EXAMPLE VIII. Analyze three hundred billions, seventy millions, nine thousand and eight. EXAMPLE IX. Analyze sixty-five trillions, three hundred millions, nine hundred and forty thousand, six hundred and five. EXAMPLE X. Analyze three hundred and twenty-one billions, nine hundred and forty-six millions, seven hundred and fifty-one thousand, four hundred and twenty-five. 1. In the analysis of numbers, it becomes evident that some numbers are regular, and others irregular, according as they have not, or have, vacant places; hence, a number may be said to be regular when its orders of units proceed in a regular, or unbroken series; as, seven thousand three hundred and twenty-one; but irregular when the series is broken, or some of the orders left out, i. e., some of the places vacant; as, seven thousand and twenty-one. Numbers consisting of a single order of units, are sometimes called round numbers; as, one thousand, five thousand, fisty millions, &c. 2. It is also evident that the orders of units composing any number, occur in regular succession, the largest on the left, and the smallest on the right, and that the places of these orders of units increase tenfold from right to left, and decrease in the same ratio from left to right; hence, the process of writing numbers will be greatly simplified, by adopting nine characters, or figures, which may severally represent as many distinct quantities of units as are allowed to each order. This is called Notation. ARTICLE II. NOTATION is the art of writing by figures. In arithmetic it teaches to represent numbers by the following characters, viz. 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, which taken alone, represent the numbers preceding them, thus: one 1, two 2, three 3, four 4, five 5, six 6, seven 7, eight 8, nine 9, nought 0, or cipher, so called, because of itself, it represents nothing, and is used in Notation to fill those places which were left vacant in the number to be represented; the others are called significant figures because each has a specific, or particular value. Now, since the numbers from one to nine are expressive of all the particular quantities that may be formed in any order of units, and since the figures from 1 to 9, are the representatives of these numbers, therefore, these figures, may represent each order of units, as it occurs in any given number, both in quantity and place, since the places occur in regular succession, as has already been shown. This process will be rendered simple by a few examples, after having studied carefully the following RULE for writing numbers. Having analyzed the given number, represent the given places, or orders of units, by significant figures, and the vacant places by ciphers, as they occur in order from left to right. EXAMPLES.* 2 1. Write ten. Here we have 1 unit, and 1 O unit; hence the 2 1 2. Write eleven, i. e., 1 unit, and 1 unit. 1 4. Write nineteen, or 1 unit, and 9 units. 5. Write twenty, or 2 units, and 0 units. * Hereafter it will be more convenient to designate the orders of units by 1 a figure placed above the word unit; thus: unit signifies a unit of the first 2 order; units, a unit of the second order, &c. 6. Write ninety-nine, or 9 units, and 9 units. Ans, is 99. 7. Write one hundred, or 1 unit, 0 unit, and 0 unit. Ans. 100 3 8. Write nine hundred and ninety-nine, or 9 units, 1 9 units, and 9 units. Ans. 999. 4 3 9. Write seven thousand and five, i. e., 7 units, O units, O units, and 5 units. Ans. 7005. 10. Write five billions, sixty millions, three hundred, and five thousand and nine. Analyze thus: 5 units, 0 units, 6 units, 0 units, 3 units, 0 units, 5 units, 0 units, 0 units, 9 units; hence, Ans. 5060305009. 11. Write nine hundred and eighty-seven millions, six hundred and fifty-four thousand, three hundred and twenty one. 12. Write fifty trillions, six hundred billions, nine millions, seventy-three thousand, eight hundred and ninety-six. 13. Write one quintillion, one hundred quadrillions, ten trillions, one billion, one hundred millions, ten thousand and one.* 14. Write seven sextillions, seven quintillions, seven quadrillions, seven trillions, seven billions, seven hundred millions, seventy thousand, seven hundred and seventy-seven. ROMAN NOTATION. The most improved characters employed in Notation among the ancient Romans, were, I, 1; V, 5; X, 10; L, 50; C, 100; * In this number we have, first, 1 unit; then follow, 1 unit, 0 unit, 0 unit, 8 7 15 14 13 O unit, 1 unit, 0 unit, 0 unit, 0 unit, 1 unit, 1 unit, 0 unit, 0 unit, 0 unit, 1 unit, O unit, 0 unit, 0 unit, 1 unit; hence, the answer is, 1100010001100010001. D, 500; M, 1000. With these characters numbers were at first represented by placing the largest on the left, and the rest in order toward the right, according to their value; as, VIIII, 9; XVIIII, 19; XXXX, 40; LXXXX, 90, &c.; showing that the smaller characters must be added to the larger ones which precede; but latterly smaller characters were placed before larger ones, signifying that their difference must be taken, instead of their sum; as, IV, 4; IX, 9; XIX, 19; XL, 40; XC, 90, &c.; hence, I, 1; II. 2; III, 3; IV, 4; V, 5; VI, 6; VII, 7; VIII, 8; IX, 9; X, 10; XI, 11; XII, 12; XIII, 13; XIV, 14; XV, 15; XVI, 16; XVII, 17; XVIII, 18; XIX, 19; XX, 20; XXX, 30; XL. 40; L, 50; LX, 60; LXX, 70; LXXX, 80; XC, 90; C. 100; CC, 200, &c.; D, 500; DC, 600, &c. ; M, 1000; MM, 2000, &c.; DCLVIII, 658; MCI, 1101; MDCCCXLVII, 1847, &c. EXAMPLES. Write in Roman characters: 1. Fifty-three; 2. Ninetynine; 3. One hundred and forty-nine; 4. One thousand seven hundred and seventy-six. ARTICLE III. NUMERATION. NUMERATION is the art of numbering; and, in Arithmetic, it teaches to read, in proper language, any number which has been written according to the laws of Notation, as directed in the following RULE for reading numbers. Separate the figures into portions of three, from right to left; call the first three units, second, thousands, &c., (in the order of the Periods,) that the first three may represent the Period of units, second, thousands, &c., then the numbers in each period, read in order from left to right, giving the name of each period as it occurs, will be the numeration required. EXAMPLES. Periods. Bill. Mill. Thous. Units. 1. Read 7895486095. Separate thus: 7,895,486,095, hence, we have 7 Billions, 895 millions, 486 thousand, 095 units, i. e., Seven billions, eight hundred and ninety-five millions, four hundred and eighty-six thousand, and ninety-five. Bill. Mills. Thous. Units. 2. Read 12065090686, thus: 12,065,090,686, making 12 billions, 065 millions, (i. e. 65 millions,) 090 thousand, (i. e., 90 thousand,) 686 units; hence, we have Twelve billions, sixty-five millions, ninety thousand, six hundred and eighty-six, for the numeration required. 3. Read 650198760065930809; Quads. Trills. Bills. Mills. Thous. Units. Thus: 650,198,760,065,930,809, i. e., 650 quadrillions, 198 trillions, 760 billions, 65 millions, 930 thousand, 809. Ans. Six hundred and fifty quadrillions, one hundred and ninety-eight trillions, seven hundred and sixty billions, sixtyfive millions, nine hundred and thirty thousand, eight hundred and nine. Having explained Notation and Numeration, there yet remains four principal Rules, viz. Addition, Subtraction, Multiplication, and Division, the operations of which are exhibited in the following signs, viz. = two parallel lines signify equality; as, 3+4=7. +plus signifies more, or Addition; as, 5+3=8. minus signifies Subtraction; as, 5—3=2. X signifies multiplied by; as, 4x 3=12. or minus signifies Division; as, 8÷4-2. or =2. |