ARITHMETIC.

ARTICLE 1. ARITHMETIC is the science of numbers, and the art of computation.

A NUMBER is a unit or a collection of units, a unit being one, i. e. a single thing of any kind; thus, in the number six the unit is one; in ten days the unit is one day.

2. All numbers are concrete or abstract.

A CONCRETE NUMBER is a number that is applied to a particular object; as six books, ten men, four days.

An ABSTRACT NUMBER is a number that is not applied to any particular object; as six, ten, seventeen.

3. Arithmetic employs six different operations, viz. Notation, Numeration, Addition, Subtraction, Multiplication, and Division.

NOTATION AND NUMERATION

4. NOTATION is the art of expressing numbers and their relations to each other by means of figures and other symbols.

5. NUMERATION is the art of reading numbers which have been expressed by figures.

ART. 1. What is Arithmetic? What is a Number? A Unit? 2. What is a Concrete Number? An Abstract Number? 3. How many operations in Arithmetic? What are they? 4. What is Notation? 5. Numeration?

6. Two methods of notation are in common use: the Arabic and the Roman.

7. The ARABIC NOTATION, or that brought into Europe by the Arabs, employs ten figures to express numbers, viz.:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Naught, One, Two, Three, Four, Five, Six, Seven, Eight, Nine. These figures are called digits, from the Latin digitus, a finger; a term probably applied to figures from the custom of counting upon the fingers.

8. The first Arabic figure, 0, is called a cipher, naught, or zero, and, standing alone, it signifies nothing.

Each of the remaining nine figures represents the number placed under it, and for convenience in distinguishing them from 0, they are called significant figures.

9. No number greater than nine can be expressed by a single Arabic figure, but by repeating the figures, and arranging them differently, all numbers may be represented.

Ten is expressed by writing the figure 1 at the left of the cipher; thus, 10. In like manner, twenty, thirty, forty, etc., are expressed by placing 2, 3, 4, etc., at the left of 0; thus,

80, 90.

20, 30, 40, 50, 60, 70, Twenty, Thirty, Forty, Fifty, Sixty, Seventy, Eighty, Ninety. 10. The numbers from 10 to 20 are expressed by placing the figure 1 at the left of each of the significant figures; thus,

11, 12,

13,

14,

15, 16,

17, etc. Eleven, Twelve, Thirteen, Fourteen, Fifteen, Sixteen, Seventeen, etc.

In a similar manner all the numbers, up to one hundred, may be written; thus,

21, Twenty-one,

36,
66,
Thirty-six, Sixty-six,

98, etc. Ninety-eight, etc.

6. How many methods of Notation? What? 7. How many figures in the Arabic Notation? What called? Why? 8. What is the first figure, 0, called? The others? Why? 9. The largest number expressed by one figure? Ten, how expressed? Twenty? 10. Numbers from ten to twenty, how expressed?

11. One hundred is expressed by placing the figure 1 at the left of two ciphers; thus 100. In like manner two hundred, three hundred, etc., are written; thus,

200,

300,

600,

800, etc.

Two hundred, Three hundred, Six hundred, Eight hundred, etc.

12. The other numbers, up to one thousand, may be expressed by putting a significant figure in the place of one or each of the ciphers in the above numbers; thus,

Two hundred and three, expressed in figures, is
Six hundred and eighty, expressed in figures, is

203,

680,

Nine hundred and ninety-eight, expressed in figures, is 998. 13. The PLACE of a figure is the position it occupies with reference to other figures; thus, in 436, the 6, counting from the right, is in the first place, 3 is in the second place, and 4 in the third place.

The figure in the first place represents simple units, or units of the first order; the second figure represents tens, or units of the second order; the third, hundreds, or units of the third order; the fourth, thousands, or units of the fourth order, etc.; thus, in the number 3576, the 6 is 6 units of the first order; the 7 tens is 7 units of the second order; the 5 hundreds is 5 units of the third order, etc.

14. From the foregoing it will be seen that each significant figure has two values; one of which is constant (i. e. always the same), the other variable; thus, in each of the numbers 2, 20, and 200, the left hand figure is two; but in the first it is two units; in the second, two tens; and in the third, two hundreds. The former of these values is the inherent or simple value, and the latter is the local or place value.

15. It is also evident that the value of a figure is made ten fold by removing it one place toward the left; a hundred fold by removing it two places, etc.; i. e. ten units of the first order

11. One hundred, how expressed? Two hundred? 12. Other numbers, how expressed? 13. What is the place of a figure? What does the figure in the first place represent? Second place? Third? 14. How many, and what values, has a figure? 15. How does moving a figure towards the left affect its value?

make one ten, ten tens make one hundred, ten hundreds make one thousand, and, in short, ten units of any order make one unit of the next higher order.

16. The cipher, when used with other figures, fills a place that would otherwise be vacant; thus, in 206 the cipher occupies the place of tens, because there are no tens expressed in the given number.

17. The figures of large numbers, for convenience in reading, are often separated by commas into periods or groups.

There are two methods of numerating: the FRENCH and the ENGLISH. By the French method a period consists of three figures; by the English, of six. The French method is most convenient, and principally used in this country.

18. By the FRENCH METHOD OF NUMERATION the first or right-hand period contains units, tens, and hundreds, and is called the period of units; the second period contains thousands, tens of thousands, and hundreds of thousands, and is called the period of thousands; etc., as in the following

2d period, 1st period, Millions, Thousands, Units,

7th period, 6th period, 5th period, 4th period, 3d period, Quintillions.Quadrillions, Trillions, Billions,

16. For what is the cipher used? 17. How many methods of numerating? What are they? Which is generally used in this country? 18. Name the diff. erent periods in the French Numeration Table. Repeat the table.