ARITHMETIC.

ART. 1. ARITHMETIC is the art of computing by numbers. Any single thing is called a unit. Number signifies a unit, or a collection of units; as, one book, two slates, five plums.

2. The ten following characters, called the Arabic figures, or digits, are used in writing numbers. 1, one; 2, two; 3, three; 4, four; 5, five; 6, six; 7, seven; 8, eight; 9, nine; and 0, naught.

They are called digits, from the Latin word digitus, which signifies a finger.

The first nine figures are called significant figures, to distinguish them from the tenth, 0, which has no value when standing alone. It is sometimes called cipher, or zero.

3. Arithmetic consists of five fundamental operations; viz., Numeration, Addition, Subtraction, Multiplication and Division.

SECTION I. — NUMERATION.

4. Numeration is the art of expressing numbers by figures, and of reading them when so expressed.

As nine is the largest number that can be expressed by any single figure, all numbers larger than nine are expressed by combinations of two or more of the ten Arabic figures. (2.)

Thus, to express ten, we combine 1 and 0, 10; to express eleven, we combine 1 and 1, thus, 11; to express twelve, 1 and 2, 12; thirteen, 13, &c. Twenty, or two tens, is written 20; twenty-one, 21; twenty-two, 22, &c., two figures being combined to express any number between nine and one hundred.

1. Write upon the slate all the numbers from ten to ninetynine, inclusive.

To express numbers larger than ninety-nine, and less than one thousand, three figures are combined.

Thus, 100 is one hundred, 101 is one hundred and one, 102 is one hundred and two, 110 is one hundred and ten, 120 is one hundred and twenty, 125 is one hundred and twenty-five, 200 is two hundred, 300 is three hundred.

2. Write on the slate all the numbers from one hundred to two hundred; from three hundred to four hundred; from seven hundred and fifty to eight hundred and fifty.

5. TO TEACHERS. If the pupil is a beginner, he should write on a slate or blackboard, numbers between ten and one thousand, until he can readily write and read any of them, when they are dictated to him. He should do the same with larger numbers. Let each succeeding exercise in the book be extended by the teacher, till the pupil is perfectly familiar with the principle and its application.

6. The value of any figure, when combined with others, depends upon the place it occupies. Its value becomes ten times as large by removing it one place to the left, and consequently, one tenth as large by removing it one place to the right.

Thus 1, when alone, represents a unit, or one, and is called a unit of the first order. On being removed one place to the left, as in 10, it represents one ten, or ten, and is then called a unit of the second order. On being again removed one place to the left, as in 100, the 1 represents ten tens, or one hundred, and it is then called a unit of the third order. Removing the 1 again one place further to the left as in 1000, it represents ten hundreds, or one thousand, and is called a unit of the fourth order.

So also 5, when alone, or at the right hand of other figures, represents 5 units, or five. 50 is 5 tens, or fifty. 500 is 50 tens, or 5 hundred. 5000 is 50 hundred, or 5 thousand.

7. In whole numbers, we call the place of the first right hand figure the units' place; of the second figure from the right, the tens' place; of the third, the hundreds' place.

Thus, 854 is eight hundreds, five tens, and four units; or eight hundred and fifty-four. 609 is six hundreds, no tens, and nine units; or six hundred and nine. The figure 0 being always used to fill a place not occupied by a significant figure.

8. Numbers containing more than three figures, are divided into periods of three figures each, by commas, counting from the right; the first or right hand period contains units, tens, and hundreds; the second period contains units of thousands, tens of thousands, and hundreds of thousands; the third contains units of millions, tens of millions, &c., as in the following table.

6th period. 5th period. 4th period. 3d period. 2d period. 1st period.
Quadrillions. Trillions. Billions. Millions. Thousands. Units.

II. The periods above quadrillions are quintillions, sextillions, septillions, octillions, nonillions, decillions, undecillions, duodecillions; and by continuing to adopt a new name for every three figures, the number of periods may be increased indefinitely.

Dividing numbers into periods of three figures is the French method of Numeration. The English method has been to divide them into periods of six figures each; but the French method is more convenient than the other, and has been almost universally adopted.

9. TO READ NUMBERS. Count off the figures, beginning at the units' place, into periods of three figures; and, beginning at the left, read the numbers standing in each period, adding the name of each period, except the right hand period.

The numbers in the above table are read: Six quadrillions, eight hundred and one trillions, eighty billions, four hundred and eightynine millions, one hundred and sixty thousand and eighty-four.

Examples to be written in words, and read by the learner.

(1.) 10. (2.) 100. (2.) 100. (3.) 1000. (4.) 10000. (5.) 100000. (6.) 101. (7.) 180. (8.) 1001. (9.) 1012. (10.) 2084. (11.) 7804. (12.) 10001. (13.) 30805. (14.) 38050. (15.) 30085. (16.) 500005. (17.) 3050601. (18.) 850160804. II. (19.) 4016080900. (20.) 1851608090504. (21.) 50080090607010.

