Discover the elegance of mathematics...: Common Math Errors-Written by Paul Dawkins

Originally the intended audience for this was my Calculus I students as pretty much every error listed here shows up in that class with alarming frequency. After writing it however I realized that, with the exception of a few examples, the first four sections should be accessible to anyone taking a math class and many of the errors listed in the first four sections also show up in math classes at pretty much every level. So, if you haven’t had calculus yet (or never will) you should ignore the last section and the occasional calculus examples in the first four sections.

I got the idea for doing this when I ran across Eric Schechter’s list of common errors located at http://www.math.vanderbilt.edu/~schectex/commerrs/. There is a fair amount of overlap in the errors discussed on both of our pages. Sometimes the discussion is similar and at other times it’s different. The main difference between our two pages is I stick to the level of Calculus and lower while he also discusses errors in proof techniques and some more advanced topics as well. I would encourage everyone interested in common math errors also take a look at his page.

· General Errors

· In fact, that was the original title of this section, but I did not want to leave people with the feeling that I’m trying to imply that math is easy and that everyone should just “get it”! For many people math is a very difficult subject and they will struggle with it. So please do not leave with the impression that I’m trying to imply that math is easy for everyone. The intent of this section is to address certain attitudes and preconceptions many students have that can make a math class very difficult to successfully complete.

· Putting off math requirements

· I don’t know how many students have come up to me and said something along the lines of :

· “I’ve been putting this off for a while now because math is so hard for

· me and now I’ve got to have it in order to graduate this semester.”

· This has got to be one of the strangest attitudes that I’ve ever run across. If math is hard for you, putting off your math requirements is one of the worst things that you can do! You should take your math requirements as soon as you can. There are several reasons for this.

· The first reason can be stated in the following way : MATH IS CUMULATIVE. In other words, most math classes build on knowledge you’ve gotten in previous math classes, including your high school math classes. So, the only real effect of putting off your math requirement is that you forget the knowledge that you once had. It will be assumed that you’ve still got this knowledge when you finally do take your math requirement!

· If you put off your math requirement you will be faced with the unpleasant situation of having to learn new material AND relearn all the forgotten material at the same time. In most cases, this means that you will struggle in the class far more than if you had just taken it right away!

· The second reason has nothing to do with knowledge (or the loss of knowledge), but instead has everything to do with reality. If math is hard for you and you struggle to pass a math course, then you really should take the course at a time that allows for the unfortunate possibility that you don’t pass. In other words, to put it bluntly, if you wait until your last semester to take your required math course and fail you won’t be graduating! Take it right away so if you do unfortunately fail the course you can retake it the next semester.

· This leads to the third reason. Too many students wait until the last semester to take their math class in the hopes that their instructor will take pity on them and not fail them because they’re graduating. To be honest the only thing that I, and many other instructors, feel in these cases is irritation at being put into the position at having to be the bad guy and failing a graduating senior. Not a situation where you can expect much in the way of sympathy!

· Doing the bare minimum

· I see far too many students trying to do the bare minimum required to pass the class, or at least what they feel is the bare minimum required. The problem with this is they often underestimate the amount of work required early in the class, get behind, and then spend the rest of the semester playing catch up and having to do far more than just the bare minimum.

· You should always try to get the best grade possible! You might be surprised and do better than you expected. At the very least you will lessen the chances of underestimating the amount of work required and getting behind.

· Remember that math is NOT a spectator sport! You must be actively involved in the learning process if you want to do well in the class.

· A good/bad first exam score doesn’t translate into a course grade

· Another heading here could be : “Don’t get cocky and don’t despair”. If you get a good score on the first exam do not decide that means that you don’t need to work hard for the rest of the semester. All the good score means is that you’re doing the proper amount of for studying for the class! Almost every semester I have a student get an A on the first class and end up with a C (or less) for the class because he/she got cocky and decided to not study as much and promptly started getting behind and doing poorly on exams.

· Likewise, if you get a bad score on the first exam do not despair! All the bad score means is that you need to do a little more work for the next exam. Work more problems, join a study group, or get a tutor to help you. Just as I have someone go downhill almost every semester I also have at least one student who fails the first exam and yet passes the class, often with a B and occasionally an A!

· Your score on the first exam simply doesn’t translate into a course grade. There is a whole semester in front of you and lot’s of opportunities to improve your grade so don’t despair if you didn’t do as well as you wanted to on the first exam.

