This section is intended as a review of the algebra necessary to understand the rest of this book, allowing the student to gauge his or her mathematical sophistication relative to what is needed for the course. The individual without adequate mathematical training will need to spend more time with this chapter. The review of algebra is presented in a slightly different manner than has probably been experienced by most students, and may prove valuable even to the mathematically sophisticated reader.
Algebra is a formal symbolic language, composed of strings of symbols. Some strings of symbols form sentences within the language (X + Y = Z), while others do not (X += Y Z). The set of rules that determines which strings belong to the language and which do not, is called the syntax of the language. Transformational rules change a given sentence in the language into another sentence without changing the meaning of the sentence. This chapter will first examine the symbol set of algebra, which is followed by a discussion of syntax and transformational rules.
The symbol set of algebra includes numbers, variables, operators, and delimiters. In combination they define all possible sentences that may be created in the language.
Numbers are analogous to proper nouns in English, such as names of dogs - Spot, Buttons, Puppy, Boots, etc. Some examples of numbers are:
Numbers may be either positive (+) or negative (-). If no sign is included the number is positive. The two numbers at the end of the example list are called universal constants. The values for these constants are = 3.1416... and e = 2.718....
Variables are symbols that stand for any number. They are the common nouns within the language of algebra. Common nouns within the English language include words such as dog and cat, that describes a general category of animals and not a particular dog or cat. Letters in the English alphabet most often represent variables, although Greek letters are sometimes used. Some example variables are:
Other symbols, called operators, signify relationships between numbers and/or variables. Operators serve the same function as verbs in the English language. Some example operators in the language of algebra are:
Note that the "*" symbol is used for multiplication instead of the "x" or "· " symbol. This is common to many computer languages. The symbol "³ " is read as "greater than or equal to" and "£" is read as "less than or equal to."
Delimiters are the punctuation marks in algebra. They let the reader know where one phrase or sentence ends and another begins. Example delimiters used in algebra are:
In this course, only the "( )" symbols are used as delimiters, with the algebraic expressions being read from the innermost parenthesis out.
Many statements in algebra can be constructed using only the symbols mentioned thus far, although other symbols exist. Some of these symbols will be discussed either later in this chapter and later in the book.
Sentences in algebra can be constructed using a few simple rules. The rules can be stated as replacement statements and are as follows:
Delimiters (parentheses) surround each phrase in order to keep the structure of the sentence straight. Sentences are constructed by creating a lower-order phrase and sequentially adding greater complexity, moving to higher-order levels. For example, the construction of a complex sentence is illustrated below:
Statements such as this are rarely seen in algebra texts because rules exist to eliminate some of the parentheses and symbols in order to make reading the sentences easier. In some cases these are rules of precedence where some operations (* and /) take precedence over others (+ and -).
The following rules permit sentences written in the full form of algebra to be rewritten to make reading easier. Note that they are not always permitted when writing statements in computer languages such as PASCAL or BASIC.
1. The "*" symbol can be eliminated, along with the parentheses surrounding the phrase if the phrase does not include two numbers as subphrases. For example, (X * (Y - Z)) may be rewritten as X (Y - Z). However, 7*9 may not be rewritten as 79.
2. Any phrase connected with "+" or "-" may be rewritten without parentheses if the inside phrase is also connected with "+" or "-". For example, ((X + Y) - 3) + Z may be rewritten as (X + Y) - 3 + Z. Continued application of this rule would result in the sentence X+Y-3+Z.
Sequential application of these rules may result in what appears to be a simpler sentence. For example, take this sentence created from the earlier example:
Apply Rule 1
Apply Rule 2
Often these transformation are taken for granted and already applied to algebraic sentences before they appear in algebra texts.
Transformations are rules for rewriting sentences in the language of algebra without changing their meaning, or truth value. Much of what is taught in an algebra course consists of transformations. Transformations involve a range of operations from the simple addition of two numbers through the simplification of complex algebraic sentences. In all cases one algebraic sentence is replaced with another algebraic sentence that has the same meaning.
When a phrase contains only numbers and an operator (i.e. 8*2), that phrase may be replaced by a single number (i.e. 16). These are the same rules that have been drilled into grade school students, at least up until the time of new math. The rule is to perform the operation and delete the parentheses. For example:
The rules for dealing with negative numbers are sometimes imperfectly learned by students, and will now be reviewed.
1. An even number of negative signs results in a positive number; an odd number of negative signs results in a negative number. For example:
2. Adding a negative number is the same as subtracting a positive number.
A second area that sometimes proves troublesome to students is that of fractions. Fractions are an algebraic phrase involving two numbers connected by the operator "/"; for example, 7/8. The top number or phrase is called the numerator, and the bottom number or phrase the denominator. One method of dealing with fractions that has gained considerable popularity since inexpensive calculators have become available is to do the division operation and then deal with decimal numbers. In what follows, two methods of dealing with fractions will be illustrated. The student should select the method that is easiest for him or her.
