MATH 21
 Lesson One

CHAPTER #1 - BASIC CONCEPTS AND PROPERTIES (pp. 1 - 43)

LESSON #1.1 - The General Objectives of this lesson are to:

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to:

GENERAL REMARKS - Before starting this course there are several things you should know about its organization.

PHILOSOPHY - Math 21 is a continuation of the work done in Math 4, and it is assumed that you recently passed Math 4 or its equivalent. If this is not the case, see me after class tonight so I can assess your standing.

In this course, the solution of word problems will be emphasized, and we will begin many class sessions by solving a word problem. You are encouraged to solve these problems in any way you can, but it is important for diagnostic purposes that you write your method of solution down.

Most of us learn more rapidly by active involvement, and your contributions will help all of us. You should also follow my work on the board carefully, and if you have any questions, ask me to explain what I am doing. Also, if you see a more elegant method of solution or an error in my work, please point it out as quickly as possible.

WORD PROBLEM:

  1. The length of a rectangle is 4 inches more than five times its width and the perimeter is 32 inches.  Find the dimensions of the rectangle.

1.1: Sets, Numbers, and Numerical Expressions - According to Webster's 7th New Collegiate Dictionary, algebra is defined to be the generalization of arithmetic in which letters representing numbers are combined according to the rules of arithmetic. Isaac Asimov expresses it more succinctly when he says,

"Algebra is just a variety of arithmetic."

In algebra, the concept of a variable provides the basis for generalizing arithmetic ideas. Using symbols such as x and y to represent any two numbers, the expression (x+y) represents the sum of these two numbers. Here the x and y are referred to as variables, and (x+y) is called an algebraic expression.

Referring to the table on page 2 of the text, a summary of the four basic arithmetic operations, addition, subtraction, multiplication and division are given, along with examples in arithmetic and algebraic form.

Sets - Since we are going to be manipulating these variables or 'generalized numbers' we should know something about them. Since some elements of set notation is used to describe these numbers let us briefly review set theory.

Definitions -

1. A set is a collection of objects.

2. The objects in the set are called the elements of the set.

3. A set can be represented symbolically as

4. Clearly listing all the elements is called set enumeration.

5. Using a defining characteristic of each element is called set builder notation.

6. The empty set is called the null set, and is expressed as

Real Number System - The real number system, which we will work with in this course, can be put into the form of a hierarchy going from the simplest type to the most complicated one.

1. The set N of natural numbers -

N = {1, 2, 3, 4, ...}

These are numbers for counting things, and early man hundreds of thousands of years ago probably used them. It wasn't until about 200 BC., however, that Greek mathematicians made the jump from finite numbers to infinite numbers. This jump is suggested by the three dots placed after the number 4.

(Discuss lack of closure under subtraction.)

2. The set I of integers -

I = {…-3, -2, -1, 0, 1, 2, 3, ...}

The set of integers consists of the union of the natural numbers, their negatives, and the number zero. Oddly enough, the incorporation of negative numbers was a long time coming. Certainly the concept was understood with the use of black and red entries in the ledgers of traders, but negative numbers were not fully incorporated into mathematics until the Italian mathematician, Girolamo Cardano, used them in 1545 AD. The zero was introduced much earlier, some say around 700 to 800 AD, by Hindu mathematicians.

(Discuss lack of closure under division.)

3. The set Q of rational numbers -

This set includes fractions as well as integers. Fractional numbers are quite ancient. They appear in the earliest mathematical writings, and were discussed at some length as early as 1550 BC. in the Rhind Papyrus of Egypt. All rational numbers are characterized by having a repeating decimal form.

 

Example 1: Express the fraction, 2/11, as a repeating decimal.  This is done by dividing the numerator (2) by the denominator (11).  When this operation is carried out we obtain the following result -

   Here we see that the repeating element is 18, and this can be expressed by overlining the  repeating element 18.

 

Example 2: Show that 0.1818… can be written as a rational number.

N = 2/11.

 

The Number Line - Before continuing our study of the real number system, let us consider a geometric interpretation of numbers called the number line. It allows us to set up a one-to-one correspondence between numbers and points on the line.

