Common Errors in College Math
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-- Eric
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THE
MOST COMMON ERRORS IN UNDERGRADUATE
MATHEMATICS
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 ellipsis
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 square root error,
xx.
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.
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 error.)
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 teacher needs.
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 phrases.
For instance, one student sent me this example
from combinatorics, a topic that requires somewhat awkward
English:
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,
ò (5x4+2)dx
![[image: equals sign with questionmark over it]](http://www.math.vanderbilt.edu/~schectex/commerrs//equalqn.gif)
x5+7x+C
(should be x5+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 the t, 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 problem.
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:
3
ò (5x4+7)dx
3x5+7x+C
But that should be
3 ò (5x4+7)dx =
3(x5+7x)+C =
3x5+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 above.
A partial loss of parentheses results in
unbalanced parentheses. For example, the expression
"3(5x4+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 either
"3(5x4+2x)+7"
or
"3(5x4+2x+7)"; those are two different answers.
Loss of parentheses is particularly common with
minus signs and/or with integrals; for instance,
–ò (5x4–7)dx
–x5–7x+C
(should be –x5+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 ellipsis.
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
interpretations:
"1, 2, 3, 4, ..., 64" (all the integers from 1 to 64)
"1, 4, 9, 16, 25, 36, 49, 64" (that's n2 as n goes from 1 to 8)
"1, 2, 4, 8, 16, 32, 64" (that's 2n 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
[*n]
12 + 22 + 32 + ... + n2
= 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
[i] 12 + 22 + 32 + ... + n2 + 2n+1
= (2n+1) + n(n+1)(2n+1)/6.
But that says
[ii] 12 + 22 + 32 + ... + (n+1)2
= (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
[*(n+1)] 12 + 22 +
32 + ... + (n+1)2
= (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:
[i] 12 + 22 + 32 + ...
+ (n-2)2 + (n-1)2 + n2 + 2n+1
= (2n+1) + n(n+1)(2n+1)/6.
[ii] 12 + 22 + 32 + ...
+ (n-2)2 + (n-1)2
+ (n+1)2
= (2n+1) + n(n+1)(2n+1)/6.
And now you can see that the left side is missing its n2 term, so
the left side of [ii] is not equal to the left side of [*(n+1)].
Algebra Errors
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" function.
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:
![[image: absolute value of x is
x if x is greater than or equal
to 0, or -x if x is less than 0]](http://www.math.vanderbilt.edu/~schectex/commerrs//absvalue.gif)
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,
 order=0
alt="[image:
sin(x+y) is NOT equal to sin x+sin y,
(x+y)^2 is NOT equal to x^2+y^2,
sqrt(x+y) is NOT equal to sqrt x+sqrt y,
1/(x+y) is NOT equal to (1/x)+(1/y).]">
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 = p/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 as
6(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,
 order=0
alt="[image:
log(sqrt x) is NOT equal to sqrt(log x),
sin(3x) is NOT equal to 3(sin x),]">
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
ò (uv) does not equal
(ò u)(ò 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,
![[image: double integral, from a to b and from c to d, of
u(x)v(y)dydx, is equal to the product of <b>these</b> two
integrals: integral from a to b of u(x)dx and integral
from c to d of v(y)dy.]](http://www.math.vanderbilt.edu/~schectex/commerrs//recintgl.gif)
(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
ò (uv) and
(ò u)(ò 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.
(3x+7)(2x–9) +
(x2+1) (3x+7)
(2x–9) + (x2+1) (2x–9) +
(x2+1)
f(x) =
=
(3x+7)(x3+6) (3x+7)
(x3+6) (x3+6)
In a sense, this is the reverse of the "loss of invisible
parentheses"
mentioned earlier; you might call this error
"insertion of invisible parentheses."
To see why, compare the preceding computation
(which is wrong)
with the following computation (which is correct).
(3x+7) [ (2x–9) + (x2+1)]
(3x+7)
[ (2x–9) + (x2+1) ] (2x–9) +
(x2+1)
g(x) = = =
(3x+7)
(x3+6) (3x+7)
(x3+6) (x3+6)
Apparently some students think that f(x) and g(x) are
the same thing -- or perhaps they simply don't bother to
look carefully enough at the top line of f(x), to discover
that not everything in the top line of f(x) has a factor
of (3x+7). If you still don't see what's going on,
here is a correct computation involving
that first function f :
x2+1
2x–9 +
(3x+7)(2x–9) +
(x2+1) 3x+7
f(x) = =
(3x+7)(x3+6) x3+6
Why would students make errors like these? Perhaps it is partly because they don't
understand some of the basic concepts of fractions. Here are some things worth noting:
Multiplication is commutative -- that is, xy = yx. Consequently, most rules about
multiplication are symmetric. For instance, multiplication distributes over addition
both on the left and on the right:
(x1+x2)y=(x1y)+(x2y)
and
x(y1+y2)=(xy1)+(xy2) .
Division is not commutative -- in general, x/y is not equal to y/x.
Consequently, rules about division are not symmetric (though perhaps some students expected
them to be symmetric). For instance,
(x1+x2)/y = (x1/y)+(x2/y)
but in general
x/(y1+y2)
¹
(x/y1)+(x/y2) .
Fractions represent division and grouping (i.e., parentheses). For instance,
the fraction
a+b
c+d
is the same thing as (a+b)/(c+d). If you omit either pair of parentheses from
that last expression, you get something entirely different.
(Thanks to Mark
Meckes for pointing out this possible explanation
of the origin of such errors.) Perhaps some of
the students' errors stem from such an omission of parentheses? or a
lack of understanding of how important those parentheses are? That would
seem to be indicated by the prevalance of another type of error described
elsewhere on this page,
"loss of invisible parentheses".
Dimensional errors. Most of this web page is devoted
to things that you should not do, but dimensional
analysis is something that you should do. Dimensional
analysis doesn't tell you the right answer, but it does
enable you to instantly recognize the wrongness of some
kinds of wrong answers. Just keep careful track of your
dimensions, and then see whether your answer looks right.
Here are some examples:
If you're asked to find a volume, and your answer is some number of
square inches, then you know you've made an error somewhere
in your calculations.
(If you find this kind of error in your answer, don't
just change "square inches" to "cubic inches" in your
answer and leave the numerical part unchanged. The step in
your calculation where you got the wrong units may also be a step
where you made a numerical error. Try to find that step,)
If you're asked to find an area or a volume, and your answer is a
negative
number, then you know you've made an error somewhere.
(Again,
don't just change the sign in your answer -- there
may be more to your error than that.)
If the question is a word-problem, think about whether
your answer makes sense. For instance, if you're given
the dimensions of a coin and you're asked
to find its surface area, and you come up with an
answer of 3000 square miles, you should realize
that you've probably made an error (even though your
answer has the right units), and you should look
for that error. (This is not really an example of dimensional
analysis, but I didn't know where else to put it.
Thanks to Sandeep Kanabar for this example.)
Even if you don't remember the formula for the volume
of a sphere of radius r, you know that it has to have
a factor of r3, not
r2 , so the answer couldn't possibly be
pr2.
Even if you don't remember the formula for the surface
area of a sphere of radius r, you know it has to get
small when r gets small. So it couldn't possibly be
something like
2+3r2.
Here is a cute example of dimensional analysis
(submitted by Benjamin Tilly).
Problem:
Where has my money gone? My dollar seems to have turned into
a penny:
$1 = 100¢
= (10¢)2
= ($0.10)2
= $0.01
= 1¢
Explanation:
Of course, the problem is a disregard for dimensional
units. Strictly speaking, if you square a dollar,
you should get a square dollar. I don't know what
a "square dollar" is, but I still know how to
compute with it, and I know that a
"square dollar" must be equal to
10,000 "square pennies", since one dollar is 100 pennies.
Dimensional computations will not yield errors if we
handle the dimensional units correctly.
Here is a correct computation:
$21 = ($1)2 =
(100¢)2
= 1002ٖ
= 10,000ٖ.
