fifty-nine, eleven thousand, fourteen thousand and sixteen, twenty thousand one hundred and three, thirty-three thousand and forty, sixty-four thousand and five, nine hundred and ninety-nine thousand nine hundred and ninety-nine.

One hundred thousand, two hundred thousand three hundred and eleven, seven hundred thousand and eighty, one million sixty thousand two hundred and seven, thirty-four millions one hundred and eight thousand and six, fifty millions three hundred thousand four hundred and one, eight hundred and three millions five hundred and ten thousand and ninety.

II. TO READ OR EXPRESS IN WORDS ANY GIVEN NUMBER IN FIGURES.

Begin at the right, and, going towards the left, name the order or rank of each figure of the given number; thus-units, tens, hundreds, thousands, &c.—that is, the first figure at the right is units, the next tens; and so on. Having in this way ascertained the rank of each figure, or its position in the Numeration Table, express the whole sum in words, reading in the usual way from left to right. After a little practice, it will become unnecessary to name the order or rank of the figures before reading them.

Exercises.

Read or write in words the following numbers, keeping in mind that the value of each figure depends on its place in the

Numeration Table. Thus, the first figure at the right always means units; the second from the right, tens; the third, hundreds; the fourth, thousands; and so on:

13, 17, 24, 36, 49, 82, 94, 100, 110, 117, 134, 166, 199, 200, 201, 273, 219, 349, 428, 494, 511, 660, 777, 813, 979, 1,000, 1,107, 1,212, 1,347, 2,051, 3,003, 4,011, 5,100, 10,336, 20,109, 37,640, 61,420, 98,012, 100,000, 735,640, 813,105, 901,027, 2,891,563, 40,200,400, 315,070,050, 500,630,107, 850,111,005, 900,301,206.

ROMAN NOTATION.

The Romans made use of the following letters, with their combinations, to express numbers. They are still in use among ourselves for some purposes, such as the headings of chapters, divisions, &c. :

Two or more of the same letter placed together, mark two or more of the same number; thus-II. means twice I., or two.

A letter of inferior value placed before one of superior value, means that the inferior is to be deducted from the superior; thus in IX., the I placed before the X means that I is to be taken from X, and IX. therefore expresses 9.

A letter of inferior value placed after one of superior value, means that the inferior is to be added to the superior-thus in LX., the X placed after L means that X is to be added to L, and LX. therefore expresses 60.

A line drawn above a letter increases its value a thousand timesas X., 10,000; D., 500,000.

The number I (=D. or 500) is increased in value ten times for every annexed; thus-I00. means 5000. The number CI) (= M. or 1000) is increased in value ten times for every C and joined to it; thus CI., by joining C and D, becomes CCIOD., or 10,000. The letters I are not now in use.

SIMPLE ADDITION.

ADDITION is the adding together of several numbers, for the purpose of finding their united amount, or what they all come to. We add or sum numbers together when we say 1 and 1 make 2; 2 and 3 make 5, &c. The amount of the numbers, when added, is called the sum.

Simple addition is the adding of numbers of the same kind-as, for instance, where the numbers all mean pounds, or all shillings. The rule for simple addition is given below.

Compound addition is where the numbers are partly of one kind, and partly of another-as, for instance, when some mean pounds, and some shillings. Compound addition is treated of afterwards, page 37.

The same distinction of simple and compound applies to the subsequent rules of Subtraction, Multiplication, and Division.

For the sake of saving words, certain signs are often employed in the various rules of arithmetic.

Addition is denoted by the figure of a cross, of this shape +. Thus, 76 means 7 added to 6; and in order to express the sum resulting, the sign =, which means equal to, is employed, as 7 + 6 = 13; that is, 7 and 6 are equal to 13.

ADDITION TABLE.

The following table is to be committed to memory, and should be frequently repeated forward and backward, till a readiness in adding is acquired; as-2 and 1 are 3, or 1 and 2 are 3; and so on:

1. Write the numbers to be added under each other, figure directly under figure, in such a way that the right-hand figures will all be straight under each other, forming one even column. Thus-units will stand under units, tens under tens, hundreds under hundreds; and so on.

2. After all the numbers are thus placed, draw a line under