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. 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. 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 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. 20. The sign of multiplication is usually omitted between two letters representing numbers, and between figures and letters; thus, a × b, is usually written ab; b× 4, is written 4 b. 6 ab, means, 6 times a times b, or 6 × a × b. 21. EXERCISE. Find the numerical value of each of the following expressions, if a = 8, b = 5, and c = 2: Find the numerical value of each of the following expressions if a = 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. line, every 7. The perimeter of a regular pentagont is a 8. The circumference of a circle is a 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 stovepipe. Can you draw a straight line upon the surface 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? 9. If a = 6, b = 2, and d = 8, what is the numerical value of the following? 12 a+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 dollars had each ? Let x= the number of dollars John had. 5x the number of dollars James had, and x+5x= *The expression " 354 dollars. 6 x 354 dollars. x= 5x = 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. ADDITION. 29. Addition (in arithmetic) is the process of combining two or more numbers into one number. NOTE 1.-The word number, as here used, stands for measured magnitude, or number of things. NOTE 2.-Addition (in general) is the process of finding the sum of two or more magnitudes. 30. The sum is the number obtained by adding. 31. The addends are the numbers to be added. 32. The sign, +, which is read plus, indicates that the numbers between which it is placed are to be added; thus, 6 + 4, means that 4 is to be added to 6. 33. The sign, =, which is read equal or equals, indicates that that which is on the left of the sign equals that which is on the right of the sign; thus, 3 + 4 = 7. 7. 5+ 4+ 2 = 6 + 5. 34. PRINCIPLES. 1. Only like numbers can be added. 2. The denomination of the sum is the same as that of the addends. 35. PRIMARY FACTS OF ADDITION. There are forty-five primary facts of addition. They are given in the Werner Arithmetic, Book II., p. 273. The nine primary facts which many pupils fail to memorize perfectly are given below. |