It should now be evident what was wrong with the first
calculation:
100¢ is not equal to
(10¢)2. It's true that the 100 is equal to
the 102, but the ¢ is not equal to
ٖ.
Likewise, later in the computation, $2 is not
equal to $.
Confusion about Notation
Idiosyncratic inverses.
We need to be sympathetic about the student's difficulty in
learning the language of mathematicians. That language is
a bit more consistent than English, but it is not entirely
consistent -- it too has its idiosyncrasies, which (like those
of English) are largely due to historical accidents,
and not really anyone's fault. Here is one such
idiosyncrasy: The expressions
sinn and tann
get interpreted in different ways, depending on what n is.
sin2x = (sin x)2
and
tan2x = (tan x)2;
but
sin–1x = arcsin(x)
and
tan–1x = arctan(x).
Some students get confused about this; some even end up
setting arctan(x) equal to 1/(tan x).
When I teach, I try to reduce confusion by always writing
arcsin or arctan, rather than
sin–1 or tan–1.
But the sin–1 and
tan–1 notation still
needs to
be discussed, as it is used on nearly all handheld calculators.
Thanks to
Ian Morrison and John Armerding
for pointing this one out.
Confusion about the square root symbol.
Every positive number b has two square roots.
The expression
Öb
actually means "the nonnegative
square root of b," but unfortunately some students think that
that expression means
"either of the square roots of b" -- i.e., they think it represents
two numbers.
...
This error is made more common because
of the unfortunate fact that we math teachers are merely human,
and sometimes a little sloppy: When
we write
Öb
on the blackboard, what we say aloud might
just be "the square root of b." But that's just laziness.
If you ask us specifically about that, we'll tell you
"Oh, I'm sorry, of course I meant
the nonnegative square root of b;
I thought that goes without saying." ...
If you really do want to indicate both square roots
of b, you use the plus-or-minus sign, as in this expression:
±Öb.
Problems with order of operations. It is customary to
perform certain mathematical operations in certain orders,
and so we don't need quite so many parentheses. For
instance, everyone agrees that "6w+5" means
"(6w)+5", and not "6(w+5)" -- the
multiplication is performed before the addition, and so the
parentheses are not needed if "(6w)+5" is what
you really mean to say. Unfortunately, some students have
not learned the correct order of some operations.
Here is an example from
Ian Morrison:
What is
–32 ? Many students think that the
expression means (–3)2, and so they arrive
at an answer of 9. But that is wrong. The convention
among mathematicians is to perform the exponentiation before
the minus sign, and so –32 is
correctly interpreted as
–(32), which yields –9.
Ambiguously written fractions.
In certain common situations with fractions, there is a
lack of consensus about what order to perform
operations in. For instance,
does "3/5x" mean "(3/5)x" or "3/(5x)" ?
For this confusion, teachers must share
the blame. They certainly mean well --
most math teachers believe that they
are following the conventional order
of operations. They are not aware
that several conventions are widely used,
and no one of them is universally accepted.
Students may learn one method from one teacher
and then go on to another teacher who expects
students to follow a different method.
Both teacher and student may be unaware of
the source of the problem.
Here are some of the most widely used
interpretations:
The "BODMAS interpretation"
(bracketed operations, division, multiplication, addition, subtraction):
Perform division before multiplication. For instance, the function
f(x) = 3/5x gets interpreted as
(3/5)x =  order=0
alt="[image: horizontal bar, 3 on top,
5 on bottom, x to the right]" align=center>.
In particular,
f(5) = 3 and f(1/5) = 3/25.
The "My Dear Aunt Sally" interpretation
(multiplication, division, addition, subtraction):
Perform multiplication before division.
For instance, the function
f(x) = 3/5x gets interpreted as
3/(5x) =  order=0
alt="[image: horizontal bar, 3 on top,
5x on bottom]" align=center>.
In particular,
f(5) = 3/25 and f(1/5) = 3. Likewise,
"ax/by" would be interpreted as
"(ax)/(by)".
The interpretation used by FORTRAN and some other
computer languages (as well as some humans): Multiplication
and division are given equal priority; a string of such operations
is processed from left to right. For instance,
"ax/by" would be interpreted as
"((ax)/b))y", or more simply
"(axy)/b".
Some students think that their electronic calculators
can be relied upon for correct answers. But some calculators
follow one convention, and other calculators follow another
convention. In fact, some of the Texas Instruments calculators
follow two conventions, according to whether multiplication
is indicated by juxtaposition or a symbol:
3 / 5x is interpreted as 3/(5x), but
3 / 5 * x is interpreted as (3/5) x.
(Thanks to Chris
Phillips and
Thomas
Cowdery
for some of these examples and comments.)
Because there is no consensus of interpretation,
I recommend that
you do not write expressions like "3/5x" -- i.e.,
do not write a fraction involving a diagonal slash followed by a product,
without any parentheses. Instead, use one of these four
nonambiguous expressions:
(3/5)x , order=0
alt="[image: horizontal bar, 3 on top,
5 on bottom, x to the right]" align=center>, 3/(5x),
order=0
alt="[image: horizontal bar, 3 on top,
5x on bottom]" align=center>.
In some cases, additional information is evident from the context --
if one is familiar with the context. For instance, an experienced
mathematician will recognize dy/dx as a derivative;
it is the quotient of two differentials.
The letter d represents the differential operator, not a variable. The
expression dx represents the differential of x, not the product of two
variables. Thus, parentheses are not needed, and would look rather
strange if used. We do not write dy/(dx) or (dy)/(dx).
Here is another common error in the writing of fractions:
If you write
the horizontal fraction bar too high, it can be misread. For instance,
or
are
acceptable expressions (with different meanings), but
![[image: horizontal bar with 3 on top and 5 on
bottom, followed by an x that is to the right of
the bar but lowered to the same height as the 5]](http://www.math.vanderbilt.edu/~schectex/commerrs//halfx6.gif) is
unacceptable -- it has no conventional meaning, and could be interpreted
ambiguously as either of the previous fractions.
I will not give full credit for ambiguous answers on any quiz or test.
In this type of error,
sloppy handwriting is the culprit. When you write an expression
such as ,
be sure to write carefully, so that the horizontal bar is aimed at the
middle of the x.
Here's one more example of interest.
When entered as 2 ^ 3 ^ 4 without parentheses, the TI85 calculator
shows 4096 and the TI89
shows 2.41785163923 E24. (Those are the answers to
(2 ^ 3) ^ 4 and 2 ^ (3 ^ 4), respectively.)
Thus, even the calculators
made by one company don't all agree on their orders of
operations. When in doubt, use parentheses! Thanks
to Bill Dodge for this example.
Stream-of-consciousness equalities and implications.
(My thanks to H. G. Mushenheim for identifying this type of error and
suggesting a name for it.) This is an error in the intermediate
steps in students' computations. It doesn't often lead to an erroneous
final result at the end of that computation, but it is tremendously
irritating to the mathematician who must grade the student's paper.
It may also lead to a loss of partial credit, if the student makes
some other error in his or her computation and the grader
is then unable to decipher the student's work because of this stream-of-consciousness
error.
To put it simply: Some students (especially college freshmen)
use the equals sign (=) as a symbol
for the word "then" or the phrase "the next step is."
For instance, when asked to find the third derivative of
x4+7x2–5, some students will write
"x4+7x2–5 =
4x3+14x =
12x2+14 = 24x." Of course, those
four
expressions are not actually equal to one another.
A slight
variant of this error consists of connecting several different
equations with equal signs, where the intermediate
equals signs are intended to convey "equivalent to" --- for example,
x = y – 3 = x+3 = y. This is very confusing and
altogether wrong, because equality is transitive --- i.e., if
a=b and b=c then a=c, but
x certainly is not equal to x+3.
It would be better to replace that middle equals sign with
some other symbol. The most obvious symbol for this purpose
is º, which means "is equivalent to,"
but that symbol has the disadvantage of looking too much like
an equals sign, and thus possibly leading to the same
confusion. Thus, a better choice would be
« or
Û, both of which mean
"if and only if." Thus, I would rewrite the example above as
x = y – 3 Û x+3 = y.