10. FOR BEGINNERS. 1. Write every third number from 1000 to 1100; thus, 1000, 1003, 1006, &c. Write every fourth number from 1300 to 1400. Write every fifth number from 2000 to 2100.

To WRITE WHOLE NUMBERS. Commence with the highest or left hand period, which may require one, two, or three figures, and write all the other periods in their order, allowing three places for each period, filling the vacant places with naughts.

and ten.

Numbers to be expressed in Figures.

1. Eighty-four. 2. Nine hundred and four. 3. Nine hundred and forty. 4. One thousand and one. 5. One thousand 6. One thousand one hundred and one. 7. Eighty thousand and eight. 8. Eighty thousand and eighty. 9. Eighty thousand and eighty-eight. 10. Five hundred and seventy-five thousand, six hundred and thirty-seven. 11. Eight millions and thirty-five. 12. Thirty-four millions, thirty-four thousand and thirty-four. 13. Five hundred millions and fifty. 14. Six billions, six thousand and six hundred. II. 15. Fifteen quintillions, forty millions, eight thousand and forty. 16. Sixty octillions, ninety trillions, and three thousand. 17. Fifteen decillions, eight sextillions, four hundred billions, and eight millions.

NUMERATION OF DECIMALS.

11. If a number be divided into 10 equal parts, one of the parts is called one tenth of that number. Thus, one tenth of 10 apples is one apple, because if ten apples be divided into 10 equal parts, one part will be one apple. If 20 apples be divided into 10 equal parts, what will you call one of the parts? Ans. One tenth of twenty apples. How many apples is one tenth of twenty apples? Why? One tenth of 30 apples? Why? One tenth of 50 apples? Why? Of 60 apples? Why? 70 apples? Why? 80 apples? Why?

If 100 apples be divided into ten equal parts, what part of 100 apples will one part be? Ans. One tenth of 100 apples. Why? How many apples is one tenth of 100 apples? Why? Of 200? Why? 300? Why? 400 500 600 apples? Why?

How many apples is one tenth of 1000 apples? Why? 2000 apples? Why? 3000? Why? 4000 apples? Why?

If a single thing be divided into 10 equal parts, one of the parts is called one tenth of that thing; two such parts are called two tenths of it; 3 such parts, three tenths, &c. If it be divided into 100 equal parts, one of the parts is called 1 hundredth of that thing; two such parts are called 2 hundredths of it; three such parts, 3 hundredths, &c.

What is meant by 1 tenth of 1 apple? 2 tenths? 3 tenths? 4 tenths? What is meant by 1 hundredth of one apple? 2 hundredths? 3 hundredths? &c.

If 2 apples be divided into 10 equal parts, what part of 1 apple will one of the parts be? How much is 1 tenth of 2 apples? Why? Of 3 apples? Why? 5? 7? Why?

How much is one tenth of 1 unit? Why? Of 2 units? Why? Of 3 units? Why?

If an apple be divided into 10 equal parts, and then each of these parts be divided into 10 equal parts, what part of 1 tenth will one of these parts be? Why?

Into how many equal parts will the whole apple be divided? What part of the whole apple will one part be? Why?

If 2 apples be divided in the same manner, what part of 1 apple will one part be? Why? What is one hundredth of 1 unit? Of 2 units? Why? Of 3 units? Why?

12. As tens are tenths of one hundred, and are written one place to the right of hundreds; and as units are tenths of one ten, and are written one place to the right of tens; so tenths of units are written one place to the right of units; and tenths of tenths, or hundredths, are written one place to the right of tens; thousandths, one place to the right of hundredths, &c.

Such parts of units as tenths, hundredths, thousandths, &c., are called decimals, from the Latin decem, ten; because their value diminishes toward the right in a tenfold ratio. A point called the decimal point, is placed to the left of the tenths' place to separate the decimals from whole numbers. Thus, 3.6 is 3 and 6 tenths; 3.06 is 3 and 6 hundredths; 3.25 is 3 and 25 hundredths; 3.005 is 3 and 5 thousandths; 3.465 is 3 and 465 thousandths. .3 is 3 tenths; .08 is 8 hundredths; .015 is 15 thousandths.

TO READ DECIMALS. Read the figures as in whole numbers, and add the name of the lowest decimal place. In reading whole numbers and decimals, read the whole numbers first, and then the decimals.

Thus .7 is 7 tenths; .18 is 18 hundredths; .00108 is 108 hundred thousandths: 15.006 is 15 and 6 thousandths; 16.0064 is 16 and 64 ten thousandths. The number in the table is three thousand and