THE MOST COMMON ERRORS IN
This web page describes the errors that I have seen most frequently in
undergraduate mathematics, the likely causes of those errors, and their
remedies. I am tired of seeing these same old errors over and over again. (I
would rather see new, original errors!) I caution my undergraduate students
about these errors at the beginning of each semester. Outline of this web page:
? ERRORS IN COMMUNICATION, including teacher hostility or
arrogance, student shyness, unclear wording, bad handwriting, not
reading directions,loss of invisible parentheses, terms lost inside an
? ALGEBRA ERRORS, including sign errors, everything is
additive, everything is commutative, undistributed
cancellations, dimensional errors
? CONFUSION ABOUT NOTATION, including idiosyncratic
inverses, square roots, order of operations, ambiguously written
fractions, stream-of-consciousness notations.
? ERRORS IN REASONING, including going over your
work, overlooking irreversibility, not checking for extraneous
roots, confusing a statement with its converse, working
backward, difficulties with quantifiers, erroneous methods that
work, unquestioning faith in calculators.
? UNWARRANTED GENERALIZATIONS, including Euler's xsquare root error, x.
? OTHER COMMON CALCULUS ERRORS, including jumping to
conclusions about infinity, loss or misuse of constants of
integration, loss of differentials.
(There is some overlap among these topics, so I recommend reading the
whole page.) ... Of related interest: Paul Cox's web page, and the books of Bradis, Minkovskii, and Kharcheva and E. A. Maxwell. Ultimately, what are the sources of errors and of misunderstanding? What
kinds of biases and erroneous preconceptions do we have? Two of my
favorite historic discoveries are Einstein's discovery of relativity and
Cantor's discoveries of some of the most basic rules of infinities. These
discoveries are remarkable in that neither involved long, involved,
complicated computations. Both are fairly simple, in retrospect, to anyone
who has studied them. But both involved "thinking outside the box" par
excellence -- i.e., seeing past the assumptions that were inherent in our
culture and our language. As philosopher John Culkin said, "We don't know
who discovered water, but we are certain it wasn't a fish." That certain
mathematical errors are common among students may be partly a
consequence of biases that are built into our language and culture, some of
which we aren't even aware of.
Errors in Communication
Some teachers are hostile to questions. That is an error made by teachers.
Teachers, you will be more comfortable in your job if you try to do it well,
and don't think of your students as the enemy. This means listening to your
students and encouraging their questions. A teacher who only lectures, and
does not encourage questions, might as well be replaced by a book or a
movie. To teach effectively, you have to know when your students have
understood something and when they haven't; the most efficient way to
discover that is to listen to them and to watch their faces. Perhaps you
identify with your brightest students, because they are most able to
appreciate the beauty of the ideas you are teaching -- but the other students
have greater need of your help, and they have a right to it.
A variant of teacher hostility is teacher arrogance. In its mildest form, this
may simply mean a teacher who, despite being polite and pleasant, is unable
to conceive of the idea that he/she could have made an error, even when that
error is brought directly to his/her attention. Actually, most of the errors
listed below can be made by teachers, not just by students. (However, most
teachers are right far more often than their students, so students should
exercise great caution when considering whether their teachers could be in
If you're a student with a hostile teacher, then I'm afraid I don't know what
advice to give you; transfer to a different section or drop the course
altogether if that is feasible. The remarks on communication in the next few
paragraphs are for students whose teachers are receptive to questions. For
such students, a common error is that of not asking questions.
When your teacher says something that you don't understand, don't be shy
about asking; that's why you're in class! If you've been listening but not
understanding, then your question is not a "stupid question." Moreover, you
probably aren't alone in your lack of understanding -- there are probably a
dozen other students in your classroom who are confused about precisely the
same point, and are even more shy and inarticulate than you. Think of
yourself as their spokesperson; you'll be doing them all a favor if you ask
your question. You'll also be doing your teacher a favor -- your teacher
doesn't always know which points have been explained clearly enough and
which points have not; your questions provide the feedback that your
If you think your teacher may have made a mistake on the chalkboard, you'd
be doing the whole class a favor by asking about it. (To save face, just in
case the error is your own, formulate it as a question rather than a statement.
For instance, instead of saying "that 5 should be a 7", you can ask "should
that 5 be a 7?")
And try to ask your question as soon as possible after it comes up. Don't
wait until the very end of the example, or until the end of class. As a teacher,
I hate it when class has ended and students are leaving the room and some
student comes up to me and says "shouldn't that 5 have been a 7?" Then I
say "Yes, you're right, but I wish you had asked about it out sooner. Now all
your classmates have an error in the notes that they took in class, and they
may have trouble deciphering their notes later."