There is also a more "advanced" form of this error. Some more advanced
students (e.g., college seniors) use the implication symbol
(Þ) as a symbol
for the phrase "the next step is."
A string of statements of the form
A
Þ
B
Þ
C
Þ
D
should mean that A by itself implies B, and B by itself implies C, and C by itself implies D;
that is the coventional
interpretation given by mathematicians. But some
students use such a string to mean merely
that if we start from A, then
the next step in our reasoning is B
(using not only A but other information as well)
and then the next step is C (perhaps using both A and B), etc.
Actually, there is a symbol for "the next step is." It looks like this:
![[image: symbol for leads to]](http://www.math.vanderbilt.edu/~schectex/commerrs//leadsto.gif)
It is also called "leads to," and in the LaTeX formatting language
it is given by the code \leadsto. However, I haven't
seen it used very often.
Errors in Reasoning
Going over your work.
Unfortunately, most textbooks do not devote a lot of attention to
checking your work, and some teachers also skip this topic. Perhaps
the reason is that there is no well-organized body of theory
on how to check your work. Unfortunately, some students end up
with the impression that it is not necessary to check your work --
just write it up once, and hope that it's correct. But that's
nonsense. All of us make mistakes sometimes. In any
subject, if you want to do good work, you have to work carefully,
and then you have to check your work. In English, this
is called "proofreading"; in computer science, this is called
"debugging."
Moreover, in mathematics, checking your work is
an important part of the learning process. Sure, you'll learn
what you did wrong when you get your homework paper back from
the grader; but you'll learn the subject much better if
you try very hard to make sure that your answers
are right before you turn in your homework.
It's important to check your work, but
"going over your work" is
the worst way to do it.
I have twisted some words here, in order to make a point. By "going
over your work" I mean reading through the steps that you've just done,
to see if they look right. The drawback of that method is you're quite
likely to make the same mistake again when you read through your steps!
This is particularly true of conceptual errors -- e.g., forgetting to check
for extraneous roots (discussed later on this web page).
You would be much more likely to catch your error if, instead, you
checked your work by some method that is different from your
original computation. Indeed, with that approach, the only way your
error can go undetected is if you make two different errors that
somehow, just by a remarkable coincidence, manage to cancel each other
out -- e.g., if you arrive at the same wrong answer by two different
incorrect methods. That happens occasionally, but very seldom.
In many cases, your second method can be easier, because it can
make use of the fact that you already have an answer. This type
of checking is not 100% reliable, but it is very highly reliable,
and it may take very little time and effort.
Here is a simple example. Suppose that we want to solve 3(x–2)+7x = 2(x+1)
for x. Here is a correct solution:
3(x–2)+7x = 2(x+1)
3x–6+7x = 2x+2
3x+7x–2x = 2+6
8x = 8
x = 1
Now, one easy way to check this work is to plug
x = 1 into each side of the
original equation, and see if the results come out the same.
On the left side, we have
3(x–2)+7x = 3(1–2)+7(1) = 3(–1)+7(1) = (–3)+7 = 4. On the right side, we have
2(x+1) = 2(1+1) = 2(2) = 4.
Those are the same, so the check works.
It's easier than the original computation, because in the original computation
we were looking for x; in the check, we already have a candidate for x.
Nevertheless,
this computation was by a different method than our original
computation, so the answer is probably right.
Different kinds of problems require different kinds of checking.
For a few kinds of problems, no other method of checking besides
"going over your work" will suggest itself to you. But for
most problems, some second method of checking will be evident
if you think about it for a moment.
If you absolutely can't
think of any other method, here is a last-resort technique:
Put the paper away somewhere. Several hours later (if you can
afford to wait that long), do the same problems over -- by
the same method, if need be -- but on a new sheet of paper,
without looking at the first sheet. Then compare the answers.
There is still some chance of making the same error twice,
but this method reduces that chance at least a little.
Unfortunately,
this technique doubles the amount of work you have to do,
and so you may be reluctant to employ this technique.
Well, that's up to you; it's your decision. But
how badly do you want to master the material and get the
higher grade? How much importance do you attach to the
integrity of your work?
One method that many students use to check their homework is this:
before turning in your paper, compare it with a classmate's paper;
see if the two of you got the same answers. I'll admit that this
does satisfy my criterion: If you got the same answer for
a problem, then that answer is probably right. This approach
has both advantages and disadvantages. One disadvantage is that
it may violate your teacher's rules about homework being an
individual effort; perhaps you should ask your teacher what
his or her rules are. Another concern is: how much do you learn
from the comparison of the two answers? If you discuss the
problem with your classmate, you may learn something. With or
without a classmate's involvement, if you think some more about
the different solutions to the
problem, you may learn something.
When you do find that your two answers differ, work very
carefully to determine which one (if either) is correct.
Don't hurry through this crucial last part of the process. You've
already demonstrated your fallibility on this type of problem, so
there is extra reason to doubt the accuracy of any further work
on this problem; check your results several times.
Perhaps the error occurred through mere carelessness, because you
weren't really interested in the work and you were in a hurry to
finish it and put it aside. If so, don't compound that error.
You now must pay for your neglect -- you now must put in even
more time to master the material properly! The problem
won't just go away or lose importance if you ignore it.
Mathematics, more than any other subject, is vertically
structured: each concept builds on many concepts that preceded
it. Once you leave a topic unmastered, it will haunt you
repeatedly throughout many of the topics that follow it, in all
of the math courses that follow it.
Also, if discover that you've made an error, try to discover
what the error was. It may be a type of error that you
are making with some frequency. Once you identify it, you may
be better able to watch out for it in the future.
Not noticing that some steps are irreversible.
If you do the same thing to both sides of a true equation,
you'll get another true equation. So if you have an equation
that is satisfied by some unknown number x, and you do the same
thing to both sides of the equation, then the new equation will
still be satisfied by the same number x. Thus, the new equation
will have all the solutions x that the old equation had --
but it might also have some new solutions.
Some operations are reversible -- i.e., we have the same set
of solutions before and after the operation. For instance,
The operation "multiply both sides of the equation
by 2" is reversible. For example,
the set of all numbers x that satisfy
x2–3x–2 = –4
is the same as the set of all numbers x that satisfy
2x2–6x–4 = –8. In fact, to reverse the operation,
we just have to multiply both sides of an equation by 1/2.
The operation "subtract 7 from both sides" is reversible.
To reverse it, just add 7 to both sides.
Some operations are not reversible, and so we may get
new solutions when we perform such an operation. For instance,
The operation "square both sides" is not reversible.
For instance, the equation x = –3 has only one solution,
but when we square both sides, we get
x2 = 9, which has two solutions.
The operation "multiply both sides by x–4" is not
reversible. The resulting equation will have for its solutions
all of the solutions of the original equation plus the additional
new solution x=4.
A commonly used method for solving equations is this: Construct a sequence
of equations, going from one equation to the next by doing the same
thing to both sides of an equation, choosing the operations to gradually
simplify the equation, until you get the equation down to something
obvious like "x=5". This method is not bad for discovery, but as a method
of certification it is unreliable. To make it reliable, you need to add one more rule:
if any of your steps are irreversible,
then you must check for extraneous roots when you get to the
end of the computation.
That's because, at the end of your computational
procedure, you'll have not only the solution(s) to the original
problem, but possibly also some additional numbers that do not solve
the original problem. How do you check for them? Just plug each of your
answers into the original problem, to see whether it works. Many
students, unfortuntely, omit that last step.
First example:
The given
problem is
![[image: square root of (2x+12)]](http://www.math.vanderbilt.edu/~schectex/commerrs//sqrt2x12.gif) – 2 = x.
To begin solving this problem,
add 2 to both sides (a
reversible step); we obtain
= x + 2.
Square both
sides -- an irreversible step; we obtain
2x+12 = x2+4x+4.