· Expecting to instantly understand a concept/topic/section

· Assuming that if it’s “easy” in class it will be “easy” on the exam

· Don’t know how to study mathematics

· The first two are really problems that fall under the last topic but I run across them often enough that I thought I’d go ahead and put them down as well. The reality is that most people simply don’t know how to study mathematics. This is not because people are not capable of studying math, but because they’ve never really learned how to study math.

· Mathematics is not like most subjects and accordingly you must also study math differently. This is an unfortunate reality and many students try to study for a math class in the same way that they would study for a history class, for example. This will inevitably lead to problems. In a history class you can, in many cases, simply attend class memorize a few names and/or dates and pass the class. In a math class things are different. Simply memorizing will not always get you through the class, you also need to understand HOW to use the formula that you’ve memorized.

· General Errors

· Putting off math requirements

· I don’t know how many students have come up to me and said something along the lines of :

· “I’ve been putting this off for a while now because math is so hard for

· me and now I’ve got to have it in order to graduate this semester.”

· Doing the bare minimum

· Remember that math is NOT a spectator sport! You must be actively involved in the learning process if you want to do well in the class.

· A good/bad first exam score doesn’t translate into a course grade

· Expecting to instantly understand a concept/topic/section

· Assuming that if it’s “easy” in class it will be “easy” on the exam

· Don’t know how to study mathematics

Algebra Errors

The topics covered here are errors that students often make in doing algebra, and not just errors typically made in an algebra class. I’ve seen every one of these mistakes made by students in all level of classes, from algebra classes up to senior level math classes! In fact a few of the examples in this section will actually come from calculus.

If you have not had calculus you can ignore these examples. In every case where I’ve given examples I’ve tried to include examples from an algebra class as well as the occasion example from upper level courses like Calculus.

I’m convinced that many of these are mistakes given here are caused by people getting lazy or getting in a hurry and not paying attention to what they’re doing. By slowing down, paying attention to what you’re doing and paying attention to proper notation you can avoid the vast majority of these mistakes!

Division by Zero

Everyone knows that the problem is that far too many people also say that or ! Remember that division by zero is undefined! You simply cannot divide by zero so don’t do it!

Here is a very good example of the kinds of havoc that can arise when you divide by zero. See if you can find the mistake that I made in the work below.

We’ll start assuming this to be true.
Multiply both sides by a.
Subtract from both sides.
Factor both sides.
Divide both sides by .
Recall we started off assuming .
Divide both sides by b.

So, we’ve managed to prove that 1 = 2! Now, we know that’s not true so clearly we made a mistake somewhere. Can you see where the mistake was made?

The mistake was in step 5. Recall that we started out with the assumption . However, if this is true then we have ! So, in step 5 we are really dividing by zero!

That simple mistake led us to something that we knew wasn’t true, however, in most cases your answer will not obviously be wrong. It will not always be clear that you are dividing by zero, as was the case in this example. You need to be on the lookout for this kind of thing.

Remember that you CAN’T divide by zero!

Bad/lost/Assumed Parenthesis

This is probably error that I find to be the most frustrating. There are a couple of errors that people commonly make here.

The first error is that people get lazy and decide that parenthesis aren’t needed at certain steps or that they can remember that the parenthesis are supposed to be there. Of course the problem here is that they often tend to forget about them in the very next step!

The other error is that students sometimes don’t understand just how important parentheses really are. This is often seen in errors made in exponentiation as my first couple of examples show.

Example 1 Square 4x.

Correct	Incorrect

Note the very important difference between these two! When dealing with exponents remember that only the quantity immediately to the left of the exponent gets the exponent. So, in the incorrect case, the x is the quantity immediately to the left of the exponent so we are squaring only the x while the 4 isn’t squared. In the correct case the parenthesis is immediately to the left of the exponent so this signifies that everything inside the parenthesis should be squared!

Parenthesis are required in this case to make sure we square the whole thing, not just the x, so don’t forget them!

Example 2 Square -3.

Correct	Incorrect

This one is similar to the previous one, but has a subtlety that causes problems on occasion. Remember that only the quantity to the left of the exponent gets the exponent. So, in the incorrect case ONLY the 3 is to the left of the exponent and so ONLY the 3 gets squared!

Many people know that technically they are supposed to square -3, but they get lazy and don’t write the parenthesis down on the premise that they will remember them when the time comes to actually evaluate it. However, it’s amazing how many of these folks promptly forget about the parenthesis and write down -9 anyway!

