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 cipher, or nothing. The first nine are called significant figures, as distinguished from the cipher, 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. 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 1831, 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 cipher, 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 successively 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. 123,456 ; 789,123;456,123;456,123;123,456;789,789;323,456;789,712;333,345;789,123;137,390; 711,716;371,712;456,711. ENGLISH NUMERATION TABLE. Thousands. To enumerate any number of figures, Tridecillions. they must be separated by semicolons into divisions of six figures each, and Thousands. each division by a comma, as in the an nexed table. Each division will be Duodecillions. known by a different name. The first Thousands. three figures in each division will be so Undecillions. many thousands of that name, and the next three will be so many of that name, Thousands. that is over its unit's place. The value Decillions. of the numbers in the annexed table is, One hundred twenty-three thousand, Thousands. four hundred fifty-six tridecillions ; sevNonillions. en hundred eighty-nine thousand, one hundred twenty-three duodecillions ; Thousands. four hundred fifty-six thousand, one Octillions. hundred twenty-three undecillions; four hundred fifty-six thousand, one hundred Thousands. twenty-three decillions ; one hundred Septillions. twenty-three thousand, four hundred fifty-six nonillions ; hundred Thousands. eighty-nine thousand, seven hundred Sextillions. eighty-nine octillions; three hundred twenty-three thousand, four hundred Thousands. fifty-six septillions; seven hundred eigh ty-nine thousand, seven hundred twelve Quintillions. sextillions ; three hundred thirty-three Thousands. thousand, three hundred forty-five quintillions ; hundred eighty-nine Quatrillions, thousand, one hundred twenty-three Thousands. quatrillions ; one hundred thirty-seven thousand, eight hundred ninety trillions; Trillions. seven hundred eleven thousand, seven Thousands. hundred sixteen billions ; three hundred Billions. seventy-one thousand, seven hundred twelve millions ; four hundred fifty-six Thousands. thousand, seven hundred eleven. Millions. Note. — The student must be familiar with Thousands. the names from Units to Tridecillions, and from Tridecillions to Units, so that he may repeat Units. them with facility either way. seven seven 876,789,335, 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 Duodecillions. every three figures have a different name. Their value would be thus exUndecillions. pressed, Eight hundred seventy-six triDecillions. decillions, seven hundred eighty-nine duodecillions, eight hundred thirty-five Nonillions. undecillions, one hundred twenty-three Octillions. decillions, three hundred sixty-nine no 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 milBillions. lions, four hundred seventy-eight thou sands, six hundred thirty-eight. Units. 376 143,776,711,333 41,771,6:31,147,671 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. Section 2. ADDITION. MENTAL EXERCISES. 1. John had two cents and Samuel gave him two more, how many has he ? 2. Thomas had three nuts and James gave him three more, how many has he ? 3. A boy had four apples, and he found two more, how many in all ? 4. I have six dollars, and a man has paid me three more, how many have I ? 5. Enoch had seven marbles, and John gave him two - more ; how many has he ? 6. Benjamin has four dollars, and his sister has three ; how many have both ? 17. Paid five dollars for a barrel of flour, and seven dollars for sugar ; how much for both ? 8. James had two cents and Samuel gave him six more ; how many has he ? 9. How many are five apples and six apples ? 10. How many are four dollars, and eight dollars ? 11. How many are 2 and 3 ? 2 and 5 ? 2 and 7 ? 2 and 9 ? 12. How many are 3 and 3 ? 3 and 5 ? 3 and 7 ? 3 and 9 ? 13. How many are 4 and 3? 4 and 5 ? 4 and 8 ? 4 and 9 ? 14. How many are 5 and 3 ? 5 and 4 ? 5 and 7 ? "5 and 8? 5 and 9 ? 15. How many are 6 and 2 ? 6 and 4? 6 and 3 ? 6 and 5? 6 and 7 ? 6 and 9 ? 16. How many are 7 and 3? 7 and 5 ? 7 and 7? 7 and 6 ? 7 and 8? 7 and 9 ? 17. How many are 8 and 2? 8 and 4? 8 and 5 ? 8 and 7 ? 8 and 9 ? 8 and 8 ? 18. How many are 9 and 1 ? 9 and 3 ? 9 and 5 ? 9 and 4 ? 9 and 6 ? 9 and 8? 9 and 9 ? 19. How many are 11 and 3 ? 11 and 2 ? 11 and 4 ? 11 and 6 ? 11 and 7 ? 11 and 9 ? 11 and 11 ? 11 and |