  Assignment 35

 "In most sciences, one generation tears down what others have built, and what one has established, another undoes. In mathematics alone each generation adds a new story to the old structure." -- (Hermann Hankel, 1839-1873) Johann Kepler (1571-1630): Despite a life that involved may misfortunes (he was accused of heresy, his child died of smallpox, his wife went mad and died, his mother was accused of witchcraft), the German-born Kepler made many contributions to mathematics and astronomy. Deeply religious, he angered the Church by supporting the Copernican theory that the earth was not the center of the universe. His greatest accomplishment was establishing three laws of planetary motion. He was the first to suggest that the paths of the planets are elliptical rather than circular. When Herkimer was a lawyer, where did he suggest a mermaid take a grievance? Answer: To Small Clams Court. Things Herky would like to know: If your conscious is always clear, is that the sign of a bad memory? If Barbie is so popular, why do you have to buy her friends? ASSIGNMENT #35 Reading: See ITEMS FOR REFLECTION (below). Written: See ITEMS FOR REFLECTION (below) Mathematical word analysis:PARENTHESES : From the Greek word parentithenai (putting in beside). In math, we have expressions like 3(5+7), which we, by established order of operations, simplify to 3(12) = 36. Here we want to multiply 3 by the sum of 5 and 7. We "put the sum beside" the 3, using parentheses.

The assignment is a series of review problems that will be handed out in class. The problems do involve two new ideas relating to exponents. Note that a positive integer exponent indicates how many times a base as used as a factor. For instance,

34 = 3x3x3x3 = 81

Please note that 34 does not mean 3 multiplied by itself four times (a common misinterpretation). The express 32 =3x3 represent 3 multiplied by itself once (not twice), 33 = 3x3x3 represents 3 multiplied by itself twice (not three times), etc. USE THE LANGUAGE CORRECTLY.

Note that the following definitions require that you understand what a positive integer represents:

x-N = 1/xN if x is not zero.

X0 = 1 if x is not zero.

Examples: 2-3 = 1/23 = 1/8 and 20 = 1. There is a logic to these definitions.

You should convince yourself that your calculator adheres to these definitions. Experiment, and convince yourself that this is true.

MATH IS POWER.

 Problem: Evaluate (-3)2 + 5-2 + 860. Solution (with communication): (-3)2 + 5-2 + 860 = 9 + 1/25 + 1 = 10 + .04 = 10.04. Problem: Simply, and express answer with positive exponents only: (x2y-3)2(xy2)-1 Solution (with communication): (x2y-3)2(xy2)-1 = (x4y-6)(x-1y-2) = x3y-8 = x3/y8. Problem: Solve x-2 = 16. Solution (with communication): x-2 = 16 ==> 1/x2 = 16 ==> x2 = 1/16 ==> x = 1/4 or x = -1/4.