Marc Mims sent me this anecdote about unasked questions:
In the early 1980s, I managed a computer retail store. Several of my employees were college students. One bright your man was having difficulty with his Freshman college algebra class. I tutored him and he did very well, but invariably, he would say, "the professor worked through this problem on the board, and it was nothing like this. I sure hope we got the correct answer."
I accompanied him to class one morning and discovered the source of his frustration. The professor was from the music department, and didn't normally teach college algebra --- he had been pressed into duty when over enrollment forced the class to be split.
During the class, he picked a problem from the assignment to work out on the board. Very early in the problem, he made an error. I don't recall the specifics, but I'm sure it was one of the many typical algebra errors you list.
Because of the error, he eventually reached a point from which he could no longer proceed. Rather than admitting an error and going to work to find it, he paused staring at the board for several seconds, then turned to the class and said, "...and the rest, young people, should be obvious."
Unclear wording. The English language was not designed for mathematical
clarity. Indeed, most of the English language was not really designed at all --
it simply grew. It is not always perfectly clear. Mathematicians must build
their communication on top of English [or replace English with whatever is
your native or local language], and so they must work to overcome the
weaknesses of English. Communicating clearly is an art that takes great
practice, and that can never be entirely perfected.Lack of clarity often comes in the form of ambiguity -- i.e., when a
communication has more than one possible interpretation.
Miscommunication can occur in several ways; here are two of them:
? One of the things that you've said has two or more possible meanings,
and you're aware of that fact, but you're satisfied that it's clear which
meaning you intended -- either because it's clear from the context, or
because you've added some further, clarifying words. But your
audience isn't as knowledgeable as you about this subject, and so the
distinction was not clear to them from the context or from your
further clarifying words. Or,
? One of the things that you've said has two or more possible meanings,
and you're not aware of that fact, because you weren't watching your
own choice of words carefully enough and/or because you're not
knowledgeable enough about some of the other meanings that those
words have to some people.
One way that ambiguity can occur is when there are multiple conventions. A
convention is an agreed-upon way of doing things. In some cases, one group
of mathematicians has agreed upon one way of doing things, and another
group of mathematicians has agreed upon another way, and the two groups
are unaware of each other. The student who gets a teacher from one group
and later gets another teacher from the other group is sure to end up
confused. An example of this is given under "ambiguously written
fractions," discussed later on this page.
Choosing precise wording is a fine art, which can be improved with practice
but never perfected. Each topic within math (or within any field) has its own
tricky phrases; familiarity with that topic leads to eventually mastering those
For instance, one student sent me this example from combinatorics, a topic that requires somewhat
How many different words of five letters can be formed from seven different consonants and four
different vowels if no two consonants and vowels can come together and no repetitions are allowed?
How many can be formed if each letter could be repeated any number of times?
There are a number of places where this problem is unclear. In the first sentence, I'm not sure what
"can come together" means, but I would guess that the intended meaning is
How many different words of five letters can be formed from seven different consonants and four
different vowels if no two consonants can occur consecutively and no two vowels can occur
consecutively and no repetitions are allowed?
The second sentence is a bit worse. The student misinterpreted that sentence to mean How many different words of five letters can be formed from seven different consonants and four
different vowels if each letter could be repeated any number of times? But usually, when a math book asks two consecutive questions related in this fashion, the second
question is intended as a modification of the first question. We are to retain all parts of the first question that are compatible with the new conditions, and to discard all parts of the first question that
would be contradicted by the new conditions. Thus, the second sentence in our example should be
interpreted in this rather different fashion, which yields a different answer: How many different words of five letters can be formed from seven different consonants and four
different vowels if no two consonants can be consecutive, no two vowels can be consecutive, and each
letter could be repeated any number of times?
Bad handwriting is an error that the student makes in communicating with
himself or herself. If you write badly, your teacher will have difficulty
reading your work, and you may even have difficulty reading your own
work after some time has passed!
Usually I do not deduct points for a sloppy handwriting style, provided that
the student ends up with the right answer at the end -- but some students
write so badly that they end up with the wrong answer because they have
misread their own work. For instance,
455? (5x+2)dx x+7x+C (should be x+2x+C) This student's handwriting was so bad that he misread his own writing; he
took the "2" for a "7". You'll have to use your imagination here, since this
electronic typesetting cannot duplicate sloppy handwriting. You do not need
to make your handwriting as neat as this typeset document, but you need to
be neat enough so that you or anyone else can distinguish easily between
characters that are intended to be different. Most students would fare better
if they would print their mathematics, instead of using cursive writing.
