8380 IQB ARITHMETIC Defined ...... ................... Numeration .......................... ...... Numeration Table to nine places of Figures .............. Numeration Table exhibiting both the French and English methods .. Addition of Simple Numbers ....................... Proof of Addition ............ Subtraction Table . ............... Proof of Subtraction .....:::::::::: Questions involving Addition and Subtraction ... Multiplication of Simple Numbers .......... Multiplication Table ............... Proof of Multiplication ......... Exercises in Multiplication ...... Division of Simple Numbers ... pe Numbers ...................... Proof of Division .............. Exercises in Division : ...... Questions exercising the Four Ground Rules . .... Fractions Defined ..................... Vulgar Fractions Defined ........... Reduction of Vulgar Fractions ... Greatest Common Divisor ....... Least Common Multiple ......... Subtraction of Fractions ..... Multiplication of Fractions ... Exercises in Vulgar Fractions ............. Decimal Fractions .............................. Numeration Table of Whole Numbers and Decimals ................ 101 Addition of Decimal Fractions ..... Subtraction of Decimal Fractions ...... Muitiplication of Decimal Fractions ......... Division of Decimal Fractions ....... Numeration Table of Federal Money .. Some fractional parts of a Dollar .. Questions wrought by Decimals. ...S. Denominate Numbers ........... 1340 . .. . .. 1775 . 214 PAGR W ine Measure ......................... .... Circular Measure, or Motion ....... Some additional Measures ....... Names of Bcoks ............ • .. 141 Reduction .......... . ....... Reduction Ascending .............. Addition of Denominate Numbers ........ Subtraction of Denominate Numbers ..... Exercises in Addition and Subtraction .. Multiplication of Denominate Numbers ... Division of Denominate Numbers ......... Questions involving the preceding Rules .... Denominate Fractions ............. Reduction of Denominate Fractions...... . Addition of Denominate Fractions .... Subtraction of Denominate Fractions ... Vulgar Fractions reduced to Decimals.. Reduction of Denominate Decimals .... Addition and Subtraction of Duodecimals .... *Multiplication of Duodecimals .......... Reduction ot Currencies .............. Value of Foreign Coins at Custom House ... Rule of Three ................... Compound Proportion ................. Practice ...................................... Table of Aliquot Parts ......... Simple Interest ............... Interest by Aliquot Parts ......................... Interest when time is Estimated in Days ..... Partial Payments ......... ...... Double Fellowship ............... ........ .... Assessment of Taxes .................. Equation of Payments ............... Involution ...................... Extraction of the Square Root .. Examples involving the Square Root ... Extraction of the Cube Root ....... Examples involving the Cube Root ... Arithmetical Progression ............... Geometrical Progression ........................... Summation of a Descending Geometrical Progression, continued to infirity 317 Alligation Alternate .................. ............. 319 Mensuration ...................... Promiscuous Questions ............. ARITHMETIC. ARTICLE 1. ARITHMETIC is the science of numbers. The operations of arithmetic are performed by the aid of five distinct rules, viz.: Numeration, Addition, Subtraction,Multiplication, and Division. These are usually called the FUNDAMENTAL RULES of arithmetic, because all other niles are founded upon them. What is Arithmetic ? How many distinct rules has it for its operations ? Repeat their names. What are these usually called ? Why are they so called ? NUMERATION. 2. NUMERATION explains the method of reading written numbers. Notation is the writing down of numbers. Various methods of notation and numeration were used by the ancients. We shall content ourselves with mentioning two, the common or Arabic method, and the Roman method. In the common method ten characters are employed. These characters when written are, 1, 2, 3, 4, 5, 6, 7, 8, 9,0 When printed, they become, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. They have the following names: 1 is called One, or a Unit, O is called Naught, Cipher, or Zero. Each of these characters, except the zero, is called a digit* ; and the first nine, when taken together, are called the nine digits. Any digit is called a significant figure. What is numeration? How is the common method sometimes called ? In this method how many characters are employed ? What are the names of these char acters ? What are called digits ? What is a significant figure ? 3. The significant figures have unchanging values ; that is, they always represent units or ones ; but the units which they represent differ in value. When a significant figure stands disconnected from other figures, the value of its unit is called its simple value. When such figure stands in connection with other figures, the value of its unit will depend upon the place which it occupies, and is therefore called its local value. Thus, in the number 3456, which consists of four sig. nificant figures standing in connection with each other, each figure expresses units; but units of different values. The right-hand figure, 6, expresses six units, whose value is their simple value; that is, each unit is a single one. The second figure, 5, expresses five units; but each unit is ter times greater than each unit of the first figure; therefore the 5 may be read 5 tens, equal to fifty units of simple value. The units expressed by the third figure, 4, are ten times greater than the units expressed by the second figure, and one hundred times greater than those expressed by the first figure; the third figure is therefore read 4 hundreds. The last figure, 3, expresses units ten times greater than the units in 4, and one thousand times greater than the units in 6, and is read 3 thousands. * From the Latin, digitus, a finger; because the ancients used to do they reckon on their fingers. Originally 10 was also called a digit. Hence this property: When figures are connected in a line as in the number 3456, the units which they express are said to be of different orders. Thus, 6 occupies the first place, and its units are of the first order, that is, they have their simple value. The 5 occupies the second place, and its units are of the second order, or tens. The 4 occupies the third place, and its units are of the third order, or hundreds The 3 occupies the fourth place, and its units are of the fourth order, or thousands. Hence the above number is three thousand four hundred and fifty-six. To numerate and read the numbers in the following table, proceed thus: Begin with the upper line 3. The first place only being occupied, you numerate Units. Then read, three units, or simply three. In the second line two places are occupied—then numerate Units, Tens-read fifty-four. In the third line three places are occupied ; then numerate Units, Tens, Hundreds—read two hundred and sixty-seven, and so proceed. |