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NOTATION AND NUMERATION 7 Fractional Forms , 122
Arabic Notation

8 Fractional Relation of
Roman Notation

17
Numbers.

123 ADDITION.

19 Review Exercises. 125 SUBTRACTION.

33 DECIMAL FRACTIONS . 129 MULTIPLICATION

43 Reduction of Decimals. 134 DIVISION

57 Addition of Decimals 138 Analysis and Review.

71 Subtraction of Decimals. 139 PROPERTIES OF NUMBERS. 75 Multiplic'n of Decimals. 141

Divisibility of Numbers. 76 Division of Decimals. 143
Factoring.
78 Short Processes.

146 Multiplicat’n by Factors. 79 Accounts and Bills

151 Division by Factors 80 Review Exercises . 155 Cancellation.. 82 DENOMINATE NUMBERS.

157 Common Divisors

84 Measures of Value. 158 Multiples .

88 Reduction Descending 160 FRACTIONS. 93 Reduction Ascending.

162 Reduction of Fractions . 96 Measures of Space. 164 Addition of Fractions 104 Linear Measures

165 Subtraction of Fractions. 107 Surface Measures ,

167 Multiplication of Frac

Measures of Volume . 171 tions 109 Board Measure.

174 Division of Fractions. 115 Measures of Capacity. 175

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Measures of Weight : · 178
Measures of Time . 182
Measures of Angles 184
Reduction of Denominate
Fractions.

186 Review Exercises.

190 Addition of Denominate Numbers .

192 Subtraction of Denominate Numbers

194 Multiplication of De

nominate Numbers 196 Division of Denominate Numbers .

197 Longitude and Time . 199 Metric System

202 PERCENTAGE.

Commercial Discount. 237
True Discount..

238 Bank Discount.

240 Review Exercises . 245 Profit and Loss.

247 Commission .

254 Review Exercises . 258 Taxes

262 Duties .

265 Stocks.

267 Insurance.

277 Exchange.

283 Average of Payments. 290 Average of Accounts .

295 PARTNERSHIP

298 RATIO

303 PROPORTION

306 INVOLUTION

315 EVOLUTION

319 PROGRESSIONS

336 MENSURATION

341 MISCELLANEOUS EXAMPLES 356 TEST QUESTIONS

363 ANSWERS

372

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207 Interest

218 Method by Aliquot Parts. 222 Six Per Cent. Method.

223 Compound Interest

225 Annual Interest.

228 Partial Payments

229 Problems in Interest. . 232

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303 306

2. A Number is a unit or collection of units.

315

3. In counting a large number of objects it is natural to

319

group them.

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Thus, coins are put in piles and the piles counted, envelopes in packages and the packages-counted, etc. These piles and packages may themselves be piled and put in larger packages, and the process continued indefinitely.

4. To express numbers so that all may understand what is meant by the characters which represent them, the system of grouping by tens has been adopted. There are, therefore, single things, or units; groups of ten units, or tens; groups containing ten tens, or hundreds, etc.

5. The method of grouping by tens is called the Deci· mal System. Decimal is derived from the Latin word decem, which means ter.

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Numbers may be expressed by words or other characters, viz: figures and letters.

6. Notation is the method of expressing numbers by figures and letters.

The Arabic Notation is the method of expressing numbers by means of figures. Its name is derived from the Arabs, by whom it was introduced into Europe. The Roman Notation is the method of expressing numbers by means of letters. It is so called because it was used by the ancient Romans.

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

ARABIC SYSTEM.

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8. In this system ten figures are employed to express numbers, viz:

Figures. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Names. Naught, One, Two, Three, Four, Five, Six, Seven, Eight, Nine.
Each of these, except naught, is called a significant figure.
Naught is also called zero and cipher.

9. By combining these figures according to certain principles, we can express any number.

10. PRINCIPLE. — When figures are written side by side, the one at the right expresses units, the next tens, and the next hundreds.

EXERCISES.

11. 1. In 79, what does the 7 express? What does the 9 express ? Read the number, beginning at the left.

2. In 58, what does the 5 express? What does the 8 express? Read the number, beginning at the left.

3. In 740, what does the 7 express? What does the 4 express? What does the 0 express ? Read the number.

4. Begin at the left and read 76, 176, 106, 360, 203.

12. Figures in units' place express units of the first order; those in tens' place, units of the second order; those in hundreds' place, units of the third order, etc.

13. Numbers between 1 ten and 2 tens are named thus:

1 ten and 1 unit or 11, eleven.
1 ten and 2 units or 12, twelve.'
1 ten and 3 units or 13, thirteen.
1 ten and 4 units or 14, fourteen.
1 ten and 5 units or 15, fifteen.
1 ten and 6 units or 16, sixteen.
1 ten and 7 units or 17, seventeen.
1 ten and 8 units or 18, eighteen.

1 ten and 9 units or 19, nineteen. The words thirteen, fourteen, fifteen, etc., mean three and ten, four and ten, five and ten, etc.

14. The units of the second order are named as follows:
2 tens or 20, twenty. 6 tens or 60, sixty.
3 tens or 30, thirty. 7 tens or 70, seventy.
4 tens or 40, forty.

8 tens or 80, eighty. 5 tens or 50, fifty.

9 tens or 90, ninety. The suffix ty means len. Thus, forty means four tens, etc.

The other numbers between 20 and 100 are read without the word and between the tens and units. Thus, 27 is read twenty-seven, instead of twenty and seven.

EXERCISES.

41

15. Read the following:

28 99 43
64 83 75
39 78 60
47 32 21

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92 25

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