By the way, write your plus sign (+) and lower-case letter Tee (t) so that they don't look identical! One easy way to do this is to put a little "tail" at
the bottom of thet, just as it appears in this typeset document. (I assume that
the fonts you're using on your browser aren't much different from my fonts.)
Not reading directions. Students often do not read the instructions on a test
carefully, and so in some cases they give the right answer to the wrong
Loss of invisible parentheses. This is not an erroneous belief; rather, it is a
sloppy technique of writing. During one of your computations, if you think a
pair of parentheses but neglect to write them (for lack of time, or from sheer
laziness), and then in the next step of your computation you forget that you
omitted a parenthesis from the previous step, you may base your subsequent
computations on the incorrectly written expression. Here is a typical
computation of this sort:
453 ? (5x+7)dx 3x+7x+C But that should be 4553 ? (5x+7)dx = 3(x+7x)+C = 3x+21x+C That's an entirely different answer, and it's the correct answer. To see where
the error creeps in, just try erasing the last pair of parentheses in the line
A partial loss of parentheses results in unbalanced parentheses. For 4example, the expression "3(5x+2x+7" is meaningless, because there are
more left parentheses than right parentheses. Moreover, it is ambiguous -- if
we try to add a right parenthesis, we could get
44either "3(5x+2x)+7" or "3(5x+2x+7)"; those are two different answers.
Loss of parentheses is particularly common with minus signs and/or with
integrals; for instance,
455–? (5x–7)dx –x–7x+C (should be –x+7x+C)
Terms lost inside an ellipsis. An ellipsis is three dots (...), used to denote
"continue the pattern". This notation can be used to write a long list. For
instance,"1, 2, 3, ..., 100" represents all the integers from 1 to 100; that's
much more convenient than actually writing all 100 numbers. And for some
purposes, an ellipsis is not just a convenience, it's a necessity. For
instance, "1, 2, 3, ..., n" represents all the integers from 1 to n, where n is
some unspecified positive integer; there's no way to write that without an
The ellipsis notation conceals some terms in the sequence. But can only be
used if enough terms are left unconcealed to make the pattern evident. For
instance,"1, ..., 64" is ambiguous -- it might have any of these
? "1, 2, 3, 4, ..., 64" (all the integers from 1 to 64) 2? "1, 4, 9, 16, 25, 36, 49, 64" (that's n as n goes from 1 to 8) n? "1, 2, 4, 8, 16, 32, 64" (that's 2 as n goes from 0 to 6) Of course, in some cases one of these meanings might be clear from the
context. And just how much information is needed "to make a pattern
evident" is a subjective matter; it may vary from one audience to another.
Best to err on the safe side: give at least as much information as would be
needed by the least imaginative member of your audience. I have seen many errors in using ellipses when I've tried to teach induction
proofs. For instance, suppose that we'd like to prove
2222[*n] 1 + 2 + 3 + ... + n = n(n+1)(2n+1)/6 for all positive integers n. The procedure is this: Verify that the equation is
true when n=1 (that's the "initial step); then assume that [*n] is true for some
unspecified value of n and use that fact to prove that it's true for the next
value of n -- i.e., to prove [*(n+1)] (that's the "transition step"). Here is a
typical error in the transition step: Add 2n+1 to both sides of [*n]. Thus we
obtain 2222[i] 1 + 2 + 3 + ... + n + 2n+1 = (2n+1) + n(n+1)(2n+1)/6.
But that says 2222[ii] 1 + 2 + 3 + ... + (n+1) = (2n+1) + n(n+1)(2n+1)/6.
We've made a mistake already, in the left side of the equation. (Can you find
it? I'll explain it in a moment.) Now make some algebra error while
rearranging the right side of the equation, to obtain 2222[*(n+1)] 1 + 2 + 3 + ... + (n+1) = (n+1)(n+2)(2n+3)/6.
And now it appears that we're done. But there was an algebra error on the
right side: (2n+1) + n(n+1)(2n+1)/6 actually is not equal to (n+1)(n+2)(2n+3)/6. (You can check that easily.) The error on the left side was more subtle. It is based on the fact that too
many terms were concealed in the ellipsis, and so the pattern was not
revealed accurately. To see what is really going on, let's rewrite equations [i]
and [ii], putting more terms in:
222222[i] 1 + 2 + 3 + ... + (n-2) + (n-1) + n + 2n+1 = (2n+1) + n(n+1)(2n+1)/6.
222222 + 2 + 3 + ... + (n-2) + (n-1) + (n+1) = (2n+1) + n(n+1)(2n+1)/6.
