NOTE. Frequent dictation exercises should be given with successively larger numbers until pupils have acquired proficiency in writing numbers.

15. Before writing the following numbers, tell how each will appear when written.

Illustration. Three thousand eight hundred seven is expressed by writing the following: three, comma, eight, cipher,

seven.

1. Forty thousand six.

2. Ninety-seven thousand five hundred twelve.

3. Three hundred sixty-nine thousand twenty-four.

4. Four million eight thousand two.

5. Fifty-six million nineteen thousand thirty-three.

6. Eighty-one million five hundred thirteen thousand two hundred fifty-one.

7. Three hundred million ninety thousand four.

8. Five billion six million seven thousand eight.

9. Seventy-two billion six hundred thirty-five thousand two hundred fifty-one.

10. One hundred three billion two million seventeen thousand one hundred four.

11 Two trillion three billion four million five thousand six. 12. Ninety-one trillion two hundred seven billion sixtynine million four thousand three

13 Eighty-six trillion one million twenty-three.

14. Two hundred sixteen trillion five hundred thousand. 15. Nineteen trillion four.

NOTE. 1. Teachers should supply dictation exercises in writing num bers until a good degree of proficiency is acquired.

2. Observe which of the number names are compound words.

3 Note that the word "and" is not used in these exercises.

16. THE ROMAN NOTATION.

This method expresses number by the use of certain print letters. They are I, V, X, L, C, D, M.

I = 1, V = 5, X= 10, L= 50, C100, D = 500, M = 1000. Their use is determined by the following

PRINCIPLES.

1. Repeating a letter repeats its value. II=2, XX = 20. 2. When a letter is placed after one of greater value, the two express a number equal to the sum of their values. XV = 15, CI = 101.

3. When a letter is placed before one of greater value, the two express a number equal to the difference of their values. IX = 9, XC = 90. (Limited to IV, IX, XL, and XC.)

4. When a letter is placed between two, each of greater value, its value is taken from the sum of their values. XIX 19, XIV = 14.

5. Placing a dash over a letter multiplies its value by a thousand. X= 10,000, M = 1,000,000.

EXERCISES.

1. Express by the Roman characters all numbers from one to one hundred

17. REDUCTION.

1. Reduction is the process of changing the unit of a number without changing its value.

2. Express 6 pints in quarts; 8 quarts in gallons; 6 feet in yards; 32 ounces in pounds; 20 mills in cents; 300 cents in dollars.

3. Have these numbers been changed to a larger or to a smaller unit?

The process of reducing a number to larger units is called Reduction Ascending.

4. How are pints reduced to quarts? quarts to gallons? feet to yards? ounces to pounds? mills to cents? ones to tens? tens to hundreds? tens to thousands? cents to dollars? Give many similar examples.

DIRECTION.

5. To reduce a number to a number of larger units, divide it by the number of the given units which makes one of the larger units.

6. Express 3 quarts in pints; 3 gallons in quarts; 4 yards in feet; 3 dimes in cents; 5 tens in ones; 6 hundreds in tens; 7 thousands in hundreds, in tens.

7. Have these numbers been changed to a larger or to a smaller unit?

The process of reducing a number to smaller units is called Reduction Descending.

8. How are quarts reduced to pints? gallons to quarts? yards to feet? dimes to cents? tens to ones? hundreds to tens? thousands to hundreds? to tens?

DIRECTION.

9. To reduce a number to a number of smaller units, multiply it by the number of the smaller units to which one of the larger units is equal.

NOTE. - A knowledge of the fundamental nature of the decimal system is of the utmost importance in arithmetical operations. This is obtained through practice in the two forms of Reduction. Multiply examples like the following

18. 1. In 50,000 there are how many tens? hundreds? thousands? ten-thousands?

2. In 17,000,000 there are how many thousands? hundred-thousands? tens? hundreds? ones? ten-thousands? 3. In 38,000 mills there are how many cents? dimes? dollars?

4. In 65 dollars there are how many dimes? cents? mills?

19. 1 is what part of 10? One ten is what part of 100? One hundred is what part of 1000? In 1,111 each unit is what part of the unit standing in the first place at its left?

20. It is customary to fix the place of ones by placing a period at its right, thus: 1. When the period is thus used it is called the decimal point. From what has been observed what kind of unit will the right-hand figure in 1.1 express? What is one tenth of one tenth? What, then, is the kind of unit expressed by the right-hand figure in 1.11? What is one tenth of one hundredth? What kind of unit is expressed by the right-hand figure in 1.111? Similarly, show what kind of units is expressed by a figure in the fourth place at the right of the decimal point; in the fifth place; in the sixth place.

21. Table of six places at the right of the decimal point.

22. The name of any number is the same as the place in which its right-hand figure stands. Read the following

31.

16.07 (Read as 16 and 7 hundredths or as 1607

23. Perfect familiarity with the names and numbers of the places at the right of the decimal point is necessary for accurate and rapid writing. The name of a number is determined by the place of its right-hand figure. In writing numbers like the foregoing correctly the first time two things must be known: the number of figures required to express the number; the place in which the right-hand figure must stand to express the kind of units.

24. Write the following numbers: 1. Twenty-three hundredths.

2. Seven tenths.

3. Sixty-nine hundredths.