By adding and subtracting appropriate amounts to both sides of
the equation (a reversible step), we obtain
x2+2x–8 = 0. Now
solve that quadratic equation by your favorite method
-- by the quadratic formula, by completing the square,
or by factoring by inspection. We obtain
x = 2 or
x = – 4.
Unfortunately, many students stop at this step; they
believe they're done; they write {2, –4}
for their answer.
A correct solution continues as follows:
Since at least one of the steps in our procedure was
irreversible, we must check for extraneous roots.
Check each of the numbers 2 and –4, to see if it
satisfies the equation originally given in the problem.
When x=2, then
– 2 =
(4+12)1/2 – 2 =
4 – 2 = x,
so we
have a correct solution. However,
When x= – 4, then
– 2 =
(–8+12)1/2 – 2 =
2 – 2 ¹ x,
so we
have an incorrect solution.
Thus the correct answer is {2}.
Second example.
The given problem is
![]() order="0"
src="http://www.math.vanderbilt.edu/~schectex/commerrs//belfrac1.gif"
alt="[image: (x^3-3)/x + (3x^2+x+7)/x = 1 + (4/x)]">.
By adding and subtracting (reversible), we obtain
![]() order="0"
src="http://www.math.vanderbilt.edu/~schectex/commerrs//belfrac2.gif"
alt="[image: (x^3+3x^2)/x=0]">.
By factoring (reversible), we obtain
![]() order="0"
src="http://www.math.vanderbilt.edu/~schectex/commerrs//belfrac3.gif"
alt="[image: x^2(x+3)/x=0]">.
Cancel out an x (irreversible); we obtain x(x+3)=0.
The solutions appear to be x=0 and x = –3. Some
students unfortunately stop here, but we shouldn't --
one of our steps was irreversible. Checking reveals that
x=0 doesn't work in the given problem, so it's extraneous.
The correct answer is just x = –3.
Of course, even aside from the issue of extraneous
roots, another reason to check your answers is to avoid
arithmetic errors. This is a special case of "checking
your work," mentioned elsewhere on this web page.
We all make computational mistakes;
we can catch most of our computational mistakes with a
little extra effort.
The extraneous roots error was brought to my attention by
Dr. Richard Beldin.
Professor Beldin
tells me that he gave a test heavily laced with
extraneous roots problems, and warned the students that
such problems would appear on the test, and
the appearance of an extraneous root in an
answer would cost half the credit for a problem, so
the students should check for extraneous roots.
Professor Beldin reports that, nevertheless, about a third
of the students neglected to check,
on so many problems that they lost two letter grades on
the overall the test score.
Confusing a statement with its converse.
The implication "A implies B" is not the
same as the implication "B implies A." For instance,
if I went swimming at the beach today, then
I got wet today
is a true statement. But
if I got wet today, then I went swimming at the beach
today
doesn't have to be true -- maybe I got wet by taking
a shower or bath at home. The difference is easy
to see in concrete examples like these, but it may
be harder to see in the abstract setting of mathematics.
Some technical terminology might be helpful here.
The symbol
Þ means "implies."
The two statements
"A Þ B"
and
"B Þ A"
are said to be converses of each other. What
we've just explained is that an implication and
its converse generally are not equivalent.
I should emphasize the word "generally" in the
last paragraph. In a few cases the implications
"A Þ B"
and
"B Þ A"
do turn out to be equivalent. For instance,
let p,q,r be the lengths of the sides of a triangle,
with r being the longest side; then
p2+q2=r2
if and only if
the triangle is a right triangle.
The "if" part of that statement is the well-known
Pythagorean Theorem; the "only if" part is its
converse, which also happens to be true but is
less well known.
Some students confuse a statement with its converse.
This may stem partly from the fact that, in many
nonmathematical situations, a statement is
equivalent to its converse, and so in everyday "human"
English we often use the word "if" interchangeably
with the phrase "if and only if". For instance,
I'll go to the vending machine and buy a snack if I get hungry
sounds reasonable. But most people would figure that if
I do not get hungry, then I won't go buy a snack.
So, evidently, what I really meant was
I'll go to the vending machine and buy a snack if and only if
I get hungry.
Most people would just say the shorter sentence, and mean the
longer one; it's a sort of verbal shortcut.
Generally you can figure out from the context
just what the real meaning is, and usually you don't even
think about it on a conscious level.
To make matters more confusing, mathematicians are humans too.
In certain contexts, even mathematicians use "if" when they
really mean "if and only if." You have to figure this
out from the context, and that may be hard to do if you're
new to the language of mathematics, and not a fluent
speaker. Chiefly, mathematicians use the verbal shortcut
when they're giving definitions, and then you have a hint:
the word being defined usually is in italics or boldface.
For instance, here is the definition of continuity of a real-valued
function f:
f is continuous if for each real number p and each
positive number e there exists
a positive number d (which may depend on p and e) such that,
for each real number q, if | p - q | < d, then
| f(p) - f(q) | < e.
The fourth word in this very long sentence is an "if" that
really means "if and only if", but we know that because
"continuous" is in boldface; this is the definition
of the word "continuous".
Converses also should not be confused with
contrapositives. Those two words sound similar
but they mean very different things.
The contrapositive of the implication
"A Þ B"
is the implication
"(~B) Þ (~A)",
where ~ means "not." Those two statements
are equivalent. For instance,
if I went swimming at the beach today, then
I got wet today
has exactly the same meaning as the more complicated
sounding statement
if I didn't get wet today, then I
didn't go swimming at the beach today.
Sometimes we replace a statement with its contrapositive,
because it may be easier to prove, even if it is
more complicated to state.
(Thanks to Valery Mishkin
for bringing this class of errors to my attention.)
Working backward. This is an unreliable method of proof used,
unfortunately,
by many students. We start with the statement that we
want to prove, and gradually replace it with consequences,
until we arrive at a statement that is obviously true
(such as 1 = 1). From that some students conclude that the
original statement is true. They overlook the fact that some
of their steps might be irreversible.
Here is an example of a successful and correct use
of "working backward":
we are asked to prove that the cube root of 3 is greater than the
square root of 2. We write these steps:
Start by assuming the thing that we're
trying to prove:
31/3 > 21/2.
Raise both sides to the power 6; that yields
(31/3)6 > (21/2)6.
Simplify both sides, using the rule of exponents that
says (ab)c = a(bc). Thus we obtain
32 > 23.
Evaluate. That yields 9 > 8, which is clearly true.
Some students would believe that we have now proved
31/3 > 21/2.
But that's not a proof -- you should never begin a proof
by assuming the very thing that you're trying to prove.
In this example, however, all the steps happen to be
reversible, so those steps can be made into a proof. We just
have to rewrite the steps in their proper order:
9 > 8 is obviously true.
Rewrite that as 32 > 23.
Rewrite that as (31/3)6 > (21/2)6,
by using the rule of exponents that
says (ab)c = a(bc).
Now take the sixth root on both sides. That leaves
31/3 > 21/2.
Working backward can be a good method for discovering proofs,
though it has to be used with caution, as discussed below.
But it is an unacceptable method for presenting proofs
after you have discovered them. Students must distinguish between
discovery (which can be haphazard, informal, illogical) and
presentation (which must be rigorous).
The reasoning used in
working backward is a reversal of the reasoning needed for
presentation of the proof -- but that means replacing
each implication
"A Þ B"
with its converse,
"B Þ A".
As we pointed out a few paragraphs ago, those
two implications are sometimes not equivalent.
In some cases, the implication is reversible -- i.e.,
some reversible operation (like multiplying both
sides of an equation by 2, or raising both sides of an inequality
to the sixth power when both sides were already positive) transforms statement
A into statement B. Perhaps the students have
gotten into the habit of expecting all implications
to be reversible, because early in their education
they were exposed to many reversible transformations
-- adding three, multiplying by a half, etc.
But in fact, most implications of mathematical
statements are not reversible, and so
"working backward" is almost never acceptable
as a method of presenting a proof.
Working backward can be used for discovering a proof
(and, in fact, sometimes it is the only discovery method available),
but it must be used with appropriate caution.
