CHARACTERS USED IN THIS WORK. $ Contraction, for U. S., United States' currency, and is prefixed to dollars and cents. = Sign of equality ; as 12 inches = 1 foot, signifies, that 12 inches are equal to one foot. + Sign of addition ; as 8+6=14, signifies, that 8 add ed to 6 is equal to 14. Sign of subtraction ; 8–6=2, that is, 8 less 6 is equal to 2. X Sign of multiplication; as 7X6=42, that is, 7 multi plied by 6 is equal to 42. • Sign of division; as 42:6–7, that is, 42 divided by 6 is equal to 7. 4 Numbers placed in this manner imply, that the upper line is to be divided by the lower line. ::: : Signs of proportion ; thus, 2:4 :: 6 : 12, that is, 2 has the same ratio to 4, that 6 has to 12 ; and such numbers are called proportionals. 15—5+3=13. Numbers placed in this manner show, that 5 is to be taken from 15, and 3 added to the remainder. The line at the top is called a vinculum, and connects all the numbers, over which it is drawn. 9 Implies, that 9 is to be raised to the second power ; that is, multiplied by itself. 8 Implies, that 8 is to be multiplied into its square. 3 ARITHMETIC. Section 1. ARITHMETIC is the art of computing by numbers. Its five principal rules are Numeration, Addition, Subtraction, Multiplication, and Division. NUMERATION. Numeration teaches to express the value of numbers either by words or characters. The numbers in Arithmetic are expressed by the following ten characters, or Arabic numeral figures, which the Moors introduced into Europe about nine hundred years ago ; viz. 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine, 0 cypher, or nothing. The first nine are called significant figures, as distinuished from the cypher, which is, of itself, insignificant. Besides this value of those figures, they have also another, which depends on the place in which they stand, when connected together; as in the following table. А wo Hundreds of Millions. coco Tens of Millions. coc v Hundreds of Thousands. Voor Tens of Thousands. co voer A Thousands. coco Torico Hundreds. 4 3 9 8 5 4 3 6 5 4 9 Here any figure in the first place, reckoning from right to left, denotes only its simple value ; but that in the second place, denotes ten times its simple value ; and that in the third place a hundred times its simple value ; and so on ; the value of any successive place being always ten times its former value. Thus in the number 1834, the 4 in the first place denotes only four units, or simply 4; 3 in the second place signifies three tens, or thirty ; 8 in the third place signifies eighty tens or eight hundred ; and the 1, in the fourth place, one thousand ; so that the whole number is read thus, one thousand eight hundred and thirty-four. As to the cypher, 0, though it signify nothing of itself, yet being joined to the right hand of other figures, it increases their value in a tenfold proportion ; thus 5 signifies only five, but 50 denotes 5 tens or fifty ; 500 is five hundred ; and so on. NOTE. The idea of number is the latest and most difficult to form. Before the mind can arrive at such an abstract conception, it must be familiar with that process of classification, by which we suce cessively remount from individuals to species, from species to genera, from genera to orders. The savage is lost in his attempts at enumeration, and significantly expresses his inability to proceed, by holding up his expanded fingers or pointing to the hair of his head. See Lacroix. ENGLISH NUMERATION TABLE. Thousands. To enumerate any number of figures, they must be separated by semicolons into divisions of six figures each, and each division by a comma, as in the annexed table. Each division will be known by a different name. The first three figures in each division will be so many thousands of that name, and the next three will be so many of that name, that is over its units place. The value of the numbers in the annexed table is, One hundred twenty-three thousand, four hundred fifty-six tridecillions ; seven hundred eighty-nine thousand, one hundred twenty-three duodecillions ; four hundred fifty-six thousand, one hundred twenty-three undecillions; four hundred fifty-six thousand, one hundred twenty-three decillions ; one hundred twenty-three thousand, four hundred fifty-six nonillions ; seven hundred eighty-nine thousand, seven hundred eighty-nine octillions ; three hundred twenty-three thousand, four hundred fifty-six septillions; seven hundred eighty-nine thousand, seven hundred twelve, sextillions ; three hundred thirty-three thousand, three hundred forty-five quintillions ; seven hundred eighty-nine thousand, one hundred twenty-three quatrillions ; one hundred thirty-seven thousand, eight hundred ninety trillions; seven hundred eleven thousand, seven hundred sixteen billions; three hundred seventy-one thousand, seven hundred twelve millions ; four hundred fifty-six thousand, seven hundred eleven. 123,456;789,123;456,123;456,123;123,456; 789,789;323,456;789,712; 333,345;789,123;137,890; 711,716;371,712;456,711. Billions. Thousands. Millions. Note. The student must be familiar with the names from Units to Tridecillions, and from Tridecillions to Units, so that he may repeat them with facility either way. Units. 876,789,835, 123,369,873,777, 127,894, 237,867, 123, 678,478,638. FRENCH NUMERATION TABLE. Tridecillions. It will be seen by the annexed table, that every three figures have a differDuodecillions. ent name. Their value would be thus Undecillions. expressed, Eight hundred seventy-six tridecillions, seven hundred eighty-nine Decillions. duodecillions, eight hundred thirty-five Nonillions. undecillions, one hundred twenty-three decillions, three hundred sixty-nine noOctillions. nillions, eight hundred seventy-three Septillions. octillions, seven hundred seventy-seven Sextillions. septillions, one hundred twenty-seven sextillions, eight hundred ninety-four Quintillions. quintillions, two hundred thirty-seven Quatrillions. quatrillions, eight hundred sixty-seven trillions, one hundred twenty-three bilTrillions. lions, six hundred seventy-eight mil lions, four hundred seventy-eight thouBillions. sands, six hundred thirty-eight. 2 Millions. Thousands. 376 611,711 3,131,671 637,313,789 63,113,716,716 143,776,711,333 44,771,631, 147,671 3,761,716, 137,716,716 871,137,637,471,378,637 3,761,716,137,716,167,138 611,167,637,896,431,617,761,617 671,386,131,176,378, 171,714,563,813 137,471,716,756,378,817,371,767,386,389,716,473 Note. Although the French method of enumeration is generally used, yet it may be well for the pupil to understand both the English and the French. |