  Assignment 53

 "To us probability is the very guide of life." -- (Bishop Butler, Preface to Analogy.) Emilie du Chatelet (France, 1706-1749). Du Clatelet was born into a family that enjoyed all of the privileges of life at the royal court in France. Still, French society regarded women as intellectually inferior to men, so she had to overcome many obstacles to achieve fame in mathematics. Her major work, Mathematical Principles of Natural Philosophy, was a French translation on Newton's Principia. She also produced a scholarly manuscript entitled Dissertation on the Nature and Propagation of Fire. Du Chatelet was brilliant with numbers. The French philosopher Voltaire claimed he witnessed her find the product of two nine-digit numbers using only mental calculation. What question did Herkimer ask when he took his pet termites into a saloon? Answer: "Is the bar tender here?" Herky's friends: DAN D. LYON...a fellow who liked yellow flowers BARB D. WYRE...she made protective fences that nobody wanted to climb over. ASSIGNMENT #53 Reading: Section 7.6, pages 437-440. Written : Page 441/4-15. Write these up neatly. Communicate. Mathematical word analysis:DIGIT: From digitus , a Latin word meaning "finger." Humans have always used fingers in the counting process.
Sometimes a logical equation-solving process can lead to extraneous solutions. That is, the problem-solving process introduces solutions that do not solve the original equation. If you use mathematics properly, you will not lose any of the original solutions, but you might introduce solutions that do make the original conditional statement true. This is why communication is important.

For instance, the solution to x = 2 is simple to obtain. It is, obviously, x = 2. Note, however, that you could square each side and not destroy the equality. That is, x = 2 ==> x2 = 4. Note that x2 = 4 ==> x = 2 or x = -2. But x = -2 does not satisfy the original equation x = 2. By squaring both sides, the potential solution x = -2 has been introduced. So, you should always check your answers to see if they satisfy the original equation.

Example: ÷(x2+5) = x + 3

==> x2+5 = x2 + 6x + 9

==> 6x = -4

==> x = -2/3.

Check: Substituting x = -2/3 into the left side of the original equation yields ÷(4/9 + 5) = ÷(49/9) = 7/3.

Substitution into the right side yields -2/3 + 3 = -2/3 + 9/3 = 7/3. Hence x = -2/3 is a solution for the original conditional statement.

Example: (x+40)1/3 = -5

==>x+40 = (-5)3 = -125

==> x = -145.

Check: (-145+40)1/3 = (-125)1/3 = -5, so x = -145 is a solution for the original conditional statement.

 Problem: Solve x = ÷(6 - x). Solution (with communication): x = ÷(6 - x) ==> x2 = 6 - x ==> x2 + x - 6 = 0 ==> (x - 2)(x + 3) = 0 ==> x = 2 or x = -3. Checking, we find x = -3 is not a solution to the original equation. The only solution is x = 2. Note: The communication states that if there are solutions to the original equation, then they must come from the set {-3,2}. No other solutions are possible. Problem: Solve ÷(-x) = x + 20. Solution (with communication): ÷(-x) = x + 20 ==> -x = (x+20)2 = x2 + 40x + 400 ==> x2 + 41x + 400 = 0 ==> (x + 16)(x + 25) = 0 ==> x = -16 or x = -25. Checking, we find that x = -25 does not solve the original equation. The only solution is x = -16. Note: An alert problem solver would realize that no positive numbers could satisfy the original equation.