At each step in the discovery process, you start from
some statement A, and you create a related statement B;
it may be the case that the implication A implies B is obvious.
But you have to think about whether B implies A.
If you can find a convincing demonstration that B implies A,
then you can proceed. If you can't find a demonstration
of B implies A, then you might as well discard statement B,
because it is of no use at all to you; look for some other statement
to use instead.
Beginners often make mistakes when they use "working backward,"
because they don't notice that some step is irreversible.
For instance, the statement
x > ![[image: square root of (x^2-1)]](http://www.math.vanderbilt.edu/~schectex/commerrs//sqrtmin1.gif)
is not true for all real numbers x. But
if we didn't know that, we might come up with this
proof:
Start with what we want to prove:
x > .
Since means
the nonnegative square root of
x2–1, we know that it is
nonnegative. Since x is even larger, it
is also nonnegative.
Squaring both sides of an inequality is
a reversible step if both sides are nonnegative.
Thus we obtain
x2 > x2–1.
Subtract x2 from both sides;
0 > –1. And that's clearly true, regardless of
what x is.
"Therefore
x >
for all real numbers x."
But that conclusion is wrong. The right
side of the inequality is undefined when
x = 0.5. And when x = –2, then both sides
of the inequality are defined, but the
inequality is false. See if you can find
where the reasoning went awry.
Well then, if reasoning backward is not acceptable
as a presentation of a proof, what is acceptable?
A direct proof is acceptable. A theorem has
certain hypotheses (assumptions) and certain conclusions.
In a direct
proof, you start with the hypotheses, and you generate
consequences -- i.e., you start making sentences, where
each sentence is either a hypothesis of the theorem,
an axiom (if you're using an abstract theory), or
a result deduced from some earlier theorem using
sentences you've already generated in the proof.
They must be in order -- i.e., if one sentence A
is used to deduce another sentence B, then sentence
A should appear before sentence B. The goal is to eventually
generate, among the consequences, the conclusion of
the desired theoreom.
Some variants on
this are possible, but only if the explanatory language
is used very carefully; such variants are not recommended
for beginners. The variants involve phrases like
"it suffices to show that...".
These phrases are like foreshadowing in a story, or
like direction signs on a highway.
They intentionally appear out of chronological order,
to make the intended route more understandable.
But in some sense they are
not really part of the official proof; they are just
commentaries on the side, to make the official proof
easier to understand. When you pass a sign that says
"100 miles to Nashville," you're not actually in Nashville
yet.
Perhaps the biggest failure in the proofs of beginners
is a severe lack of words. A beginner will write down
an equation that should be accompanied by either
the phrase "we have now shown" or the phrase
"we intend to show", to clarify just where we are
in the proof. But the beginner writes neither phrase,
and the
reader is expected to guess which it is. This is like
a novel in which there are many flashbacks and
also much foreshadowing,
but all the verbs are in present tense; the reader
must try to figure out a logical order
in which the events actually occur.
One easy method that I have begun recommending to
students is this: Put a questionmark over any relationship
(equals sign, greater than sign, etc.)
that represents an assertion that you want to
prove, but have not yet proved. An equals-sign
without a questionmark will then be understood to
represent an equation that you have already proved.
Later you can put a checkmark
next to the equations whose doubt has been removed.
This method may help the student writing the work, but
unfortunately it does not greatly help the teacher or
grader who is reading the work -- the order of steps is
still obscured.
Another common style of proof is the indirect proof,
also known as proof by contradiction.
In this proof, we start with the hypotheses of the desired
theorem; but we may also add, as additional hypothesis,
the statement that "the desired theorem's conclusion is
false." In other words, we really want to prove
A Þ B, so we start by
assuming both A and ~B (where ~ means "not").
We then start generating consequences,
and we try to generate a contradiction among
our consequences. When we do so, this establishes that
A Þ B
must have been true after all.
This kind of proof is harder to read, but it is actually
easier to discover and to write: we have more hypotheses
(not only A, but also ~B),
so it is easier to generate consequences. I recommend
that beginners avoid indirect proofs as long as possible;
but if you continue with your math education, you will
eventually run into some abstract theorems in higher
math that can only be proved by indirect proof.
Difficulties with quantifiers.
Quantifiers are the phrases "there exists" and "for every."
Many students -- even beginning graduate students in mathematics!
-- have little or no understanding of the use of quantifiers. For
instance, which of these statements is true and which is false,
using the standard real number system?
For each positive number a there exists a positive number b
such that b is less than a.
There exists a positive number b such that for each positive
number a we have b less than a.
Difficulty with quantifiers may be common, but
I'm not sure what causes the difficulty. Perhaps it
is just that mathematical sentences are grammatically
more complicated than nonmathematical ones. For instance,
a real-valued function f defined on the real line
is continuous if
for each point p and
for each number epsilon greater than zero, there
exists a number delta greater than zero such that,
for each point q, if the distance from p to q is less
than delta, then the distance from f(p) to f(q) is
less than epsilon.
This sentence involves several nested clauses,
based on several quantifiers:
for each point p ...
for each number epsilon ...
there exists delta such that ...
for each point q ...
Nonmathematical grammar generally doesn't involve
so many nested clauses and such crucial attention
to the order of the words.
I think that many students would benefit from thinking
of quantifiers as indicators of a competition between
two adversaries, as in a court of law. For instance,
when I assert that the function f is continuous,
I am asserting that
no matter what point p and what positive number epsilon you specify,
I can then specify a corresponding positive number delta, such that,
no matter what point q you then specify, if you demonstrate that
your q has distance from your p less than my delta, then I can
demonstrate that the resulting f(p) and f(q) are separated
by a distance less than your epsilon.
Of course, it must be understood that the two adversaries
in mathematics are emotionally and morally neutral. In
a court of law (at least, as depicted on television), it is
often the case that
one side is the "good guys" and the other side is the
"bad guys," but in principle the law is supposed to be a neutral
way of seeking the truth; mathematical reasoning is too.
Some students may have an easier time avoiding errors with quantifiers
if they actually use symbols instead of words. This may make the
differences in the quantifiers more visually prominent. The symbols
to use are
![[image: for all]](http://www.math.vanderbilt.edu/~schectex/commerrs//forall.gif)
universal quantifier
"for all" (or "for each")
![[image: there exists]](http://www.math.vanderbilt.edu/~schectex/commerrs//exists.gif)
existential quantifier
"there exists" (or "there exists at least one")
With those symbols, my earlier two statements about
real numbers can be written, respectively, as
(a > 0)
(b > 0)
(b < a).
(b > 0)
(a > 0)
(b < a).
And the definition of continuity of a real-valued function f defined
on the real line can be restated as
(p)
(e>0)
(d>0)
(q)
(if |p–q|<d,
then |f(p)–f(q)|<e).
Now
you can see the four nested quantifiers very clearly; this may
explain why the definition is so complicated -- and perhaps it will
help to clarify what the definition means.
Some readers have requested that I add a few words about negations of quantifiers.
The basic rules are these:
~=~ and
~=~, where
~ means "not". That is, you can move a negation past a quantifier,
if you just switch which type of quantifier you're using.
An example of ~=~:
Saying "not every peanut in this jar is stale"
is the same thing as saying "at least one peanut in this jar is not stale."
An example of ~=~:
Saying "there does not exist a stale
peanut in this jar"
is the same thing as saying "every peanut in this jar is non-stale."
Here is a more complicated example: Following are a few different
ways to say that
"f is not continuous". Start with the formula that I gave above, but with a
"not" in front of it. Gradually move the "not" to the right, switching
each quantifier that it passes. So all these statements are equivalent:
~ (p)
(e>0)
(d>0)
(q)
(if |p–q|<d,
then |f(p)–f(q)|<e).
(p)
~
(e>0)
(d>0)
(q)
(if |p–q|<d,
then |f(p)–f(q)|<e).
(p)
(e>0)
~
(d>0)
(q)
(if |p–q|<d,
then |f(p)–f(q)|<e).
(p)
(e>0)
(d>0)
~
(q)
(if |p–q|<d,
then |f(p)–f(q)|<e).
