APPENDIX B: Review of Basic Arithmetic

Transcription

1 APPENDIX B: Review of Basic Arithmetic Personal Trainer Algebra Click Algebra in the Personal Trainer for an interactive review of these concepts. Equality = Is equal to 3 = 3 Three equals three. 3 = +3 Three equals positive three. 4 = 4 Negative four equals negative four. 4 = (4) Negative four equals negative the quantity four. X = X X equals X. X = 4 X equals 4. Is approximately equal to is approximately equal to 3.. Not Equality Is not equal to is not equal to is not equal to negative 4. X X X is not equal to negative X. Inequality > Is greater than 4 > 3 4 is greater than ch_appb.indd 555

2 55 Appendix B Review of Basic Arithmetic 4 > 4 4 is greater than negative 4. a > b a is greater than b. < Is less than 5 < 7 5 is less than 7. < Negative is less than. X < 4 X is less than 4. Is greater than or equal to is greater than or equal to is greater than or equal to is greater than or equal to. Is less than or equal to 5 is less than or equal to 5. X X is less than or equal to Negative 7 is less than or equal to positive 7. < 3 < 5 3 is greater than and less than 5. 5 > 3 > 3 is less than 5 and greater than (a true statement, but we prefer the order < 3 < 5). 4 < X X is greater than 4 and less than or equal to. Absolute Value X The absolute value of X is the value of the number with the negative sign (if any) removed. 7 = 7 The absolute value of 7 is 7. = The absolute value of negative is. + = The absolute value of positive is. Addition and Subtraction = 8 3 plus 5 equals = 5 minus 4 equals = The order in which numbers are added does not change the result = The order in which numbers are added does not change the result. 5 + ( 3) = 5 3 = Adding a negative number is the same as subtracting the same positive number. ch_appb.indd 55

3 Appendix B Review of Basic Arithmetic = 5 Anything plus equals itself. 4 = 4 Anything minus equals itself. Multiplication and Division (3) = times 3 equals. 3() = (3) = The order in which terms are multiplied is not important. 7() = 7 Anything times equals itself. 5() = Anything times equals. ( 3) = A positive number times a negative number is always negative. = / = 3 divided by equals 3. = / = Anything divided by equals itself. a b = ( )/ = 3 The result of a negative number divided by a positive number is negative. The upper portion of a fraction (here, a) is called the numerator; the lower portion (here, b) is called the denominator. = /( 3) = The result of a positive number divided by a negative number 3 is negative; if either the numerator or the denominator (but not both) is neg ative, the result is negative. 3 = ( )/( 3) = If both numerator and denominator are negative, the result is positive. = / = Zero divided by anything is. = / (undefined) Anything divided by is undefined. = / (undefined) The result of dividing by is undefined. ( )( )( 3) = 3 When a series of numbers are multiplied, if the total number of nega tive terms is odd, the result is negative. ( )()( 3) = 3 When a series of numbers are multiplied, if the total number of nega tive terms is even, the result is positive. ( a)( b)( c) = abc a( b)( c) = abc ( )( 3) The previous two rules hold for both variables and numbers. = ( )( 3)/( ) = 9 When a series of numbers are divided, the same rules apply: If the total number of negative signs is odd, the result is negative; if the number of negative signs is even, the result is positive. ch_appb.indd 557

4 558 Appendix B Review of Basic Arithmetic Exponentiation 4 = 4(4) = 4 squared (or 4 to the second power) equals. a = a(a) Any number squared is that number times itself. 5 = 5 Any number to the first power is that number itself. 5 = Any number to the zeroth power equals. ( a) = ( a)( a) = a A negative number squared is positive. ( 3 )( ) = 3+ = 5 If two exponential quantities that have the same base are multiplied, the result can be obtained by adding the exponents. 3 = 3 = = If two exponential quantities that have the same base are divided, the result can be obtained by subtracting the exponents. 3 = raised to the third power is followed by three zeros. 5 =, raised to the fifth power is followed by five zeros. The square root of 4 =± The square root of 4 is either + or. 4( 4) = 4 The definition of the square root: The square root of a number is that value that multiplied by itself gives the original number. 4 / = 4 = ± The square root can be written as an exponent of /. 4 ( 4 ) = 4 = 4 This follows from the rule that states that multiplication is the addition of exponents, and also from the definition of the square root. ( 3 ) = 3() = =,, When a value with an exponent is itself raised to a power, the ( 3) = ( 3/ ) = 3 / ( ) = 3 = 3 exponents are multiplied. 4 is undefined The square root of any negative number is not defined (in the real number system); if your computation results in taking the square root of a negative number, you have made a mistake somewhere. Fractions a b 5 3 The upper portion of a fraction (here, a) is called the numerator; the lower portion (here, b) is called the denominator = = When fractions that are to be added have the same denominator, 5 5 add the numerators and divide the sum by the denominator. + 3 = 5 + = When fractions that are to be added do not have the same denominator, they must be converted to fractions that do have the same ( common ) denominator and then added. ch_appb.indd 558