Multiplication of fractions is relatively straightforward: multiply the numerators for the new numerator and the denominators for the new denominator. For example:
Using decimals the result would be:
Division is similar to multiplication except the rule is to invert and multiply. An example is:
or in decimal form:
Addition and subtraction with fractions first requires finding the least common denominator, adding (or subtracting) the numerators, and then placing the result over the least common denominator. For example:
Decimal form is simpler, in my opinion, with the preceding being:
Fractions have a special rewriting rule that sometimes allows an expression to be transformed to a simpler expression. If a similar phrase appears in both the numerator and the denominator of the fraction and these similar phrases are connected at the highest level by multiplication, then the similar phrases may be canceled. The rule is actually easier to demonstrate than to state:
The following is an INCORRECT application of the above rule:
A number of rewriting rules exist within algebra to simplify an algebraic phrase with a shorthand notation of that phrase. Exponential notation is an example of a shorthand notational scheme. If a series of similar algebraic phrases are multiplied times one another, the expression may be rewritten with the phrase raised to a power. The power is the number of times the phrase is multiplied by itself and is written as a superscript of the phrase. For example:
Some special rules apply to exponents. A negative exponent may be transformed to a positive exponent if the base is changed to one divided by the base. A numerical example follows:
A root of a number may be expressed as a base (the number) raised to the inverse of the root. For example:
When two phrases that have the same base are multiplied, the product is equal to the base raised to the sum of their exponents. The following examples illustrate this principle.
It is possible to raise a decimal number to a decimal power, that is "funny" numbers may be raised to "funny" powers. For example:
Exponents such as the ones shown above may be evaluated using most scientific calculators. Generally these calculators work by first entering the base, i.e. 3.44565, clicking on a key, perhaps "XY", entering the exponent, i.e. 1.234678, and then hitting another key, perhaps "=". Different calculators use different key combinations to achieve similar results, so consult the user's manual for the correct procedure for a given calculator. Even though most readers would not be able to simplify the above expression "by hand" because they never learned the algorithm to do this kind of simplification, doesn't mean that it cannot be done. Somebody learned the algorithm, wrote a program to implement it, and made it available on calculators so others might use it. Such expressions make perfect sense in the world of algebra and will be seen in later chapters.
A special form of exponential notation, called binomial expansion occurs when a phrase connected with addition or subtraction operators is raised to the second power. Binomial expansion is illustrated below:
A more complex example of the preceding occurs when the phrase being squared has more than two terms.
When two expressions are connected with the multiplication operator, it is often possible to "multiply through" and change the expression. In its simplest form, if a number or variable is multiplied by a phrase connected at the highest level with the addition or subtraction operator, the phrase may be rewritten as the variable or number times each term in the phrase. Again, it is easier to illustrate than to describe the rule:
If the number or variable is negative, the sign must be carried through all the resulting terms as seen in the following example:
Another example of the application of this rule occurs when the number -1 is not written in the expression, but rather inferred:
When two additive phrases are connected by multiplication, a more complex form of this rewriting rule may be applied. In this case one phrase is multiplied by each term of the other phrase:
Note that the binomial expansion discussed earlier was an application of this rewriting rule.
A corollary to the previously discussed rewriting rule for multiplication of phrases is factoring, or combining like terms. The rule may be stated as follows: If each term in a phrase connected at the highest level with the addition or subtraction operator contains a similar term, the similar term(s) may be factored out and multiplied times the remaining terms. It is the opposite of "multiplying through." Two examples follow:
Much of what is learned as algebra in high school and college consists of learning when to apply what rewriting rule to a sentence to simplify that sentence. Application of rewriting rules change the form of the sentence, but not its meaning or truth value. Sometimes a sentence in algebra must be expanded before it may be simplified. Knowing when to apply a rewriting rule is often a matter of experience. As an exercise, simplify the following expression:
A sentence in algebra is evaluated when the variables in the sentence are given specific values, or numbers. Two sentences are said to have similar truth values if they will always evaluate to equal values when evaluated for all possible numbers. For example, the sentence in the immediately preceding example may be evaluated where X = 4 and Y = 6 to yield the following result:
The result should not surprise the student who had correctly solved the preceding simplification. The result must ALWAYS be equal to 2 as long as both X and Y are not zero. Note that the sentences are evaluated from the "innermost parenthesis out", meaning that the evaluation of the sentence takes place in stages: phrases that are nested within the innermost or inside parentheses are evaluated before phrases that contain other phrases.