ARITHMETIC.

PART I.

INTEGRAL AND DECIMAL,

1. ARITHMETIC is the science that treats of the nature and use of numbers, the fundamental rules of which are, Numeration, Addition, Subtraction, Multiplication, and Division.

2. Explanation of the signs most generally used.

(1.) + plus, or more, the sign of addition. When placed between two numbers, it signifies that they are to be added together; thus 5+4 means that 5 is to be added to 4.

It

(2.) minus, or less, the sign of subtraction. signifies that the latter number is to be taken from the former; thus 5-4 means 4 subtracted from 5.

(3.) x into, the sign of multiplication, signifies that the numbers are to be multiplied together; thus 5×4 means that 5 is to be multiplied into 4.

(4.) by, the sign of division; signifies that the former number is to be divided by the latter; thus 54 means that 5 is to be divided by 4, and is usually written .

(5.)

=

equal to, signifies that the numbers are equal to one another; thus 5+4=9.

(6.).. therefore.

B

NUMERATION AND NOTATION.

3. Notation is the art of representing numbers by their proper signs.

Numeration is the converse of this, and means the expressing in words the numbers so represented.

4. The first nine numbers have each a separate character to distinguish them, and are called the nine digits. They are as follows-1, one, 2, two, 3, three, 4, four, 5, five, 6, six, 7, seven, 8, eight, and 9, nine. By the use of these, together with 0, nought, or cipher, all numbers whatever may be represented.

5. When any of these figures are placed together, they have, besides their own definite value, a relative one, depending upon their situation to the right or left of what is called the unit's place.

6. The unit's place is marked by a dot to the right of it; or, if there be no dot, then the last figure to the right is understood to be in that place.

7. The class or order of each digit is determined, by observing that every figure is ten times greater than it would be in the next place to the right, and ten times less than it would be in the next place to the left.

8. The different classes are as follows:

To the left of the unit's place, (taking the unit as the first figure,) the

2d figure means so many tens.

To the right of the unit's place, (taking the unit as

This will be better understood by an inspection of the following table.

Beginning at the 1 and going towards the left, we have,

One unit, or one.

Two tens and one unit, or twenty-one.

Three hundreds, two tens, and one unit; or, three hundred and twenty-one, &c.

In the second line we have, in like manner,

Two units, or two.

Three tens and two units, or thirty-two, &c.

Then beginning again at the 1, and going on to the right, we might say,

One unit.

One unit and two tenths, or twelve tenths.

One unit, two tenths and three hundredths, or one hundred and twenty-three hundredths.

But it is more usual to reckon from the point, and not include the unit; thus

Two tenths.

Two tenths and three hundredths; or twenty-three hundredths.

Two tenths, three hundredths, and four thousandths; or, two hundred and thirty-four thousandths.

Two tenths, three hundredths, four thousandths, and five tenths of thousandths; or, two thousand three hundred and forty-five tenths of thousandths, &c.

10. If in any particular number one or more orders or classes be wanting, the vacant places are filled up by 0.

Thus, in the last row but two of the table, 987 stands for nine hundred and eighty seven; but if we wish to put down nine hundred and seven, we must have 907. Similarly nine hundred is 900, .004 is four thousandths, 2005 is two tenths and five tenths of thousandths, or two thousand and five tenths of thousandths.

11. Observe, however, that when the unit's place is marked, ciphers added on the outside either to the right or left, do not alter the amount of a number. Thus, 12.3 is

the same as 012.30; .9 is the same as .90, or .900, or .9000, &c.

12. The numbers to the left of the point or dot are called integers, those to the right decimals.

XAMPLES.

I. Read in words the following numbers:(1.) 16, 25, 334, 729, 5003, 6072, 54037. (2.) 76541, 890031, 604050, 3535003, 9875430. (3.) 50000003, 35970236, 123456789, 987654321. (4.) .7, .07, .007, .0007, .77, .777, .7777. (5.) .06, .066, .0666, .0606, .6006, .6606, .6066. (6.) .35013, .70696, .050505, .500005, .7931542. (7.) .0005009, .79314502, .20541397, .00000078. ́(8.) 3.5, 5.3, 7.4, 4.7, 5.19, 19.5, 51.9, 1.95. (9.) 4.4, .44, 4.04, 40.4, 7.35, 73.5, .735. (10.) 123.456, 654.321, 107.701, 600.006, 34.5678, 45.6789.

(11.) 34.56789, 23.45678, 12.3456789, 98.7654321. (12.) 99999.88888, 9898.989898, 540673.8900402. (13.) 5437210403.2, 234579047.049, 3.706548751. (14.) 445603.85920176, 43790500.00003021,

.003003003.

(15.) 70000007.00000007, 521521521.521521521. (16.) 324506490051.350, 52107.3457900145.

II. Write in figures the following numbers:(1.) Twenty-five; forty-seven; three hundred and five.

(2.) Four hundred and fifty; nine hundred and twelve. (3.) One thousand six hundred and four; three thou

sand and three.

(4.) Eight thousand seven hundred and ninety-four;

ten thousand and eleven.

(5.) Fifty-four thousand six hundred and eighty; ninety-five thousand.