(p)
(e>0)
(d>0)
(q)
~
(if |p–q|<d,
then |f(p)–f(q)|<e).
(p)
(e>0)
(d>0)
(q)
(|p–q|<d but
|f(p)–f(q)|³e).
Erroneous method justified by one or two instances of correct
results.
Sometimes an erroneous method can lead (just by coincidence) to a
correct result. But that does not justify the method.
Sample problem: Simplify 16/64.
Erroneous method: Cancel the 6's.
Correct answer: 1/4.
Below is another computation like that. Can you find all of its errors? Some student
actually turned this in on an exam, and expected partial credit because he had
the right answer. (Thanks to Sean Raleigh for bringing this one to my attention; the solution was graded by Boern Lamel.)
 order=0>
Unquestioning faith in calculators. Many students believe
that their calculators are always right. But that is not true.
All calculators have limitations, and will give incorrect answers
under some circumstances (as will math teachers and math books).
Probably the most common error with calculators is simply forgetting
to switch between degrees and radians (or not understanding the need to switch).
Degrees are often used in engineering and science classes, but radians
are almost always used in calculus and higher math classes. That's because
most of the formulas involving trigonometric functions come out much simpler
with radians than with degrees -- the formulas for the derivatives, for the
power series expansions, etc.
Here is another widely occurring calculator error.
Some graphing calculators, if asked to display a graph of x^(1/3) or
x^(2/3), will only display the right half of the graph -- i.e., there
will be no points plotted in the left half-plane. But the function x^(1/3)
is odd, and the function x^(2/3) is even; both functions (if graphed
correctly) have points in both the right and left half-planes.
To
get a correct graph, you need to look in the calculator's function
menus until you find a special "button" for cube roots. Use that
to get x^(1/3); use the square of the cube root to get x^(2/3).
Why is that? Well, first you need to understand that for
some constants k, the correct graph of x^k is blank in the
left half-plane, because the function x^k is actually undefined
for x < 0. For instance, k = 1/2 is one such constant.
The numbers
1/3 and 2/3 are not such constants, but
if you simply punch in the formula x^(1/3) or
x^(2/3) using the caret symbol (^) for exponentiation, the calculator
must replace the fractions 1/3 or 2/3 with some sort of approximations
such as k=0.3333 or k=0.6667.
Those approximations turn out to have
the same property I just mentioned for k=0.5 -- the resulting
function is undefined for x less than 0. You avoid this approximation
error by using the cube root button.
Dave Rusin has put together
some
notes on the wide variety of
errors one can make by not understanding one's calculator.
By the way, I'll take this opportunity to mention that Dave Rusin has
put together a super website,
Mathematical
Atlas: A gateway to Mathematics, which offers definitions,
introductions, and links to all sorts of topics in math.
Unwarranted Generalizations
A formula or notation may work properly in one context, but
some students try to apply it in a wider context, where it may
not work properly at all. Robin Chapman
also calls this type of error "crass formalism." Here
is one example that he has mentioned:
Every positive number has two square roots: one
positive, the other negative. The notation Öb
generally
is only used when b is a nonnegative real number; it means
"the nonnegative square root of b," and not just "the
square root of b."
The notation Öb probably should not
be used at all in the context of complex numbers.
Every nonzero complex number b has
two square roots, but in general there is no natural
way to say which one should be associated with the
expression
Öb.
The formula
![]() order="0"
src="http://www.math.vanderbilt.edu/~schectex/commerrs//crass2.gif"
alt="[image: square root of (ab) equals
(square root of a) times (square root of b)]">
is correct when a and b are
positive real numbers, but it leads to errors when
generalized indiscriminately to other kinds of
numbers. Beginners in the use
of complex numbers are prone to errors such as
![]() order="0" src="http://www.math.vanderbilt.edu/~schectex/commerrs//crass3.gif"
alt="[image: 1 equals (square root of 1) equals (square root of (-1)(-1))
equals (square root of -1)(square root of -1) equals i times i equals -1.]">.
In fact, the great
mathematician Leonhard Euler published a computation
similar to this in a book in 1770, when the theory of
complex numbers was still young.
Here is another example, from my own teaching experience:
What is the derivative of xx? If you ask this during
the first week of calculus, a correct answer is "we haven't covered
that yet." But many students will very confidently tell you that the
answer is x • xx–1.
Some of them may even simplify
that expression -- it reduces to xx -- and a few students
will even remark: "Say, that's interesting -- xx
is its own derivative!" Of course, all these students are wrong. The
correct answer, covered after about a semester of calculus, is
![[image: derivative with respect to x]](http://www.math.vanderbilt.edu/~schectex/commerrs//xtox1.gif) (xx) = xx (1 + ln x).
The difficulty is that, in high school or shortly after they arrive
at college, the students have learned that
(xk) = kxk–1
That formula is actually WRONG, but in a very subtle way. The
correct formula is
(xk) = kxk–1
(for all x where the right side is defined),
if k is any constant.
The equation is unchanged, but it's now accompanied by some
words telling us when the equation is applicable.
I've thrown in the parenthetical "for all x where the right side is
defined," in order to avoid discussing the complications that
arise when x £ 0. But
the part that I really want to discuss here is the other part --
i.e., the phrase "if k is any constant."
To most teachers, that additional phrase
doesn't seem important, because in
the teacher's mind "x" usually means a variable
and "k" usually means a constant. The letters
x and k are used in different ways here, a little like the
difference between bound and free variables in logic: Fix
any constant k; then the equation states a relationship between
two functions of the variable x. So the language suggests to
us that x is probably not supposed to equal k.
But the math teacher is already fluent in this language, whereas mathematics
is a foreign language to most students. To most students, the distinction
between the two boxed formulas is one which doesn't seem important at first,
because the only examples shown to the student at first are those in which
k actually is a constant. Why bother to mention that k
must be a constant, when there are no other conceivable meanings for k?
So the student memorizes the first (incorrect) formula, rather than the
second (correct) formula.
Every mathematical formula should be accompanied by a few words of English
(or your natural language, whatever it is). The words in English tell when
the formula can or can't be applied. But frequently we neglect the words,
because they seem to be clear from the context. When the context changes,
the words that we've omitted may become crucial.
Students have difficulty with this. Here is an experiment that I have
tried a few times: At the beginning of the semester, I tell the students
that the correct answer to
(xx)
is not xx, but rather
xx (1 + ln x),
and I tell them that this problem will be on their final exam at the end
of the semester. I repeat these statements once or twice during the
semester, and I repeat them again at the very end of the semester, just before
classes end. Nevertheless, a large percentage (sometimes a third) of my
students still get the problem wrong on the final exam! Their original,
incorrect learning persists despite my efforts.
I have a couple of theories about why this happens: (i) For most students,
mathematics is a foreign language, and the student focuses his or her attention
on the part which seems most foreign -- i.e., the formulas. The words
have the appearance of something familiar ("oh, that's just English, and
I already know English"), and so the student doesn't pay a lot of attention
to the words. (ii) Undergraduate students tend
to focus on mechanical computations; they are not yet mathematically mature
enough to be able to think easily about theoretical and abstract ideas.
A sort of footnote: Here is a common error among readers of
this web page. Several people have written to me to
ask, shouldn't that formula say
"if k is any constant except 0", or
"if k is any constant except –1", or something like that?
They think some special note needs to
be made about the logarithm case. Actually, my formula is
correct as it stands -- i.e., for every constant real number
k -- but if you want to tell the whole story, you'd
have to append some additional formula(s). When
k=0, my formula just
says the derivative of 1 is 0x–1; that's
true but not very enlightening. My formula doesn't mention,
but also doesn't contradict, the fact that the derivative of
ln(x) is
1x–1. You can always say more
about any subject, but I just wanted to contrast the formulas
xk and xx as simply as possible.
... And of course, for simplicity's sake,
I haven't mentioned the complications you run
into when x is zero or
a negative number; I'm only considering
those values of x for which
xk and xx are easy to define.