Example 3 Subtract from

Correct	Incorrect

Be careful and note the difference between the two! In the first case I put parenthesis around then and in the second I didn’t. Since we are subtracting a polynomial we need to make sure we subtract the WHOLE polynomial! The only way to make sure we do that correctly is to put parenthesis around it.

Again, this is one of those errors that people do know that technically the parenthesis should be there, but they don’t put them in and promptly forget that they were there and do the subtraction incorrectly.

Example 4 Convert to fractional exponents.

Correct	Incorrect

This comes back to same mistake in the first two. If only the quantity to the left of the exponent gets the exponent. So, the incorrect case is really and this is clearly NOT the original root.

Example 5 Evaluate .

This is a calculus problem, so if you haven’t had calculus you can ignore this example. However, far too many of my calculus students make this mistake for me to ignore it.

Correct	Incorrect

Note the use of the parenthesis. The problem states that it is -3 times the WHOLE integral not just the first term of the integral (as is done in the incorrect example).

Improper Distribution

Be careful when using the distribution property! There two main errors that I run across on a regular basis.

Example 1 Multiply .

Correct	Incorrect

Make sure that you distribute the 4 all the way through the parenthesis! Too often people just multiply the first term by the 4 and ignore the second term. This is especially true when the second term is just a number. For some reason, if the second term contains variables students will remember to do the distribution correctly more often than not.

Example 2 Multiply .

Correct	Incorrect

Remember that exponentiation must be performed BEFORE you distribute any coefficients through the parenthesis!

Additive Assumptions

I didn’t know what else to call this, but it’s an error that many students make. Here’s the assumption. Since then everything works like this. However, here is a whole list in which this doesn’t work.

It’s not hard to convince yourself that any of these aren’t true. Just pick a couple of numbers and plug them in! For instance,

You will find the occasional set of numbers for which one of these rules will work, but they don’t work for almost any randomly chosen pair of numbers.

Note that there are far more examples where this additive assumption doesn’t work than what I’ve listed here. I simply wrote down the ones that I see most often. Also a couple of those that I listed could be made more general. For instance,

Canceling Errors

These errors fall into two categories. Simplifying rational expressions and solving equations. Let’s look at simplifying rational expressions first.

Example 1 Simplify (done correctly).

Notice that in order to cancel the x out of the denominator I first factored an out of the numerator. You can only cancel something if it is multiplied by the WHOLE numerator and denominator, or if IS the whole numerator or denominator (as in the case of the denominator in our example).

Contrast this with the next example which contains a very common error that students make.

Example 2 Simplify (done incorrectly).

Note that the only change in the problems is the second term in the numerator. Far too many students try to simplify this as,

In other words, they cancel the x in the denominator against one of the x’s in the first term from the numerator. THIS CAN’T BE DONE!!!!! In order to do this canceling you MUST have an x in both terms.

To convince yourself that this kind of canceling isn’t true consider the following number example.

Example 3 Simplify .

This can easily be done just be doing the arithmetic as follows

However, let’s do an incorrect cancel similar to the previous example. We’ll first cancel the two in the denominator into the eight in the numerator. This is NOT CORRECT, but it mirrors the canceling that was incorrectly done in the previous example. This gives,

Clearly these two aren’t the same! So you need to be careful with canceling!

Now, let’s take a quick look at canceling errors involved in solving equations.

Example 4 Solve (done incorrectly).

Too many students get used to just canceling (i.e. simplifying) things to make their life easier. So, the biggest mistake in solving this kind of equation is to cancel an x from both sides to get,

While, is a solution, there is another solution that we’ve missed. Can you see what it is? Take a look at the next example to see what it is.

Example 5 Solve (done correctly).

Here’s the correct way to solve this equation. First get everything on one side then factor!

From this we can see that either

In the second case we get the we got in the first attempt, but from the first case we also get that we didn’t get in the first attempt. Clearly will work in the equation and so is a solution!

We missed the in the first attempt because we tried to make our life easier by “simplifying” the equation before solving. While some simplification is a good and necessary thing, you should NEVER divide out a term as we did in the first attempt when solving. If you do this you WILL loose solutions.

Proper Use of Square Root

There seems to be a very large misconception about the use of square roots out there. Students seem to be under the misconception that

This is not correct however. Square roots are ALWAYS positive! So the correct value is

This is the ONLY value of the square root! If we want the -4 then we do the following

Notice that I used parenthesis only to make the point on just how the minus sign was appearing! In general the middle two steps are omitted. So, if we want the negative value we have to actually put in the minus sign!

