  Assignment 29

 "It is not our conclusions that betray us. It is our major premises." -- (Tom Burnam, The Dictionary of Misinformation ) Incredibly, the concept of a negative number confused mathematicians until well into the 1800's. While the Hindus and Chinese did work with negative numbers, the thought that one could have numbers less than nothing bothered even the best of mathematicians. Math texts frequently confused subtraction and the use of the minus symbol to represent opposite . Research on negative numbers will yield you some rather surprising results. The confusion is characterized by this statement from French mathematician Jean Le Rond d'Alembert (1717-1783). "A problem leading to a negative solution means that some part of the hypothesis was false but assumed to be true." Another famous French mathematician, Blaise Pascal (1623-1662) said that subtraction of a positive number from zero is "pure nonsense." He also stated: "I have known those who could not understand that to take four from zero there remains zero." When Herkimer was thirsty, why did he put ice cubes in this father's bed? Answer: So he could have an ice cold pop. Herky's friends: SAM PULL...he works for a polling organization. JOE KING...this guy likes to tell funny stories. ASSIGNMENT #29 Reading: Section 5.5, pages 282-286. Written: Page 287/47-65 (odds) Mathematical word analysis:RANDOM: From the old French root randir (to gallop). Idea is perhaps that, in full gallop, the horse or rider has abandoned control.

The process of completing the square can be used to solve quadratic equations.

Example 1:

x2 - 6x -27 = 0

==> x2 - 6x = 27

==> x2 -6x + 9 = 27 + 9 = 36

==> (x - 3)2 = 36

==> x - 3 = 6 or x - 3 = -6

==> x = 9 or x = -3.

The process works even if the solutions are complex numbers.

Example 2:

x2 -2x + 4 = 0

==> x2 - 2x = -4

==> x2 - 2x + 1 = -4 + 1 = -3

==> (x - 1)2 = -3

==> x - 1 = ũ(3)i or x - 1 = -ũ(3)i

==> x = 1 + ũ(3)i or x = 1 - ũ(3)i .

NOTE: One can check to see if the complex numbers do work. We will check x = 1 + ũ(3)i.

(1 + ũ(3)i)2 -2(1 + ũ(3)i) + 4

= 1 + 2ũ(3)i +3i2 - 2 -2ũ(3)i + 4

= 1 + 2ũ(3)i - 3 - 2 -2ũ(3)i + 4

=(1 - 3 - 2 + 4) + [2ũ(3)i - 2ũ(3)i]

= 0 + 0 = 0.

MATH IS POWER.

 Problem: Solve by completing the square: x2 - 6x - 13 = 0 Solution (with communication): x2 - 6x - 13 = 0 ==> x2 - 6x = 13 ==> x2 - 6x + 9 = 13 + 9 ==> (x - 3)2 = 22 ==> x - 3 = ũ22 or x - 3 = - ũ22 ==> x = 3 + ũ22 or x = 3 - ũ22 ==> x = 7.69 or x = -1.69. Problem: Solve by competing the square: 4x2 - 8x - 9 = 0 Solution (with communication): 4x2 - 8x - 9 = 0 ==> 4(x2 - 2x) = 9 ==> x2 - 2x = 2.25 ==> x2 - 2x + 1 = 2.25 + 1 ==> (x - 1)2 = 3.25 ==> x - 1 = ũ3.25 or x - 1 = - ũ3.25 ==> x = 1 + ũ3.25 or x = 1 - ũ3.25 ==> x = 2.803 or x = -0.803.