Other Common Calculus Errors
Jumping to conclusions about infinity. Some problems involving
infinity can be solved using "the elementary arithmetic of infinity".
Some students jump to the conclusion that all problems involving
infinity can be solved by this sort of "elementary arithmetic,"
and so they guess all sorts of incorrect answers (mainly 0 or infinity) to such
problems.
Here is an example of the "elementary arithmetic": If we use the
equation cautiously, we can say (informally) that
1/¥ = 0 -- though perhaps it would
be less misleading to write instead
1/¥ ® 0.
(My thanks to
Hans Aberg for
this suggestion and for several other suggestions on this
web page.)
What this rule really means is that if you take a medium-sized number
and divide it by an enormous number, you get a number very close to 0.
For instance, without doing any real work, we can use this rule to conclude at
a glance that
![[image: the limit, as x goes to infinity,
of 3 over x-squared, is 0]](http://www.math.vanderbilt.edu/~schectex/commerrs//infty2.gif)
Thus, the problem 1/¥
has the answer 0. The problem
¥ – ¥
does not have an answer in any analogous fashion; we might say that
¥ – ¥
is undefined. This does not mean that "Undefined" is the
answer to any problem of the form
¥ – ¥.
What it means, rather, is that each
problem involving ¥ – ¥
requires a separate analysis; different problems of this type
have different answers. For instance,
![[image: as x goes to infinity, we have
<b>these</b> limits: x^3-x goes to infinity;
(x^2+(1/x))-x^2 goes to 0;
the square root of (x^2+x) minus x
goes to 1/2.]](http://www.math.vanderbilt.edu/~schectex/commerrs//infty4.gif)
Those first two problems are fairly obvious; the last problem takes more
sophisticated analysis. Just guessing would not get you an answer of 1/2.
(If you don't understand what is going on in the last problem, try graphing
the functions
![[image: square root of (x^2_x)]](http://www.math.vanderbilt.edu/~schectex/commerrs//infty6.gif) and x
on one display screen on your graphing calculator. That
may provide a lot of insight, though it's not a proof.)
In a similar fashion, ![[image: infinity over infinity and 0 over 0]](http://www.math.vanderbilt.edu/~schectex/commerrs//infty5.gif)
do not have quick and easy answers; they too require more specialized and
sophisticated analyses.
Here is a common error mentioned by
Stuart Price:
Some students
seem to think that limn®„ (1+(1/n))n = 1.
Their reasoning is this:
"When n®„, then 1+(1/n) ® 1. Now compute limn®„ 1n = 1."
Of course, this reasoning is just a bit too simplistic.
You have to deal with both of the n's in the expression
(1+(1/n))n at the same time -- i.e., they
both go to infinity simultaneously; you can't figure that one
goes to infinity and then the other goes to infinity.
And in fact, if you let the other one go to infinity first, you'd get a
different answer:
limn®„ (1+0.0000001)n = „. So evidently the answer lies
somewhere
between 1 and „. That doesn't tell us much; my point
here is that easy methods do not work on this problem.
The correct answer is a number that is near 2.718. (It's an important
constant, known to mathematicians as "e".)
There's no way you
could get that by an easy method.
That reminds me of a related
question that seems to bother many students:
What is 00 ?
The answer is that, although it sometimes convenient to
give 00 some temporary definition in order to
simplify the notation in working on some particular problem,
those temporary definitions vary from problem to problem.
There is no one definition that always works well for
00.
To see why, think about these things:
When x is a positive number, then x0 = 1.
When y is a positive number, then 0y = 0.
If you want to investigate this question further, here
is a little project that may be instructive: Make a graph
or table that shows all the values of
xy, when x and y both run through all the values in
{ 0.4, 0.2, 0.1, 0.05, 0.025, 0.0125}.
Here is a related question: What is
limx ®
0+ xx ?
Hint: You can find it by writing
xx = ex ln x
and evaluating
limx ®
0+ x ln x using l'Hopital's Rule.
But even if you do find
limx ®
0+ xx,
its value is not a good definition of
00. A definition of that quantity should be
equal to
limx,y ®
0+ xy -- that is, the
limit of xy as x and y
both approach
0 along all possible paths in the x-y plane.
But that limit doesn't exist -- we get different
limits by approaching along different paths.
Problems with series.
Sean Raleigh reports that the most common series error he has seen is
this: If a1, a2, a3, ...
is a sequence converging to 0, then many students conclude (erroneously) that
the series a1 + a2 + a3 + ...
must be convergent (i.e., must add up to a finite number). Perhaps they hold that
belief because it is true for most of the examples that they have seen.
Most counterexamples are too advanced to be included in an elementary textbook.
Of course, every calculus book gives the simple example of the harmonic series:
1 + (1/2) + (1/3) + (1/4) + ...
= ¥
but one single example of divergence does not seem to outweigh in the students' minds
the many examples of convergence that they have seen.
Loss or misuse of constants of integration. The
indefinite integral
of a function involves an "arbitrary constant", and this causes
confusion
for many students, because the notation doesn't convey the
concept very well.
An expression such as "3x2+5x+C" really is supposed to
represent
an infinite collection of functions -- it represents all of the
functions
3x2+5x+7,
3x2+5x+19,
3x2+5x–3.19,
etc.
plus more functions of the same sort. One of the difficulties,
also, is that
the same letter "C" is customarily used for all
such arbitrary constants; but
one computation may involve several different arbitrary
constants. It
would be more accurate to put subscripts on the C's, to
differentiate one
of them from another -- i.e., write C1,
C2, C3, etc. -- and I often do
that in my lectures.
Here is an example. The formula for Integration By Parts, in
its briefest form, is
òudv = uv - òvdu;
that can be understood more easily as
òu(x)v'(x)dx = u(x)v(x) -
òu'(x)v(x)dx.
Now, that formula is correct, but it can easily be mishandled and
can lead to errors. Here is
one particularly amusing error:
Plug u(x)=1/x and v(x)=x into the
formula above. We get
ò(1/x)(1)dx = (1/x)(x) -
ò(-1/x2)(x)dx
which simplifies to
ò (1/x)dx =
1 + ò (1/x)dx.
Now, regardless of what you think is
the value of ò (1/x)dx,
you just have to subtract that amount from both sides of the
preceding equation, to obtain
0=1. Wait, how can that be????
Well, if we're very
careful, we realize that
the two ò(1/x)dx's on the two sides
of the last equation
are not actually the same. What that last equation
really says is
[ln|x| + C1] = 1 +
[ln|x| + C2].
That is a true equation, if we choose the constants C1
and C2 appropriately --
i.e., if we choose them so that
C1–C2=1.
Thus, the two constants are not independent of each other --
they are not completely "arbitrary". Perhaps a more accurate
explanation is this: The two expressions
[ln|x| + C1] and 1+[ln|x| + C2]
do not actually represent individual functions; rather, each
of those expressions represents a set of functions.
The expression [ln|x| + C1] represents the set of all the
functions of x that can be obtained by starting with
the function ln|x| and then adding a constant.
The expression 1 + [ln|x| + C2] represents the set of all the
functions of x that can be obtained by starting with
the function ln|x|, then adding a constant, and
then adding 1.
Those two descriptions may sound different, but if you
think about it, you'll see that those descriptions
are nevertheless specifying the same set. My thanks
to Antonio Ferraioli ("Ferra") for this 0=1 paradox
and its explanation.
Some students manage to make this kind of error even with
definite integrals.
They start from the formula
ò (1/x)dx =
1 + ò (1/x)dx,
which is correct; but then when they "switch to definite
integrals",
they get the formula
òab (1/x)dx
=
1 + òab (1/x)dx,
which is not correct. If you really want to "switch to
definite integrals",
you need to think of that constant 1 as a special sort of
function. When you switch to definite integrals,
any function p(x) gets replaced by p(b)–p(a). In particular,
the constant function 1 is the function given by p(x)=1 for all x.
So p(b)–p(a) becomes
1–1, or 0.
Some students may understand this better if we do the whole thing
with definite
integrals, right from the start.