5 Appendix B Review of Basic Arithmetic = =.5743 When fractions are to be multiplied, multiply the numerators together and then divide the product by the product of the denominators. Fractions may be converted to their decimal equivalents by dividing =.57 Sometimes numbers are rounded (here, to two decimal places) by pro cedures described in Chapter..57 is.57()% = 57% To convert a decimal fraction to a percent, multiply by = 5 When the same factor appears in both the numerator and the denomi nator of two fractions that are being multiplied, they may be canceled. Here, the 3 s disappear. Order of Operations + 3 = 3 + The order of addition does not matter. (3) = 3() The order of multiplication does not matter. (3 + 4) = (7) = 4 When parentheses (or brackets) are indicated, operations within the parentheses must be performed first. (3) + 4 = + 4 = When the order of operations is ambiguous, the following PEMDAS 3 (4) = 8(4) = 3 sequence must be followed: () Parentheses, () Exponentiation, 4/ 5 = 4/ 5 = 5 = 3 (3) Multiplication or Division, (4) Addition or Subtraction. + 3(4) = + = 4 Parentheses (3 + 4) = (7) = 4 When parentheses (or brackets) are indicated, operations within the parentheses must be performed first. ( ) = (3) + (4) + (5) = 4 When a sum contained in parentheses is multiplied by a factor, a(b + c + d) = ab + ac + ad the factor must be multiplied by all the terms in the sum. [(3)(4)(5)] = ()(3)(4)(5) = When a product contained in parentheses is multiplied by a factor, a(b c) = ab ac the factor multiplies the product. (a + b) = (a + b)(a + b) = a + ab + b (a + b)(a b) = a b Equations: Solving for Y Y 3 = 4 Y = Y = 7 We can add the same value (here, 3) to both sides of an equation without altering the equality. ch_appb.indd 559

6 5 Appendix B Review of Basic Arithmetic Y + = Y = Y = Y 3 Y Y = 5 = () 53 = 5 Y = Y = Y = 5 Y 4 = Y = + 4 Y = Y = Y = 5 We can subtract the same value (here, ) from both sides of an equation without altering the equality. We can multiply both sides of an equation by the same value (here, 3) without altering the equality. We can divide both sides of an equation by the same value (here, ) without altering the equality. We can perform any sequence of the above four rules without altering the equality as long as we perform the same operation (addition, subtraction, multiplication, or division) on both sides of the equal sign. 5 = 4 3Y + 5 = 4( 3Y + ) 5. = 3Y = 3Y. 5 = Y or Y =.5 Self-test for Arithmetic (answers follow) In questions, answer true or false. Equality. 7 = 7. 3 = (7) = (+7) 4. + = 5. X = (X) Not Equality. +3 +(3) a a ch_appb.indd 5

7 Appendix B Review of Basic Arithmetic 5 Inequality. > >. 3 < 3. 4 > 3 4. < < Absolute Value 8. 4 = 4 9. =. 39 = +39 What is the value of X? Addition and Subtraction. X = ( 3) = X 3. X = = X Multiplication and Division 5. X = 3(4). 4() = X 7. 4() = X 8. X = ( 3) 9. X = (5)( 5) 3. X = 9/9 3. X = 5/ 3. 4 = X = 3 X = 4 ( ) 34. X = 34/ 35. X = 34 X 3. X = ( a) ( b) (c) Exponentiation = X 39. X = 3 4. X = 7 4. X = 9 4. X = ( 4) 43. X = X = / 4. X = X = ( )( 3 ) 48. X = (4 / )(4 / ) 49. X = ( / ) 5. 3 ( 3 ) = X = X 5. X = 9 Fractions 53. X = 54. X = X = X = Order of Operations 57. X = 5 + (4) 58. X = 5() X = 4 / + 3. X = (3 + ) + Parentheses. X = 4(3 ). X = (3 + )(3 ) 3. X = ( + )( + ) In questions 4, answer true or false. 4. (5 + 4) = (5 + 4) (5 + 4) = 5 + () + 5. (5 4) = (5 4)(5 4) = 5 () +. (5 + 4) (5 4) = 5 7 ch_appb.indd 5

8 5 Appendix B Review of Basic Arithmetic Equations: Solve for Y 7. Y + = 5 8. Y = 9. 3Y + = 4 7. Y 5 7. = 5 4Y Y 8 = Answers to Self-test for Arithmetic 7. Y = Y Y + = = Y (Y ) 5 =. True. True 3. True 4. False 5. True. False 7. True 8. True 9. False. True. True. False 3. False 4. True 5. True. True 7. True 8. True 9. False. True Undefined 3. abc ± Undefined 45. ±4 4. / 47., ,, Undefined /4 = /7 = True 5. True. True /4 = or /7 = ch_appb.indd 5