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 ∫ (5x⁴+7)dx 3x⁵+7x+C

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,

–∫ (5x⁴–7)dx –x⁵–7x+C (should be –x⁵+7x+C)

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).

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:

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.

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 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,

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).

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).

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 :

• 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) ≠ (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".

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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

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,

In a similar fashion, do not have quick and easy answers; they too require more specialized and sophisticated analyses.