Let's use the formula
òab u(x)v¢(x)dx
= u(b)v(b) – u(a)v(a) – òab u¢(x)v(x)dx.
Note that this formula has one more term than my previous boxed
formula -- when we convert u(x)v(x) to the definite integral
version, we replace it with u(b)v(b)–u(a)v(a).
Now plug in u(x)=1/x and v(x)=x. We get
òab (1/x)(1)dx
= (1/b)(b) – (1/a)(a) – òab (–1/x2)(x)dx
which (assuming 0 is not in the interval [a,b]) simplifies to
[ln|b|–ln|a|]
= 1 – 1 –
[–ln|b|+ln|a|]
which is true -- i.e., there is no contradiction here.
Some students may be puzzled by the differences between
the two versions of the Integration by Parts formula
(in boxes, in the last few paragraphs). I will describe
in a little more detail how you get from the definite
integral formula (in the last box)
to the indefinite integral formula (in the first box in this section).
Think of a as a constant and b as a
variable, and you'll get
something like this:
[ò u(x)v¢(x)dx +
C1] = [u(x)v(x)
– C2] –
[ò u¢(x)v(x)dx +
C3].
Note that the u(b)v(b) term gets replaced by u(x)v(x), and the
u(a)v(a) term "disappears" because it is constant.
Finally, we can "absorb" the arbitrary constants into the
indefinite integrals -- i.e., we don't need to write
C1, C2, C3, because
any indefinite integral is only determined up to adding
or subtracting a constant anyway. Thus, we
arrive at the briefer formula
ò u(x)v¢(x)dx =
u(x)v(x) – ò u¢(x)v(x)dx.
Handling constants of integration gets even more complicated
in the first course on differential equations, and there are
even more kinds of errors possible. I won't try to list all
of them here, but here is the simplest and most common error
that I've seen: In calculus, some students get the idea
that you can just omit the "+C" in your intermediate
computations, and then tack it on at the end of your answer,
if you know which kinds of problems require an arbitrary
constant. That will usually work in calculus, but
it doesn't work in differential equations, because in
differential equations the "C" can show up anywhere --
not necessarily as a "+C" at the end of the answer.
Here's a simple example: Let's solve the differential
equation xy¢+7=y (where y¢ means dy/dx). One way to solve
it is by the following steps:
Rewrite the problem as
y¢ – (1/x)y = – 7/x, to
show that it is linear.
The integrating factor is then exp[ò (–1/x)dx] = 1/x.
Multiply both sides of the differential equation by the
integrating factor, to obtain an exact differential
equation:
(y/x)¢ = (1/x)y¢ – (1/x2)y = –7/x2.
Integrate both sides. Thus
y/x = (7/x) + C.
Solve for y. Thus y = 7 + Cx.
That's the correct answer. But if we had taken the attitude
"don't bother with C, just tack it on when you're done,"
instead
of the last two steps we'd have written:
"Integrate both sides. Thus
y/x = 7/x."
"Solve for y. Thus y = 7. "
"Tack on the "+C". Thus y = 7 + C. "
That's wrong, whether you simplify it or not.
Loss of differentials.
This shows up both in differentiation
and in integration. The "loss of differentials" is much like
the "loss of invisible parentheses" discussed earlier in this
document; it is a type of sloppy writing in intermediate steps which leads
to actual errors in the final answer.
When students first begin to learn to differentiate, they are always
differentiating with respect to the same variable, and so they see no reason
to mention that variable. Thus, in differentiating the
function y = f(x) = 7x3+5x,
they may correctly write
![[image: several correct notations]](http://www.math.vanderbilt.edu/~schectex/commerrs//diffls1.gif)
or they may incorrectly write "dy = 21x2+5."
The omission of the "dx" from this last equation makes
no real difference in the student's mind, and this slovenly omission may become
a habit. But it will cause difficulties later in the course. In fact,
I am starting to think that we could avoid a lot of difficulty if we discourage
beginning calculus students from using the notations f ¢(x)
or Dy. If we require them to use the notation dy/dx , and
penalize them for writing it as dy, we might save them a lot of
headaches later.
The difficulty, of course, shows up when we arrive at the Chain Rule.
Suddenly, the question is no longer "What is the derivative of y",
but rather, "What is the derivative of y with respect to x?
with respect to u? How are those two derivatives related?"
The student who does not make a habit of distinguishing between dy/dx
and dy/du in writing may also have difficulty distinguishing between
them conceptually, and thus will have difficulty understanding the Chain
Rule.
This also leads to difficulties with the "u-substitutions"
rule, which is just the Chain Rule turned into a rule about integrals.
For instance:
![[image: large table containing
several integral problems, common
wrong answers, and correct answers]](http://www.math.vanderbilt.edu/~schectex/commerrs//diffls2.gif)
What causes these errors?
For the first three problems, the student
is attempting to use the formula
ò (1/u)du =
ln |u|+C
(which is a correct formula, but not directly applicable).
However, the student has learned it incorrectly as
" ò (1/u) = ln |u|+C."
Substitute u = 1+x2
or u = x3
or u = cos x into that formula to get the first three
erroneous answers in the table above. The expressions ò (1/u)du
and ò (1/u)dx
have very different meanings, but you're likely to confuse them if you
write them both as ò (1/u).
For the last problem in the table above, the student is
attempting to use the formula
ò u2du = (1/3)u3+C,
which is a correct formula, but not relevant to the present problem.
The
student has probaby memorized that formula in the incorrect form
ò u2 = (1/3)u3+C.
The expressions
ò u2du
and
ò u2dx
have very different meanings, but you're likely to confuse them if you
write them both as
ò u2.
Another correct way to write the rule about logarithms is
![[image: integral of u'(x) over u(x),
dx, is equal to ln|u(x)| +C.]](http://www.math.vanderbilt.edu/~schectex/commerrs//diffls3.gif) .
Since this expresses everything in terms of the variable x, it may
make errors less likely. Admittedly, it is a complicated looking formula,
but it is preferable to a wrong formula. The first, third, and fourth problems
in the preceding table all require more complicated methods; just using
logarithms won't solve the problems for you. The problem of integrating
x –3 actually
requires a less complicated method
-- i.e., without logarithms.
We should prohibit students from writing an integral sign without a
matching differential. Just as any "(" must be matched with a
")", so too any integral sign must be matched with a "dx"
or "du" or "dt" or whatever. The expression
![[image: integral of 1 over u]](http://www.math.vanderbilt.edu/~schectex/commerrs//diffls5.gif) is
unbalanced, and should be prohibited. If we're considering a substitution
of u = 1+x2, then
ò (1/u)du
is very different from ò (1/u)dx,
and so the expression ò (1/u)
is ambiguous and meaningless. If you write
ò (1/u)
in one of your intermediate steps, you may forget whether it represents
ò (1/u)du
or
ò (1/u)dx,
and you may inadvertently switch from one to the other -- thus replacing
one mathematical quantity with another to which it is not equal.
By the way, some students get confused about whether
ò (1/u)du should be
ln|u|+C or ln(u)+C. Here is an answer.
ò (1/u)du is always
equal to ln|u|+C, but sometimes that answer
can be simplified and sometimes it can't.
In math, we generally prefer to
write our answers in simplest form (and we sometimes
insist on it). In those situations where we know
that u will only take positive values
(e.g., when u=1+x2, or when
the domain is restricted so that u can't be negative),
then ò (1/u)du should
be written as ln(u)+C. In those situations
where we don't know whether u will be positive, we
should write the answer as
ln|u|+C. (But sometimes we omit the absolute value sign
out of sheer laziness, justifying this with the excuse
that we can make the domain smaller.)
These loss-of-differentials errors in differentiation
and in integration can be caught easily by a bit of "dimensional
analysis" (discussed earlier). To do that, it is useful to think in
terms of "infinitesimals" -- i.e., numbers that are
"infinitely small" but still not zero.
Newton and Leibniz had infinitesimals in mind when they invented calculus
300 years ago, but they didn't know how to explain infinitesimals rigorously.
Infinitesimals became unfashi | |
