Assignment 1

"How can it be that mathematics, being after all a product of human thought independent of experience, is so admirable adapted to the objects of reality." -- (Albert Einstein)


Math History Tidbit:
Contrary to popular belief, Pythagoras (c. 550 B.C.) and his followers, the Pythagoreans, who contributed much to the development of mathematics, were not scientific sages. The Pythagoreans believed that all of nature could be explained by numbers. They were actually mystics who classified numbers into categories such as male and female. The Pythagoreans were among the first to see a mathematical design to the universe as they knew it. A fascinating historical fact is that during a time when women were considered intellectually inferior to men, the Pythagoreans welcomed women as equals, encouraging them to learn mathematics and become teachers.

Herkimer's Corner

Why did Herkimer have trouble making Kool-Aid?

Answer: He couldn't figure how to get two quarts of water into the little envelope.

Herky's friends:

G. HOWIE SHIVERS ...this guy is a refrigeration specialist.

HOWIE GETINDERE ... this fellow is a locksmith


Assignment 1

Reading: Sections 1.1 and 1.2, pages 3-13.

Things to note:

Number definitions on page 3.
Properties of real numbers on page 5.
Order of operations on page 11.
Definitions on pages 11-13.

Written: Problems 1-14 in Quiz 1 on page 17. (Communicate!)

Items for reflection:

Mathematics is a language. It is the language of our universe. Every shred of existing evidence points to a mathematical design of the known universe in which we exist. Be you a creationist or an evolutionist, your existence, and your survival are guided by mathematical principles and laws.

Since the time of Pythagoras (c. 550 B.C.), humans have created the magnificent language of mathematics in attempts to understand the world in which they existed. The language has been modified and changed over a period of two thousand years. Today, we know considerably more mathematics than did the Pythagoreans, Galileo, Newton, and even Einstein. But, we know the language needs further development. As modern as we think we are, one hundred years from now our descendants will see us from a historical standpoint as being relatively primitive. They will know considerably more mathematics than we know at the present time. You cannot expect to be very successful in a study of mathematics if you simply see it as a set of unchanging rules and laws. Mathematics is not a static discipline. It is forever changing, and forever being discovered.

As you begin this course, try to appreciate mathematics as a language. Like any language, it has basic definitions and rules. It is important to understand (not memorize) the basics. Obviously, if you don't learn the basics, the language becomes difficult to learn an appreciate. Try to get yourself to realize that mathematics is the language of our universe. If you develop a good attitude and appreciation for this magnificent academic discipline, learning can be both fun and rewarding.

You are studying a branch of mathematics called algebra. The word algebra comes from an Arabic word al-jabr, which appeared in the title of a mathematics book Hisab al-jabr w' al-muqabaleh, written by al-Khwarizmi around 825. Loosely translated, the book title means "the science of reunion and reduction." A loose translation of al-jabr means "to balance." There are other translations, but I like this one because in modern algebra we solve equations by a balancing process. As a simple example, consider the equation

x + 3 = 15

If we think of this equation as being balanced around the = symbol (think of a seesaw), we preserve the balance if we subtract 3 from each side of the = symbol. If we do this, we solve the equation, obtaining

x = 12

OK, you already know some of the language of mathematics. Let's get ready to learn a lot more. Since math is a language, communication in problem solving is important. Below are four examples that stress communication. Note that the last two problems stress the order of operations. This is extremely important. Among many other things, your calculator follows the order of operations in performing computations. If you don't "communicate" properly with your calculator, you many not get the results you are supposed to get. The basic hierarchy of operations is as follows:

(1) Exponents
(2) Multiplication and division in order from left to right.
(3) Addition and subtraction in order from left to right.

You can use parentheses () to alter the order of operations, if necessary. For instance, 10/2 + 3 = 5 + 3 = 8, while 10/(2 + 3) = 10/5 = 2.

Problem: Evaluate 5(x-2) + x2 when x = 6.

Solution (with communication):

If x = 6, then 5(x-2) + x2 = 5(6-2) + 62 = 5(4) + 36 = 20 + 36 = 56.

Problem: Simplify the expression

3(2-4x) - 2(x-4).

Solution (with communication):

3(2-4x) - 2(x-4) = 6 - 12x -2x + 8 = 14 - 14x.


Problem: Evaluate 12 - 8x4 - 14/7.

Solution (with communication):

12 - 8x4 - 14/7 = 12 - 32 - 2 = -20 - 2 = -22.

Problem: Evaluate 20/22 - 8/(1+3) + 5x32.

Solution (with communication):

20/22 - 8/(1+3) + 5x32 = 20/4 - 8/4 + 5x9 = 5 - 2 + 45 = 48.