Assignment 2

 "Numbers rule the universe." -- (Pythagoras, c. 550 B. C.) Greatly influenced by the Pythagoreans, Plato (c. 400 B.C.) propagated the doctrine of the mathematical design of nature. In his famous work, the Republic, Plato stated that "the knowledge at which geometry aims is knowledge of the eternal." Throughout human history, there have been interesting and conflicting views on the nature of mathematics. Platonism asserted that mathematics represents a separate universe of abstract objects existing outside of what we know as time and space. Other historical doctrines claim that mathematics is a human-created discipline, but Platonism says that humans don't invent or create mathematics. They can only discover what is already there. Why did Herkimer fail as a pharmacist? Answer: He couldn't figure how to get the little bottle into the typewriter. Herky's friends: HUGO FIRST ...a frightened sky diving instructor. SELMA HOUSE ... a real estate agent. Reading: Section 1.3, pages 19-22. Written: Pages 22-24/29-39 (odds), 43-49(odds)

 Mathematical word analysis:MULTIPLY: From the combined roots of multi (many) and pli (folds). One might think of multiplication as a number being folded on itself many times.

Practice being neat and organized. Mathematics is a language. Learn to use it correctly and efficiently.

Problem #32 on page 22 wants you to solve the equation

-4(3 + x) + 5 = 4(x + 3).

An equation is a conditional statement that has no truth value by itself. If we substitute a value for x, then the resulting statement (which in this case is a statement of equality) is either true or false. For instance, if we substitute x = 0 into the equation, we obtain

-7 = 12.

This is a perfectly meaningful statement. It is, of course, a false statement. If you can classify a statement as false, then the statement is not meaningless. Now, when we solve an equation, our purpose is to find all real number values that make the conditional statement true. We have yet to solve the equation, but we have discovered that x = 0 makes the conditional statement false. Hence, x = 0 is not a solution to the equation. It is not among the collection of numbers that make the conditional statement true.

To solve the equation above, we will produce a series of equivalent equations. In doing so, we will preserve the "balance" around the symbol = . (Remember, the Arabic word al-jabr can be translated "to balance.").

The symbol ==> means "implies." Let's learn this, because mathematics is a language, and communication is important. It is very important to note that the symbols = and ==> mean two entirely different things. Think before you write! Here is an example of a statement that uses both symbols correctly: 3x = 21 ==> x = 7. To translate, if three times a number is 21, this implies that the number is 7. Or we could say, if 3x = 21 is true, then a necessary consequence is that x = 7.

Here is a neat homework-type presentation for problem #32 on page 22.

-4(3 + x) + 5 = 4(x + 3)

==> -12 - 4x + 5 = 4x + 12

==> -7 -4x = 4x + 12

==> -7 = 8x + 12

==> 8x = -19

==> x = -2.375. (Note the period at the end of the sentence.)

Translation: If -4(3 + x) + 5 = 4(x+3), then this implies that x must be the number -2.375.

Finally, note on page 19 there are what the authors call transformations that produce equivalent equations. Basically, these are principles that you should have learned in first-year algebra. Let's make a real effort to think at a sophisticated level. These transformations, if used properly, preserve the balance around the = symbol. When solving an equation, it is important to preserve the balance.

When doing homework, write neatly. Communicate! Say what you mean, and mean what you say. Use the language of mathematics properly, effectively, and efficiently.

 Problem: Solve (1/3)(x - 5) = (3/7)(x - 7). Solution (with communication): (1/3)(x - 5) = (3/7)(x - 7) ==> 21[(1/3)(x - 5)] = 21[(3/7)(x - 7)] ==> 7(x - 5) = 9(x - 7) ==> 7x - 35 = 9x - 63 ==> 7x + 28 = 9x ==> 2x = 28 ==> x = 14. Check: Substituting x = 14 into the original equation yields 3 = 3. Problem: You have two summer jobs. In the first job, you work 28 hours and earn \$7.25 per hour. In the second job, you earn \$6.50 an hour and can work as many hours as you want. If your want to earn \$255 per week, how many hours must you work at your second job. Solution (with communication): Let x = the number of hours you need to work at the second job to earn \$255 per week. Premise ==> 28(\$7.25) + x(\$6.50) = \$255 ==> x = [\$255 - 28(\$7.25)]/\$6.50 ==> x = 8. Conclusion: You will need to work 8 hours a week at the second job if you want to earn a total of \$255 per week. Note: This problem is less time-consuming if you save the calculator computations until the final step. This does require knowledge of the order operations.