Mistakes which Frequently Occur in College Algebra, and in turn, Technical Calculus.

Below is a list of mistakes which frequently occur in College Algebra. Technical Calculus uses algebra extensively, so these mistakes occur frequently in Technical Calculus too. How many have you been guilty of making? This list is provided to give you an opportunity to learn from the mistakes of others.

Mistake 1:

Example to show that it is a mistake:

Notice .

Correct Rule:

, provided and

Mistake 2:

Example to show that it is a mistake:

Notice .

Correct Rule:

Mistake 3:

Example to show that it is a mistake:

Notice .

Correct Rule: A factor can only be canceled if it is a common factor of

each term in the numerator and the denominator of a fractional expression.

The term c is not a common factor of each term in the numerator or the

denominator.

Mistake 4:

Example to show that it is a mistake:

Notice .

Correct Rule: A factor can only be canceled if it is a common factor of

each term in the numerator and the denominator of a fractional expression.

The term d is not a common factor of each term in the numerator.

Mistake 5: The meaning of and the rules for the negative sign

The negative sign, ““ can be a source for mistakes creeping into your work. This short review is intended to make you aware of the ways in which the negative sign can be used in doing mathematics. The first way the negative ever appeared to you was its use as a symbol to denote subtraction, as in . In this setting the negative sign denoted the process of “taking away” an amount from a larger amount.

As you progressed in your mathematical studies, the negative sign was, next, introduced to signify “negative numbers,” numbers to the left of the number 0 on the number line. e.g.,

In this context the negative sign was used to denote an additional collection of numbers, the negative numbers. Addition was defined on this enlarged set of numbers and the negative numbers were recognized to be additive inverses of their positive counterpart. Recall what an additive inverse is.

Additive Inverses: The additive inverse of a number b is –b. The additive inverse is defined to be the number, when added to b, which gives the result 0. Examples of additive inverses are:

The additive inverse of 2 is –2:

The additive inverse of –2 is –(-2):

Notice that

This says that .

Notice that the notion of additive inverse and addition on negative numbers corresponds to our old definition of subtraction of one number from a larger number. For example, the addition of 17 and the additive inverse of 9, i.e., , gives the result 8:

and this matches with the old subtraction problem: The introduction of negative numbers and the operation of addition on the collection of integers (positive, zero, and negative) continues to work just as it did when we wished to subtract a positive number from a larger positive number. In mathematics we call this process the process of extending addition to the larger set of integers.

What happens if we wish to subtract two integers, regardless of their signs? Do you recall the subtraction rule for where a and b are two integers? This rule, if not remembered, can be a source of mistakes:

The Subtraction Rule

The following examples show this rule in action in a number of different cases. Examine these cases for patterns to help you avoid mistakes with the subtraction of two integers.

Another rule to remember when working with negative numbers is the rule which describes how the negative and positive signs behave when numbers are multiplied. These rules are given below.

The product of two numbers with like signs is positive, i.e.,

The product of two numbers with unlike signs is negative, i.e.,

Mistake 6: Don’t be stingy with the use of parentheses

Parentheses are an important notational device which helps to keep your

mathematical work organized and free of ambiguity. For example, what is your answer to the simple mathematical expression.

?

Many who compute this expression will work from left to right as follows:

50.

Others will apply the hierarchy (or order) of operations that they learned in high school algebra to computing the expression. The hierarchy of operations is sometimes referred to as the “My Dear Aunt Sally” rule. This rule states that in any mathematical expression which contains a mixture of arithmetic operations, all multiplications should be executed first, all divisions should be executed second, all additions third, and all subtractions last. The order or hierarchy of these four operations is Multiplication, Division, Addition, and Subtraction, in other words “My Dear Aunt Sally.” Applying the order of operations to the expression above yields:

26;

a different result then we obtain if we compute from left to right.

Parentheses can be used be effectively to clear up the inherent ambiguity of the

expression . For example,

and

Recall that parentheses are a means of grouping the order of operations. When

working with expressions involving parentheses, no matter how complicated the expression is, start your arithmetic or algebraic computations with those expressions that are found in the innermost parentheses. Then when the innermost expressions are computed, move to the next level of parentheses in order to compute the expressions contained in them. In this fashion, by moving with your computations from the innermost parentheses to the outermost parentheses you will eventually complete the computation correctly.

