Division of Common Fractions, 160 Greatest Common Divisor of Fractions, 167 Miscellaneous Examples in Profit and 253 Miscellaneous Exercises in Fractions, 169 Reduction of Fractions of Compound 170 PARTNERSHIP, OR COMPANY BUSINESS, 254 Addition of Fractions of Compound Subtraction of Fractions of Compound Questions to be performed by Analysis 176 Miscellaneous Questions by Analysis, 179 SECTION XIX. 261 Numeration of Decimal Fractions, . . 182 Notation of Decimal Fractions, Addition and Subtraction of Duodeci- Miscellaneous Exercises in Decimals, 193 mals, Multiplication of Duodecimals, INVOLUTION, . 269 Miscellaneous Éxercises in Interest, .204 EVOLUTION, 205 Extraction of the Square Root, 272 210 Application of the Square Root, . Application of the Cube Root, . COMPOUND INTEREST, 212 SECTION XXXIX. Annuities at Simple Interest by Arith- SECTION XXIV. SECTION XL. COMMISSION, BROKERAGE, AND STOOKS, 218 Annuities at Compound Interest by BANKING, 221 222 SECTION XLI. SECTION XXVII. SECTION XLII. SECTION XXVIII. SECTION XLIII. SECTION XXIX. 230 SECTION XLIV. RATIO, 237 SECTION XLV. MENSURATION OF LUMBER AND TIMBER, . 318 . 276 . 285 • 214 . . 216 . . . 806 ARITHMETIC. ARTICLE 1. QUANTITY is anything that can be measured. An abstract number is a number, whose units have no reference to any particular thing or quantity; as two, five, seven. A concrete number is a number, whose units have reference to some particular thing or quantity; as two books, five feet, seven gallons. ARITHMETIC is the science of numbers, and the art of computing by them. A rule of arithmetic is a direction for performing an operation with numbers. The introductory and principal rules of arithmetic are Notation and Numeration, Addition, Subtraction, Multiplication, and Division. The last four are called the fundamental rules, because upon them depend all other arithmetical processes. § I. NOTATION AND NUMERATION. NOTATION. ART. 2. NOTATION is the art of expressing numbers by figures or other symbols. There are two methods of notation in common use; the Roman and the Arabic. QUESTIONS. — Art. 1. What is quantity? What is a unit? What is a number? What is an abstract number? What is a concrete number? What is arithmetic? What is a rule? Which are the introductory rules ? What are the last four called ? — Art. 2. What is notation? How many kinds of notation in common use? What are they? ART. 3. The Roman notation, so called from its originating with the ancient Romans, employs in expressing numbers seven capital letters, viz. : I for one ; V for five ; X for ten ; L for fifty ; C for one hundred ; D for five hundred ; M for one thousand. All the other numbers are expressed by the use of these letters, either in repetitions or combinations ; cas, II expresses two; IV, four ; VI, six, &c. By a repetition of a letter, the value denoted by the letter is represented as repeated; as, XX represents twenty ; CCC, three hundred. By writing a letter denoting a less value before a letter de noting a greater, their difference of value is represented; as, IV represents four; XL, forty. By writing a letter denoting a less value after a letter denoting a greater, their sum is represented; as, VI represents six ; XV, fifteen. A dash (-) placed over a letter increases the value denoted by the letter a thousand times ; as, V represents five thousand; IV, four thousand. TABLE OF ROMAN LETTERS. I LXXX eighty. two. ninety. one hundred. IV four. two hundred. V five. CCC three hundred. VI six. СССС four hundred. VII D five hundred. VIII eight. six hundred. IX nine. seven hundred. X ten. eight hundred. XX twenty. DCCCC nine hundred. XXX thirty. one thousand. XL forty. fifteen hundred. L fifty. two thousand. LX sixty. X ten thousand. LXX seventy. one million. one. seven. QUESTIONS. — Art. 3. Why is the Roman notation so called ? By what are numbers expressed in the Roman notation ? What effect has the repetition of a letter? What is the effect of writing a letter expressing a less value before a letter denoting a greater ? What of writing the letter after another denoting a greater value? How many times is the value denoted by a letter increased by placing a dash over it? Repeat the tablg. The Roman notation is now but little used, except in numbering sections, chapters, and other divisions of books. EXERCISES IN ROMAN NOTATION. The learner may write the following numbers in letters : 1. Ninety-six. Ans. XCVI. 2. Eighty-seven. 3. One hundred and ten. 4. One hundred and sixty-nine. 5. Two hundred and seventy-five. 6. Five hundred and forty-two. 7. One thousand three hundred and nineteen. 8. One thousand eight hundred and fifty-cight. Art. 4. The Arabic notation, so called from its having been made known through the Arabs, employs in expressing numbers ten characters or figures, viz. : 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. one, two, three, four, five, six, seven, eight, nine, cipher. The first nine are sometimes called digits, from digitus, the Latin signifying a finger, because of the use formerly made of the fingers in reckoning. The cipher, also, has sometimes been called naught, or zero, from its expressing the absence of a number, or nothing, when standing alone. ART. 5. The particular position a figure occupies with regard to other figures is called its PLACE ; as in 32, counting om the right, the 2 occupies the first place, and the the second place, and so on for any other like arrangement of figures. The digits have been denominated significant figures, because each has of itself a positive value, always representing so many units, or ones, as its name indicates. But the size or value of the units represented by a figure differs with the place occupied by the figure. For example, there are written together to represent a number three figures, thus, 366 (three hundred and sixty-six). Each of the figures, without regard to its place, expresses units, or ones; but these units, or ones, differ in value. The 6 occupying the first place represents 6 single units; the 6 occupying the second place repre QUESTIONS. What use is now made of Roman notation ? Art. 4. How many characters are employed in the Arabic notation ? What are the first nine called, and why? What is the cipher sometimes called ? What does it represent when standing alone ? — Art. 5. What is meant by the place of a figure? What have the digits been denominated ? Why? How does the size or value of units represented by figures differ ? sents 6 tens, or 6 units each ten times the size or value of a unit of the first place; and the 3 occupying the third place represents 3 hundreds, or 3 units each one hundred times the size or value of a unit of the first place. ART. 6. The cipher becomes significant when connected with other figures, by filling a place that otherwise would be vacant ; as in 10 (ten), where it gives a ten-fold value to the 1; and 120 (one hundred and twenty), where it gives a ten-fold value to the 12; and 304, where it has the same effect, by filling an inter vening place, causing the 3 to represent three hundreds, instead of three tens. Art. 7. The simple value of a figure is the value its unit has when the figure stands alone; or, in a collection, when standing in the right-hand place. Thus 6 alone, or in 26, expresses a simple value of six single units, or ones. The local value of a figure is the value its unit has when the figure is removed from the right-hand place, and depends upon the place the figure occupies. The local value of figures will be made plain by the following table and its explanation. Hundreds of Thousands. The figures in this table are read thus : co Units. Nine. 98 Ninety-eight. 9 8 7 Nine hundred eighty-seven. 9 8 7 6 Nine thousand eight hundred seventy-six. 9 8 7 6 5 Ninety-eight thousand seven hundred sixty-five. Nine hundred eighty-seven thousand six hundred fifty-four. 9 8 7 6 5 4 3 Nine millions eight hundred seventy-six thousaná 9 8 7 6 5 4 { { five hundred forty-three. QUESTIONS. Art. 6. When does a cipher become significant ? - Art. 7. What is the simple value of a figure? What is the local value of a figure ? What is the design of this table ? |