[ii] 12And now you can see that the left side is missing its n term, so the left side
of [ii] is not equal to the left side of [*(n+1)].
Sign errors are surely the most common errors of all. I generally deduct
only one point for these errors, not because they are unimportant, but
because deducting more would involve swimming against a tide that is just
too strong for me. The great number of sign errors suggests that students are
careless and unconcerned -- that students think sign errors do not matter. But
sign errors certainly do matter, a great deal. Your trains will not run, your
rockets will not fly, your bridges will fall down, if they are constructed with
calculations that have sign errors.
Sign errors are just the symptom; there can be several different underlying
causes. One cause is the "loss of invisible parentheses," discussed in a later
section of this web page. Another cause is the belief that a minus sign means a negative number. I think that most students who harbor this belief
do so only on an unconscious level; they would give it up if it were brought
to their attention. [My thanks to Jon Jacobsen for identifying this error.] Is –x a negative number? That depends on what x is.
? Yes, if x is a positive number.
? No, if x itself is a negative number. For instance, when x = –6, then
–x = 6 (or, for emphasis, –x = +6).
That's something like a "double negative". We sometimes need double
negatives in math, but they are unfamiliar to students because we generally
try to avoid them in English; they are conceptually complicated. For
instance, instead of saying "I do not have a lack of funds" (two negatives), it
is simpler to say "I have sufficient funds" (one positive).
Another reason that some students get confused on this point is that we
read "–x" aloud as "minus x" or as "negative x". The latter reading suggests
to some students that the answer should be a negative number, but that's not
right. [Suggested by Chris Phillips.]
Misunderstanding this point also causes some students to have difficulty
understanding the definition of the absolute value function. Geometrically,
we think of |x| as the distance between x and 0. Thus |–3| = 3 and |27.3| = 27.3, etc. A distance is always a positive quantity (or more precisely,
a nonnegative quantity, since it could be zero). Informally and imprecisely,
we might say that the absolute value function is the "make it positive"
Those definitions of absolute value are all geometric or verbal or
algorithmic. It is useful to also have a formula that defines |x|, but to do that we must make use of the double negative, discussed a few sentences ago.
Thus we obtain this formula:
which is a bit complicated and confuses many beginners. Perhaps it's better
to start with the distance concept.
Many college students don't know how to add fractions. They don't know
how to add (x/y)+(u/v), and some of them don't even know how to
add (2/3)+(7/9). It is hard to classify the different kinds of mistakes they
make, but in many cases their mistakes are related to this one:
Everything is additive. In advanced mathematics, a function or
operation f is called additive if it satisfies f(x+y)=f(x)+f(y) for all numbers x and y. This is true for certain familiar operations -- for instance,
? the limit of a sum is the sum of the limits,
? the derivative of a sum is the sum of the derivatives,
? the integral of a sum is the sum of the integrals.
But it is not true for certain other kinds of operations. Nevertheless, students
often apply this addition rule indiscriminately. For instance, contrary to the
belief of many students,
We do get equality holding for a few unusual and coincidental choices
of x and y, but we have inequality for most choices of x and y. (For instance,
all four of those lines are inequalities when x = y = π/2. The student who is
not sure about all this should work out that example in detail; he or she will
see that that example is typical.)
One explanation for the error with sines is that some students, seeing the
parentheses, feel that the sine operator is a multiplication operator -- i.e., just
as6(x+y)=6x+6y is correct, they think that sin(x+y)=sin(x)+sin(y) is correct.
The "everything is additive" error is actually the most common occurrence
of a more general class of errors:
Everything is commutative. In higher mathematics, we say that two operations commute if we can perform them in either order and get the same
result. We've already looked at some examples with addition; here are some
examples with other operations. Contrary to some students' beliefs,
etc. Another common error is to assume that multiplication commutes with
differentiation or integration. But actually, in
general (uv)′ does not equal (u′)(v′) and ? (uv)does not equal (? u)(? v).
However, to be completely honest about this, I must admit that there is one
very special case where such a multiplication formula for integrals is correct.
It is applicable only when the region of integration is a rectangle with sides
parallel to the coordinate axes, and
u(x) is a function that depends only on x (not on y), and
v(y) is a function that depends only on y (not on x). Under those conditions,
(I hope that I am doing more good than harm by mentioning this formula,
but I'm not sure that that is so. I am afraid that a few students will write
down an abbreviated form of this formula without the accompanying
restrictive conditions, and will end up believing that I told them to
equate ? (uv) and (? u)(? v) in general. Please don't do that.)
Undistributed cancellations. Here is an error that I have seen fairly often,
but I don't have a very clear idea why students make it.