At this point it appears that we can completely fill up our number line with rational numbers. In fact, Greek mathematicians thought that this was the case, until in the 6th century BC., the mathematical school of Pythagoreans encountered a number that could not be written as a ratio of two integers: i.e. the number was not rational. The number was the, the length of the diagonal of a square whose sides are one unit long, the square root of 2,

 

Proof that the square root of two is not rational:

Since the Greeks originally thought that all numbers were rational, this discovery was tantamount to finding that the diagonal of a square did not have a mathematical length. Most Greek mathematicians resolved this paradox by thinking of numbers as lengths; i.e. in geometric terms, which inhibited the development of algebra which might have rivaled or surpassed their work in geometry. It took centuries of development and sophistication in mathematics to resolve this problem. What was done was to fill in the gaps of the number line, which are not filled by rational numbers, with irrational numbers. An algorithm for finding the square root of an decimal number can be found in the Math 21 Web Pages.

4. The set L of irrational numbers -

Note that the prefix 'ir' means not. Also, unlike;

We are now able to define a number in which addition, subtraction, multiplication, division, and roots of positive numbers are closed operations.

5. The set R of real numbers -

This set includes all rational and irrational numbers. For most of this course, the set of real numbers will be sufficient. One thing we can't do, however, is take even roots of negative numbers.

As you can see, we started with the most basic type of number, and through a series of abstractions, reached the level of the real numbers. We can represent this schematically with the following diagram.

Numerical Expressions - When evaluating numerical expressions it is important that you understand and follow the proper -

Order of Operations -

1. Do everything inside parentheses first. -

Please

2. Carry out indicated exponentiation. -

Excuse

3. Carry out indicated multiplication. -

My

4. Carry out indicated division. -

Dear

5. Carry out indicated addition. -

Aunt

6. Carry out indicated subtraction. -

Sally

LESSON #1.2 - The General Objective of this lesson is to review the arithmetic of real numbers.

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to:

1.2: Operations With Real Numbers- All real numbers have a magnitude and a sign. The magnitude of a number is a measure of how big it is, and is expressed as the absolute value of the number.

 

Absolute Value -The absolute value of any real number x is written as |x|, and is defined as;

1. If x 0, then |x| = x.

2. If x < 0, then |x| = - x.

What this definition means is that the absolute value of any real number is always positive. Geometrically, the absolute value of a number is equal to the distance between the number and the zero on the number line.

Addition of Real Numbers -

Subtraction of Real Numbers - If x and y are two real numbers, then

x - y = x + (-y)

Multiplication of Real Numbers -

1. The product of two positive numbers or two negative numbers is equal to the product of their absolute values.

2. The product of a positive number and a negative number is equal to the negative of the product of their absolute values.

3. The product of zero and a number a number is zero.

Division of Real Numbers -

1. The quotient of two positive numbers or two negative numbers is equal to the quotient of their absolute values.

2. The quotient of a positive number and a negative number is equal to the negative of the quotient of their absolute values.

3. The quotient of zero and a nonzero number a number is zero.

4. Division by zero is undefined.

 

Rule of Signs - When dealing with signed numbers the following rules apply-

1.  The product (or quotient) of two positive numbers is positive.

2.  The product (or quotient) of two negative numbers is positive.

3.  The product (or quotient) of two different signed numbers is negative.

4.  Two consecutive plus signs are equivalent to one plus sign.

5.  Two consecutive negative signs are equivalent to one plus sign.

6.  Two consecutive different signs are equivalent to one negative sign.

Board and Class Problems – None.

 

LESSON #1.3 - The General Objectives of this lesson are to:

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to:

1.3: Properties of Real Numbers and the Use of Exponents - Although the following properties may appear to be obvious to some of you, I think it is important to review them because the are the basic rules for the 'game of algebra'. It is not important that you know the name of each rule, it is only important that you know each rule.

Exponents - Exponents are a short hand way of indicating repeated multiplication. If n is a natural number and b is any real number, then

bn = bbb ... b , where there are n factors of b.

In this expression, b is the base, and n is the exponent.

Board and Class Problems – None

 

LESSON #1.4 - The General Objectives of this lesson is to introduce the concept of an algebraic expression.

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to:

1.4 - Algebraic Expressions- An algebraic expression is a combination of literal numbers and/or numerals joined together by a finite number ofarithmetic operations. This definition means that just about anything we see in this course, is an algebraic expression. Some examples are;

x + y 

(5y - z)/x 

x2 – 2x +1

An algebraic expression consists of parts called-

Simplifying Algebraic Expressions - Algebraic expressions are simplified by combining similar terms.

Board and Class Problems – None

Evaluating Algebraic Expressions - When the variables are assigned specific values the algebraic expression can be evaluated.

Board and Class Problems – None

Translating from English to Algebra - One of our major goals is to learn how to translate a problem in standard language into algebraic form. On page 34 a table showing comparisons between English and algebra is given. You should take some time to study this table.

Board and Class Problems – None

HOMEWORK - Lesson One Assignment.

COMMENTS -

 

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