86. Two hundred forty-five billion one hundred seven million fifty-nine thousand eight hundred seventy. 87. Twenty-five trillion two hundred seven million thirty seven thousand three hundred one. 88. One hundred forty-three trillion six hundred twenty seven billion two hundred twenty-five million three hundred forty-six thousand eleven. 89. Sixteen quadrillion three trillion seventeen billion five million thirty-four thousand two. RULE FOR NUMERATION.—1. Begin at the right, and separate the number into periods of three figures each. 2. Begin at the left, and read each period containing on er more figures as if it stood alone, adding its name. Read the following numbers: 35. In the English method of numeration six figures form a period. The first period denotes ones, the second millions, the third billions, etc. (1) 30075,807250 denotes 30075 millions, 807250 ones; read, thirty thousand seventy-five million eight hundred seven thousand two hundred fifty. (2) 50,000320,400062 denotes 50 billions, 320 millions, 400062 ones; read, fifty billion three hundred twenty million four hundred thousand sixty-two. THE ROMAN NOTATION. 36. In the Roman notation seven capital letters are employed to denote numbers, viz.: Letters: I, མ, X, L, C, D, M, Values: one, 5 ones, 1 ten, 5 tens, 1 hund., 5 hunds., 1 thousand. Placing a dash over a letter increases its value 1000 times. Thus: 5 thous. 10 thous. 5 ten-thous. 1 hund. thous. 5 hund. thous. 1 million. Hence, I, X, C, M, X, C, M, etc., represent 1 unit of the first, second, third, etc., order, called unit letters; and V, L,D,V,L, D, etc., represent 5 units of first, second, third, etc., order, called 5-unit letters. Το express other numbers the letters are combined thus: 10. Writing the unit letter 1, 2 or 3 times, expresses 1, 2 or 3 units of that order. Thus, III denotes 3 ones or 3; XX, 2 tens or 20; CCC, 3 hundreds or 300. 20. Writing the unit letter before the 5-unit letter expresses 4 units of that order. Thus, IV denotes 4 ones or 4; XL, 4 tens or 40; CD, 4 hundreds or 400. 30. Writing the unit letter 1, 2 or 3 times after the 5-unit letter, expresses 6, 7 or 8 units of that order. Thus, VI denotes 6 ones or 6; LXX, 7 tens or 70; DCCC, 8 hundreds or 800. 40. Writing a unit letter before the unit letter of the next higher order, expresses 9 units of the lower order. Thus, IX denotes 9 ones or 9; MX, 9 thousand or 9000. 50. Writing in order the letters representing the local values of the digits in any number, expresses that number in the Roman notation. Thus, 874 is denoted by DCCCLXXIV. 37. The sign of dollars is $, which is called the dollar mark. A period (.), called the decimal point, is placed before a number expressing cents. (1) $3 is 3 dollars. (4) $2.75 is 2 dollars 75 cents. In expressing dollars and cents, by figures, place $ before the number, the decimal point before the cents, and a cipher between the decimal point and any number of cents less than 10. Express by figures: 10. Five cents. 14. Sixty-one cents. 16. Three cents. 11. Nine dollars seventy-three cents. 13. Twenty dollars eleven cents. 15. Forty-one dollars twenty cents. 17. Seventy dollars six cents. 19. Five dollars sixty-five cents. seventy-eight dollars forty-two cents. QUESTIONS. What is a unit? A number? The unit of a number? Like numbers? Unlike numbers? An abstract number? A concrete number? An integer? What is arithmetic? Science? Art? Problems? Parallel problems? In this treatise how are oral problems often designated, and what are they intended to illustrate? What is notation? Nnmeration? How many characters are used in Arabic notation? On what principles are they combined so as to express any number? Give an example. What is the value of a number? In 5384 what is the value of 3? 4? 5? 8? What does a figure in the fifth place denote? In the sixth? In the seventh? What is the difference between the French and English methods of numeration? Which method is in common use? What letters are used in the Roman notation? What values do they represent? How combined so as to express other numbers? What is the rule for expressing dollars and cents by figures? ADDITION. 38. (1) There are 8 marbles in one sack, 6 marbles in another and 4 marbles in a box; how many marbles are there in all? Uniting 8 marbles and 6 marbles I have 14 marbles; now uniting 14 marbles and 4 marbles I have 18 marbles. Hence, there are 18 marbles in all. 39. Addition is uniting two or more numbers into one. 40. The parts are the numbers added, and the sum or amount is the result of addition. (1) 8 hats and 7 hats are 15 hats. Here, 8 hats and 7 hats are the parts, and 15 hats the sum. 41. A sign is a symbol used to denote an operation or relation. 42. The sign of addition is, which is called plus, meaning more. When stands between two numbers, it indicates that they are to be added. 43. The sign of equality is, which is read equals, or is equal to. Thus, 7411, is read: 7 plus 4 equals 11. It may also be read: 7 and 4 are 11. 44. PRINCIPLES.-1°. Only like numbers can be added. For, a number is a collection of like units; hence, numbers of unlike units cannot be collected into one number. (1) 5 days, 8 days and 2 days can be added. (3) Numbers composed of ones, tens, hundreds, etc., may be added by adding units of like order. Thus, 43 + 25: = 4 tens 3 ones + 2 tens 5 ones 6 tens 8 ones = 68. 2°. The sum is the same in whatever order the parts are added. (1) 8+ 5+ 7 + 3 = 13+7+ 3 (2) 8 +5 +7 +3 = 8+ 510 = 20+ 3 == 23. |