(#51-#100 of 100) STANDARD TASKS
(Every CITIZEN should be "put to the task" of such as these -- at least once.)

(To seek the answer of a given Task, click on the eyelet. To dismiss the answer, click on "OK".)

51. What is the relation between a RATIONAL decimal number (with PERIOD) and a GEOMETRIC PROGRESSION?
52. We see in Task 39 that 1/7 = 0.|142857|. In the manner of Task 40, show that this is the SUM OF A GEOMETRIC PROGRESSION (GP).
53. Task 39 shows that 1/3 = 0.|3|, with a PERIOD of a SINGLE DIGIT, while 1/7 = 0.|142857|, with a PERIOD of SIX DIGITS. Is there a Principle involved here?
54. The EUCLIDEAN ALGORITHM (EA) is perhaps the most famous of algorithms. Computer scientists use it to test the processing speed of computers. Use EA to find the GREATEST COMMON DIVISOR (gcd) and LEAST COMMON MULTIPLE (lcm) of numbers 420 and 990.
55. Given a classroom with 30 desks and 32 student, can each student be assigned a desk? What mathematical principles are involved?
56. Given 7 market-towns in your area, 5 malls in each town, 3 food-markets at each mall, how many CHOICES of marketing do you have? What principle is involved here?
57. What is THE EQUIVALENCE PRINCIPLE?
58. Given 5 cans of different colors for woodwork in a room, if CHOICE is INDEPENDENT (one color doesn't preclude another), how many CHOICES have you (including NONE or ALL). What principle is involved here?
59. Calculations of the sort in Task 47 can seem "huge". For n = 9, the number of CHOICES is vastly greater than a GOOGOL (1 followed by 100 zeros). Find an approximation for finding powers of 2 as powers of 10. Apply to n = 9.
60. Show that every number of the form abcabc can be factored by 13.
61. Find the solution of the equation x = x + 1.
62. Apply the Descartes' sign rule, to determine the possible number of positive and negative roots in the polynomial: f(x) = 2x7 + x6 - 5x4 + 4x3 - x2 + 9x - 1.
63. Given x3 + 4x2 - 3x + 5 = 0, how many real roots can it have?.
64. What is the meaning of 'n == k(mod m)'? What is the significance of 'n == k(mod p), p = prime'?
65. The answer to Task 20 notes that a number (say, 144) is a MULTIPLE OF 9 IF THE SUM OF ITS DIGITS IS A MULTIPLE OF 9 (144 -> 1+4+4 = 9, 144 = 9x16). Use the MODULUS device of Task 55 to explain this.
66. For line-convenience write division 17/3 as 3)17 |5 + 2 (dividend D=17, divisor d=3, quotient Q=5, remainder R = 2). Then what is incorrrect with 3)-17 |-5 + (-2)?
67. What is "scientific notation"?
68. Convert to "scientific notation" form: 20; 3043; 8,750,000; .339; .00000187.
69. Solve these four algebraic equations: (a) x + 2 = 2x + 3; (b) (x + 2)^1/2 = -(2x + 3)^1/2; (c) x - 2 = 2x - 3; (d) (x - 2)^1/2 = -(2x - 3)^1/2.
70. Find the value(s) satisfying this inequation: |2x + 5| > 7.
71. Write the complex number form with real value of 12 and imaginary value of 13.
72. In the complex number form, 9 - 7i, i = (-1)1/2, what is the imaginary value, what is the real value?
73. Write the absolute value or modulus or norm of the complex number form, 5 + 12i.
74. State addition rule for adding complex numbers, a + bi, and c + di.
75. State rule for subtracting complex number a + bi from complex number c + di.
76. State multiplication rule for multiplying complex numbers, a + bi, and c + di.
77. State rule for dividing complex number a + bi by complex number c + di.
78. Write the polar form of the complex number x + yi.
79. Write Euler's formula for a complex number.
80. Given the complex algebraic equation: 4x + 3yi - 5xi + 7y = 1 - 13i, solve for x and y.
81. Given the complex number -11x - 7yi, find its conjugate.
82. Minkowski Space or Lattice Space is a Grid Space with UNIT SQUARES for UNITS OF AREA. Give the FORMULA for THE AREA OF A CLOSED GEOMETRIC FIGURE in this space.
83. Show that PLANE CARTESIAN COORDINATES set forth a Minkowski Space (as in #82) by specifying its LATTICE POINTS and its SQUARE UNITS of this SPACE.
84. Describe a Minkowski Space you may see every day at home or in a building.
85. Given PLANE CARTESIAN COORDINATE SPACE as a MINKOWSKI SPACE; and COORDINATES (2,5), (11,5), (2,12), (11,12), enclosing a FIGURE in this space. Find its AREA.
86. Find AREA of IRREGULAR FIGURE in PLANE CARTESIAN COORDINATE SPACE BOUNDED by COORDINATES: (17,2), (18,3), (19,3), (20,4), (21,3), (22,5), (21,6), (22,7), (21,8), (21,9), (22,10), (22,11), (21,10), (20,9), (20,8), (19,7), (18,6), (19,5), (18,4), (17,4), (17,3), (17,2).
87. What is the DIGITAL ROOT of a NUMBER (in decimal notation), such as 144?
88. I define THE CODIGITAL ROOT of a NUMBER (in decimal notation) to be ITS REMAINDER WHEN DIVIDED BY 11. Can you find the CODIGITAL ROOT of 121 by MERELY ADDING & SUBTRACTING (that is, WITHOUT DIVIDING)?
89. FIND THE DIGITAL ROOT & CODIGITALROOT OF 693. What do the results tell you about FACTORS of 693?
90. Given a NUMBER written in DECIMAL NOTATION. You PERMUTE ITS DIGITS (INTERCHANGE THEM). What is the DIFFERENCE BETWEEN "BEFORE" AND "AFTER"?
91. What is "BOOKKEEPER'S CHECK"?
92. A bookeeper finds that DEBIT SUM AND CREDIT SUM DO NOT AGREE. Bookkeeper looks for AN ERRONEOUS PERMUTATION OF DIGITS, but cannot find one. But finds an incorrect entry of 278 dollars for correct amount 269 dollars. What does this tell the bookeeper?
93. What are the standard number systems of arithmetic?
94. What are the primary and inverse operations of arithmetic?
95. In what number systems are arithmetical inverses partial and total?
96. Why do addition and multiplication have only one inverse each (respectively, subtraction and division), while exponentiation has two inverses (logarithm, root extraction)?
97. How do the two inverses of exponentiation arise?
98. How can we know when two choices or roles are independent?
99. What is Boole's Theorem for calculating the total number of truth values or choices or roles generated by n independent cases?
100. Given 4 independent choices or roles. How many choices or roles are possible?
101. Merely by looking at the right-most digit of a natural number such as 1448, how do I know it is not the square of a natural number or integer?
102. The square of every natural number or integer is either divisible by a unique divisor, d, or leaves remainder 1 when divided by d. What is number d? And why is this so?