275,301,246,157,896,275,832,456,297,143,215,367,291,3 NOTE 1.-The names of the periods above tredecillid quatuordecillion, quindecillion, sexdecillion, septendecillio decillion, novemdecillion and vigintillion. 16. Note the number of decimal places in each of lowing expressions: 6. .000896 896 millionths. (6 decimal places.) = 7. .000,468,275 = 8. .000,000,000,462 = billionths. (9 decimal p 9. .000,000,000,000,527 = trillionths. quadrillionths. quintillionths = 10. .000,000,000,000,000,001 = places. = 15. In any number of millionths there are places. d de 16. In any number of billionths there are places. de Algebra-Notation. 17. Letters are used to represent numbers; thus, the letter a, b, or c may represent a number to which any value may be given. 18. Known numbers, or those that may be known without solving a problem, when not expressed by figures, are usually represented by the first letters of the alphabet; as, a, b, c, d. ILLUSTRATIONS. (a) To find the perimeter of a square when its side is given. Let a one side.* Then 4 a = the perimeter. Hence the rule: To find the perimeter of a square, multiply the number denoting the length of its side by 4. (b) To find the perimeter of an oblong when its length and breadth are given. Let a = the length. the breadth. Then 2a 2b, or (a + b) × 2 = the perimeter. Hence the rule: To find the perimeter of an oblong, multiply the sum of the numbers denoting its length and breadth by 2. 19. Unknown numbers, or those which are to be found by the solution of a problem, are usually represented by the last letters of the alphabet; as, x, y, z. (a) There are two three times the first. Let Then and ILLUSTRATION. numbers whose sum is 48, and the second is What are the numbers? * That is, the number of units in one side. The letter stands for the number. 21. EXERCISE. Find the numerical value of each of the following expressio Find the numerical value of each of the following expressio =20, b = 4, and c = 2: 24. A geometrical line has length, but neither breadth nor thickness. NOTE.-Lines drawn upon paper or upon the blackboard are not geometrical lines, since they have breadth and thickness. They represent geometrical lines. 25. A straight line is the shortest distance from one point to another point. 26. A curved line changes its direction at every point. 27. A broken line is not straight, but is made up of straight lines. 1. The line AB is a 2. The line CD is a 3. The line EF is a 4. The line FG is a 5. The line EK is a 6. The perimeter* of a square is a equal lines. line made up of line 7. The perimeter of a regular pentagon† is a made up of — equal 8. The circumference lines. of a circle is a line, every point in which is equally distant from a point called the center of the circle. 9. The diameter of a circle is a line. 10. Imagine a straight line drawn upon the surface of a Can you draw a straight line upon the surface stovepipe. of a sphere? * See Book I., p. 53. + See Book I., p. 63. See Book II., p. 256. 28. MISCELLANEOUS REVIEW. 1. If a equals one side of a regular pentagon, the perimeter of the pentagon is 2. If equals the perimeter of a square, the side of the square equals b + 3. If a equals a straight line connecting two points and b equals a curved line connecting the same points, then a is † than b. 4. Find the difference between two hundred seven thousandths, and two hundred and seven thousandths. ‡ 5. How many zeros in 1 million expressed by figures? 1 billion? 1 trillion? 6. How many decimal places in any number of millionths? billionths? trillionths? 7. How many decimal places in 5 thousandths? in 25 thousandths? in 275 thousandths? in 4346 thousandths? 8. A figure in the second integral place represents units how many times as great as those represented by a figure in the second decimal place? of the following? 12a+3b-5 d. 10. John had a certain amount of money and James had 5 times as much; together they had 354 dollars. How many *The expression "a equals one side" means that a equals the number of units in one side. Remember that in this kind of notation the letters employed stand for numbers. + Longer or shorter? See p. 15, Exercise 14. |