I suppose that this misconception arises because they are also asked to solve things like . Clearly the answer to this is and often they will solve by “taking the square root” of both sides. There is a missing step however. Here is the proper solution technique for this problem.

Note that the shows up in the second step before we actually find the value of the square root! It doesn’t show up as part of taking the square root.

I feel that I need to point out that many instructors (including myself on occasion) don’t help matters in that they will often omit the second step and by doing so seem to imply that the is showing up because of the square root.

So remember that square roots ALWAYS return a positive answer. If you want a negative you’ll need to put it in a minus sign BEFORE you take the square root.

Ambiguous Fractions

This is more a notational issue than an algebra issue. I decided to put it here because too many students come out of algebra classes without understanding this point. There are really three kinds of “bad” notation that people often use with fractions that can lead to errors in work.

The first is using a “/” to denote a fraction, for instance 2/3. In this case there really isn’t a problem with using a “/”, but what about 2/3x? This can be either of the two following fractions.

It is not clear from 2/3x which of these two it should be! You, as the student, may know which one of the two that you intended it to be, but a grader won’t. Also, while you may know which of the two you intended it to be when you wrote it down, will you still know which of the two it is when you go back to look at the problem when you study?

You should only use a “/” for fractions when it will be clear and obvious to everyone, not just you, how the fraction should be interpreted.

The next notational problem I see fairly regularly is people writing . It is not clear from this if the x belongs in the denominator or the fraction or not. Students often write fractions like this and usually they mean that the x shouldn’t be in the denominator. The problem is on a quick glance it often looks like it should be in the denominator and the student just didn’t draw the fraction bar over far enough.

If you intend for the x to be in the denominator then write it as such that way, , i.e. make sure that you draw the fraction bar over the WHOLE denominator. If you don’t intend for it to be in the denominator then don’t leave any doubt! Write it as .

The final notational problem that I see comes back to using a “/” to denote a fraction, but is really a parenthesis problem. This involves fractions like

Often students who use “/” to denote fractions will write this is fraction as

These students know that they are writing down the original fraction. However, almost anyone else will see the following

This is definitely NOT the original fraction. So, if you MUST use “/” to denote fractions use parenthesis to make it clear what is the numerator and what is the denominator. So, you should write it as

Trig Errors

This is a fairly short section, but contains some errors that I see my calculus students continually making so I thought I’d include them here as a separate section.

Degrees vs. Radians

Most trig classes that I’ve seen taught tend to concentrate on doing things in degrees. I suppose that this is because it’s easier for the students to visualize, but the reality is that almost all of calculus is done in radians and students too often come out of a trig class ill prepared to deal with all the radians in a calculus class.

You simply must get used to doing everything in radians in a calculus class. If you are asked to evaluate at we are asking you to use 10 radians not 10 degrees! The answers are very, very different! Consider the following,

You’ll notice that they aren’t even the same sign!

So, be careful and make sure that you always use radians when dealing with trig functions in a trig class. Make sure your calculator is set to calculations in radians.

cos(x) is NOT multiplication

I see students attempting both of the following on a continual basis

These just simply aren’t true. The only reason that I can think of for these mistakes is that students must be thinking of as a multiplication of something called cos and x. This couldn’t be farther from the truth! Cosine is a function and the cos is used to denote that we are dealing with the cosine function!

If you’re not sure you believe that those aren’t true just pick a couple of values for x and y and plug into the first example.

So, it’s clear that the first isn’t true and we could do a similar test for the second example.

I suppose that the problem is that occasionally there are values for these that are true. For instance, you could use in the second example and both sides would be zero and so it would work for that value of x. In general however, for the vast majority of values out there in the world these simply aren’t true!

On a more general note. I picked on cosine for this example, but I could have used any of the six trig functions, so be careful!

Powers of trig functions

Remember that if n is a positive integer then

The same holds for all the other trig functions as well of course. This is just a notational idiosyncrasy that you’ve got to get used to. Also remember to keep the following straight.

In the first case we taking the tangent then squaring result and in the second we are squaring the x then taking the tangent.

The is actually not the best notation for this type of problem, but I see people (both students and instructors) using it all the time. We really should probably use to make things clear.

Inverse trig notation

The notation for inverse trig functions is not the best. You need to remember, that despite what I just got done talking about above,

This is why I said that n was a positive integer in the previous discussion. I wanted to avoid this notational problem. The -1 in is NOT an exponent, it is there to denote the fact that we are dealing with an inverse trig function.