135 141 FRACTIONS — VULGAR FRACTIONS, 140 Reduction of Vulgar Fractions, Subtraction of Vulgar Fractions, Multiplication of Vulgar Fractions, 155 PROPORTION, Division of Vulgar Fractions, 160 Miscellaneous Exercises in Vulgar Reduction of Fractions of Compound PARTNERSHIP, OR COMPANY BUSINESS, 248 Addition of Fractions of Compound Subtraction of Fractions of Com. Miscellaneous Examples in Profit and Questions to be performed by Analy. Miscellaneous Questions by Analy. NUODECIMALS, 179 Addition and Subtraction of Duodeci. Multiplication of Duodecimals, 259 Numeration of Decimal Fractions, Notation of Decimal Fractions, Multiplication of Decimals, .... 186 EVOLUTION, 188 Extraction of the Square Root, 264 Application of the Square Root, 268 Miscellaneous Exercises in Decimals, 192 Extraction of the Cube Root, Application of the Cube Root,. Annuities at Simple Interest by Arith- Miscellaneous Exercises in Interest, 206 Annuities at Compound Interest by Geometrical Progression, Table, • 290 293 ARITHMETIC. ARTICLE 1. ARITHMETIC is the science of numbers and the art of computing by them. A number is a unit or an assemblage of units. A unit or unity is the number one, and signifies an individual thing or quantity. The introductory and principal rules of arithmetic are Notation, Numeration, Addition, Subtraction, Multiplication, and Division. The last four are called the principal or fundamental rules, because all arithmetical operations depend upon them. SI. NOTATION AND NUMERATION. NOTATION. Art. 2. Notation is the art of expressing numbers by fig. ures or other symbols. There are two methods of notation in common use; the Roman, and the Arabic or Indian. * Art. 3. The Roman notation employs 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. The intermediate numbers and the numbers greater than one thousand are expressed by the use of these letters in various combinations; thus, II expresses two; IV, four; VI, siz; IX, nine; XV, fifteen ; &c. * For the origin of our present numeral characters, see the History of Arithmetic in the larger work of the author. QUESTIONS. — Art. 1. What is arithmetic? What is number? What is a unit or unity? Which are the principal or fundamental rules of arithmetic ? Why are they called the principal rules ? - Art. 2. What is notation ? How many and what methods of notation are in common use? - Art. 3. What are used to express numbers in the Roman notation? What are their names ? When two or more equal numbers are united, or a less number follows a greater, the sum of the two represents their value; as, XX, twenty; VI, six. But when a less number is placed before a greater, the difference of the two represents their value; as, IV, four; IX, nine. TABLE OF ROMAN LETTERS. I LX sixty II two. LXX one. seventy III three. LXXX eighty. IV four. XC ninety. V five. С one hundred. VI six. two hundred. VII seven. CCC three hundred. VIII eight. CCCC four hundred. IX nine. five hundred. X ten. six hundred. XX twenty. DCC seven hundred. XXX thirty. DCCC eight hundred. XL forty. DCCCC nine hundred. L fifty. one thousand. Any number between unity and two thousand may be expressed by the letters in the preceding table, By first writing down the largest part of the required number, found in the table, and then annexing to this the next less, that will not make a number greater than the one required, and thus proceeding until the number is complete. 2 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 forty-eight. QUESTIONS. When is the sum of two letters taken for their value ? When the difference? Repeat the Table of Roman Letters. What direction is given for writing numbers in the Roman notation ? ART. 4. The Arabic or Indian notation employs ten distinct characters or figures, sometimes called digits, viz. : 1, 2, 3, 4, 5, 6, 7, 8, 0. one, two, three, four, five, six, seven, eight, nine, cipher. The first nine are called significant figures, because each one has a value of itself when standing alone. The cipher is also sometimes called naught or zero; and, when standing alone, it has no value and signifies nothing. NUMERATION. Art. 5. NUMERATION is the art of reading numbers, or naming the value of figures in the order of their places. Art. 6. The Arabic figures have two values, a simple and a local, and, from their convenience, are now universally used in arithmetical calculations. Art. 7. The simple value of a figure is the value it has when standing alone, thus, 6; or when standing in the right-hand place of whole numbers, thus, 26. In either case the 6 denotes six units or ones. Art. 8. The local value of a figure is the value it has when it is removed from the right-hand place toward the left, and depends on the place the figure occupies. For example, 6 standing at the left hand of 5, thus, 65, expresses ten times the value it does when standing alone, or in the right-hand place, and denotes six tens or sixty; the five at the right hand of it denotes five units, and the two figures together express sixty-five. When placed at the left of two figures, thus, 678, it expresses one hundred times its simple value, or ten times its value when standing in the second or tens' place; its value being always increased tenfold, when it is removed one place to the left. Therefore, while the 8 denotes eight units, and the 7, seven tens, the 6 denotes six hundreds, and the whole together, 678, six hundred and seventy-eight. QUESTIONS. - Art. 4. How many characters are employed in the Arabic or Indian notation? What are the first nine called? Why? What is the tenth called ? What does it represent or signify when standing alone ? - Art. 5. What is numeration ? Art. 6. What two values have the Arabic figures ? - Art. 7. What is the simple value of a figure ? – Art. 8. What is the local value? Why is this value called its local value? What effect has the removal of a figure one place to the left upon its value? Two places ? &c. Art. 9. The cipher becomes significant when connected with other figures ; as in 10 (ten), where it gives a tenfold value to the l; and 120 (one hundred and twenty), where it gives a tenfold value to the 12; and 304 (three hundred and four), where it has the same influence on the 3, causing it to represent three hundreds instead of three tens. The local value of figures will be made plain by the following table and its explanation. Hundreds of Thousands. Tens. Luci for Iuelyn The figures in this table are read thus : 9 Nine. 9 8 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. 9 8 7 6 5 4 S Nine hundred eighty-seven thousand six hundred {fifty-four. 9 8 7 6 5 4 3 Nine millions eight hundred seventy-six thousand five hundred forty-three. It will be noticed in the above table, that each figure in the right-hand or units' place expresses only its simple value, or so many units; but, when standing in the second place, it denotes so many tens, or ten times its simple value; and when in the third place, so many hundreds, or one hundred times its simple value ; when in the fourth place, so many thousands, or a thousand times its simple value, and so on; the value of any figure being always increased tenfold by each removal of it one place to the left hand. 1 QUESTIONS. — Art. 9. When does the cipher become significant? What is its effect, when placed at the right hand of a figure ? What is the design of this table? What value has a figure standing in the right-hand or units' place? What, in the second place? What, in the third ? How do figures increase from the right toward the left ? |