Art. 9. When a number is expressed by six figures, the sixth or left-hand figure denotes hundreds of thousands, or Hundred-thousands.

14. How many hundred-thousands in 534,000? 308,000? 650,430? 508,080 ?

15. How many hundred-thousands, ten-thousands, and thousands in 354,000? 607,800? 350,307? 193,240? 470,386 ?

Art. 10. In reading a number expressed by six figures, the sixth, fifth, and fourth figures are read together as thousands. Thus, 452,000 is read four hundred and fifty-two thousand.

Art. 11. The fourth, fifth, and sixth figures of ą

number constitute the Thousands' Period.

20. Read the thousands' period in the 16th, 17th, 18th, and 19th examples.

21. How many orders in units' period? In thousands' period?

22. What are the names of the three orders in units' period? In thousands' period?

23. How may the two periods be separated? Ans. By a comma.

WRITTEN EXERCISES.

1. Write in words, 3000; 4060; 3580; 7086; 6606; and 8080.

2. Write in words, 4400; 5008; 6070; 8506; 5087; 7600; and 3003.

3. Express in figures, three thousand; seven thousand; nine thousand; four thousand five hundred; eight thousand nine hundred.

4. Express in figures, two thousand four hundred and forty; four thousand six hundred and sixty; five thousand eight hundred; six thousand five hundred and twenty-five.

5. Express in figures, seventy-five; two hundred and forty; three hundred and six; five hundred and forty-five; four thousand.

6. Express in figures four hundred and forty; five hundred and ninety; seven thousand eight hundred; eight thousand and fifty.

7. Write in words, 10000; 25000; 40500; 36000; 44000; 30400; 45080; 64008; 89800.

8. Express in figures, forty-five thousand five hundred and four; sixty thousand seven hundred and ninety; thirty-eight thousand and twenty; ninety-six thousand and eighty-four.

9. Express in figures, four hundred and twenty; seven hundred and eighty-nine; four thousand and fifty-seven; seventy-five thousand; sixteen thousand and ninety-eight.

10. Express in figures, as one number, 87 thousand 327 units; 60 thousand 405 units; 70 thousand 346 units; 4 thousand 40 units; 5 thousand 5 units; 95 thousand 406 units.

11. Express in figures, as one number, 88 thousand 88 units; 8 thousand 80 units; 65 thousand 60 units; 6 thousand 600 units; 60 thousand.

12. Write in words, 300000; 440000; 334000; 245500; 304800; 450340.

13. Express in figures, four hundred thousand; six hundred thousand; eight hundred and forty thousand; seven hundred and sixty thousand.

14. Express in figures, nine hundred and fifty thousand four hundred; four hundred and fifty-five thousand two hundred and eighty.

15. Separate the following numbers into periods: 3080; 44004; 400080; 20066; 109038; 160006; 809090; 706030; 40004; 30030.

LESSON III.'

DEFINITIONS, PRINCIPLES, AND RULES.

Art. 12. Arithmetic is the science of numbers, and the art of numerical computation.

A Number is a unit or a collection of units.

A Unit is one thing of any kind.

An Integer is a whole number.

Art. 13. There are three methods of expressing numbers:

1. By words; as, five, fifty, etc.

2. By letters, called the Roman method. (Art. 23.) 3. By figures, called the Arabic method.

Art. 14. Notation is the art of expressing numbers by figures or letters.

Numeration is the art of reading numbers expressed by figures or letters.

The word Notation is commonly used to denote the Arabic method, which expresses numbers by figures.

Art. 15. In expressing numbers by figures, ten characters are used, viz.: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The first of these characters, 0, is called Naught, or Cipher. It denotes nothing, or the absence of number.

The other nine characters are called Significant Figures, or Numeral Figures. They each express one or more units. They are also called Digits.

Art. 16. The successive figures which express a number, denote successive Orders of Units. These orders are numbered from the right; as, first, sec'ond, third, fourth, fifth, and so on.

A figure in units' place denotes units of the first order; in tens' place, units of the second order; in hundreds' place, units of the third order, and so on the term units being used to express ones of any order.

I. A.-2.

Art. 17. Ten units make one ten, ten tens make one hundred, ten hundreds make one thousand; and, generally, ten units of any order make one unit of the next higher order.

NOTE. The teacher can make this principle plain by means of the illustration given on page 9. It is easily shown that 10 ones or units equal 1 ten, and that 10 tens equal 1 hundred.

Art. 18. Figures have two values, called Simple and Local.

The Simple Value of a figure is its value when standing in units place.

The Local Value of a figure is its value arising from the order in which it stands.

When 3, for example, stands alone, or in the first order, it denotes 3 units; when it stands in the second order, as in 34, it denotes 3 tens; when it stands in the third order, as in 354, it denotes 3 hundreds. Hence, the local value of figures increases from right to left in a tenfold ratio.

The local value of each of the successive figures which express a number, is called a Term. The terms of 325 are 3 hundreds, 2 tens, and 5 units.

Art. 19. The figures denoting the successive orders of units, are divided into groups of three figures each, called Periods. The first or righthand period is called Units; the second, Thousands; the third, Millions; the fourth, Billions; the fifth, Trillions; the sixth, Quadrillions; the seventh, Quintillions, etc.