WRITING LARGE NUMBERS.

9. PROP. IV.-Numbers are written one period at a time and in the order in which the periods are read.

Observe regarding this proposition:

1. Each period in a number except the one at the left must contain three figures. Hence the places for which significant figures are not given must be filled with ciphers.

Thus, three hundred seven million, four thousand, eightytwo, is written 307,004,082. Observe in this number a significant figure is given only for the hundreds and ones in the million's period, hence the ten's place is filled with a cipher. For a like reason the ten's and hundred's place in the thousand's period and the hundred's place in the unit's period are filled with ciphers.

2. When a number is read, a period in which all the orders are wanting is not named. Care must therefore be taken to notice such periods and fill their places in each case with three ciphers.

For example, in the number seven million three hundred four, the thousands period is not named, but when the number is expressed in figures its place is filled with three ciphers; thus, 7,000,304.

RULE.—Begin at the left and write the figures expressing the hundreds, tens, and ones of each period in their proper order, filling with ciphers all periods or places where no significant figures are given.

EXAMPLES FOR PRACTICE.

10. Express in figures the following numbers:

1. Two hundred seven. Four hundred fifty. Seven hundred ninety. Three hundred eighty-seven.

2. Three thousand nine. Eight thousand sixty.

3. Eleven hundred.

4. Ten tens.

Twenty-five hundred.

One hundred tens. Ten tens and three.

One hundred tens and fifteen. Eight hundred six tens.

5. Ten thousand. Ten thousand nine. Twenty thousand. Twenty thousand fifty-three.

6. Eleven thousand eleven. One million one.

7. Eighty million eighty thousand eighty.

8. Two thousand three hundred five million. 9. Eight hundred sixty-two tens.

tens. Two thousand sixty-three tens.

Five hundred seven

10. Forty two thousand million. Seven thousand and six million. Forty-four billion seven.

11. 87 million 1 thousand 2. 907 trillion 4 million 6. 12. 11 billion 108 thousand 39. 1 trillion 1 million 1.

DEFINITIONS.

11. A Unit is a single thing, or group of single things, regarded as one; as, one ox, one yard, one ten, one hundred.

12. Units are of two kinds - Mathematical and Common. A mathematical unit is a single thing which has a fixed value; as, one yard, one quart, one hour, one ten. A common unit is a single thing which has no fixed value; as, one house, one tree, one garden, one farm.

13. A Number is a unit, or collection of units; as, one man, three houses, four, six hundred.

Observe, the number is "the how many" and is represented by whatever answers the question, How many? Thus in the expression seven yards, seven represents the number.

14. The Unit of a Number is one of the things numbered. Thus, the unit of eight bushels is one bushel, of five boys is one boy, of nine is one.

15. A Concrete Number is a number which is applied to objects that are named; as four chairs, ten bells.

16. An Abstract Number is a number which is not applied to any named objects; as nine, five, thirteen.

17. Like Numbers are such as have the same unit. Thus, four windows and eleven windows are like numbers, eight and ten, three hundred and seven hundred.

18. Unlike Numbers are such as have different units. Thus, twelve yards and five days are unlike numbers, also six cents and nine minutes.

19. Figures are characters used to express numbers.

20. The Value of a figure is the number which it represents.

21. The Simple or Absolute Value of a figure is the number it represents when standing alone, as 8.

22. The Local or Representative Value of a figure is the number it represents in consequence of the place it occupies. Thus, in 66 the 6 in the second place from the right represents a number ten times as great as the 6 in the first place.

23. Notation is the method of writing numbers by means of figures or letters.

24. Numeration is the method of reading numbers which are expressed by figures or letters.

25. A Scale in Arithmetic is a succession of mathematical units which increase or decrease in value according to a fixed order.

26. A Decimal Scale is one in which the fixed order of increase or decrease is uniformly ten.

This is the scale used in expressing numbers by figures.

27. Arithmetic is the Science of Numbers and the Art of Computation.

ROMAN NOTATION.

28. Characters Used.-The Roman Notation expresses numbers by seven letters and a dash.

C,

D,

M.

Five

One

Letters.-I, V, X, L,
Values. One, Five, Ten, Fifty, Hundred, Hundred, Thousand.

One

29. Laws of Roman Notation.-The above seven letters and the dash are used in accordance with the following laws:

1. Repeating a letter repeats its value.

Thus, I denotes one; II, two; III, three; X, ten; XX, two tens, or twenty.

2. When a letter is placed at the left of one of greater value, the difference of their values is the number expressed.

Thus, IV denotés four; IX, nine; XL, forty.

3. When a letter is placed at the right of one of greater value, the sum of their values is the number expressed.

Thus, VI denotes six; XI, eleven; LX, sixty.

4. A dash placed over a letter multiplies its value by one thousand.

Thus, X denotes ten thousand; IV, four thousand; V, five thousand.

4. Four. 8. Seventeen. 12. Seventy-four. 16. Forty-four.

17. One hundred twenty-seven. Eight hundred four. 18. One hundred forty-nine. Ninety-five.

19. One thousand. Five thousand. Fifty thousand.

20. Ten thousand. One hundred thousand. Five hundred thousand. Ninety thousand.

21. 1800. 1875. 8065. 7939. 1854. 20365. 85342. 22. Read the following: MIX; MDLXIV; X; D; MM; MD; DVII; MDCCCLXXVI; ML; DLX.

REVIEW AND TEST QUESTIONS.

31. Study carefully and answer each of the following questions:

1. Define a scale. A decimal scale.

2. How many figures are required to express numbers in the decimal scale, and why?

3. Explain the use of the cipher, and illustrate by examples. 4. State reasons why a scale is necessary in expressing

numbers.

5. Explain the use of each of the three elements—figures, place, and comma-in expressing numbers.

6. What is meant by the simple or absolute value of figures? What by the local or representative value?

7. How is the local value of a figure affected by changing it from the first to the third place in a number?

8. How by changing a figure from the second to the fourth? From the fourth to the ninth?

9. Explain how the names of numbers from twelve to twenty are formed. From twenty to nine hundred ninety.

10. What is meant by a period of figures?

11. Explain how the name for each order in any period is formed.

12. State the name of the right-hand order in each of the first six periods, commencing with units.

13. State the two things mentioned in (9) which must be observed when writing large numbers.

14. Give a rule for reading numbers; also for writing numbers.