Parentheses are also valuable in keeping track of the distribution of a product across a sum, especially in algebraic expressions. Recall the Distributive Property describes how multiplication distributes across addition:

For example, as we illustrated earlier, suppose the problem is to solve the

following for x:

A student’s solution can look like this:

The correct solution with the appropriate use of parentheses is:

Notice how the use of parentheses aids the correct application of the Distributive

Property in the solution above.

Sometimes expressions can be very complicated, nested several layers deep with parentheses. A helpful tip to keep in mind when dealing with parentheses in complicated mathematical expressions is that they always come in pairs, a left parenthesis must have a right parenthesis to which it is associated. In the event that you encounter trouble computing a complicated expression, check to see that it has all of its parentheses by counting the left ones and the right ones. The count should be identical. If not, then something is wrong somewhere.

Mistake 7: Dividing by fractional expressions

You know the rule for dividing by fractions. Most students remember it this way: “invert and multiply.” But the rule is more precise than the phrase “invert and multiply.” To be precise,

By and large, the rule in this form does not give students too much trouble. Students begin to encounter difficulty with “invert and multiply” when faced with algebraic expressions like:

What does this fractional expression really mean? Does it mean

or

These are two different mathematical expressions:

and

Notice that again we have a case where the prudent use of parentheses helps to clear up an expression which is definitely ambiguous.

Mistake 8: Recopying a mathematical expression incorrectly

Have you ever been guilty of the mistake of incorrectly copying a previous line of mathematical work? For example, suppose the problem was to find the solutions of the equation . This cubic polynomial factors nicely into

yielding the solutions of , . If you had the misfortune to copy this problem down as , then the factors change giving

and solutions , 2. Your answer is wrong, but you are likely to receive some partial credit for this work on an exam since you have followed through by factoring the polynomial to find the solutions. If this has ever happened to you, consider yourself incredibly lucky to get out of messing up a sign so nicely.

Consider this, more typical example. Suppose the problem is to find the solutions of the equation . Again this cubic polynomial factors nicely into

giving as solutions , 1, 2. If, however, on this polynomial any sign is incorrectly copied then the solutions for x involve irrational numbers and complex numbers. For example, the solutions for are

and

Could you have solved for these solutions? Miscopying a single sign can change the character of a problem from a routine one into a very formidable one.

Be careful throughout your mathematical work as you go from one step to another. Working neatly will help a great deal to eliminate mistakes arising from copying mathematical expressions incorrectly.

We have illustrated what can go wrong in a problem with just a change in one sign. Think how bad it could get if a coefficient or some other aspect of a mathematical expression is not rewritten correctly.

Mistake 9: Inverse operations: Which direction do you really want to go?

The word “inverse” in mathematics generally means “opposite.” If we are discussing the operation of addition, then the additive inverse of a given number is a number when added to the given number gives 0. If we are discussing multiplication, then the multiplicative inverse of a given number is a number when multiplied by the given number gives 1.

Examples are:

and

-5 is the additive inverse of 5 and is the multiplicative inverse of 5.

The word “inverse” also applies to functions. Nice illustrating examples are the function keys on your calculator. Do you have a button labeled “?” Enter a number into your calculator, any number, say 5. Push the button. Push it again and again and again. Notice that the display reads 5, then , then 5, then , . . . . You learned about functions and their inverses in College Algebra and you will learn more in other mathematics courses that you take. But one source of difficulty for some students is keeping straight which direction you are going when working with functions and their inverses. As an example, consider the log function. Do you have log as a function on your calculator? If so you will notice that it is located on a button that has two functions. Not only does the button have the log but it also has the inverse function of the log. Experiment with this calculator by playing a game similar to the one we played with the button. Take a number, any positive number. Enter it and push the log button. Next push the second function assigned to that key and see what happens. You get back the number you started with. You have just seen another demonstration of a function and its inverse. Because these functions are so closely related it is easy to get them confused. Be alert to this potential difficulty as you study inverse functions in Technical Calculus, especially when working with trigonometric functions.