# Mathematical misconceptions and how to combat them

There are a lot of common misconceptions when it comes to math. A common one that has already been addressed on this site is $1 \neq .999\cdots$, as is that imaginary numbers "do not exist". Another one which I have encountered is that one can prove $2 = 1$ using calculus as follows:

$$2 = \frac{\frac{d}{dx}(x^2)}{x} = \frac{\left(\frac{d}{dx}\left(\sum\limits_{n=1}^x x \right)\right)}{x} = \frac{\left(\sum\limits_{n=1}^x \frac{d}{dx}(x)\right)}{x}= \frac{\left(\sum\limits_{n=1}^x 1\right)}{x} = \frac{x}{x} = 1$$

And that because of this calculus is somehow "wrong". This example can be countered by either noting that $\sum_{n=1}^x x$ isn't even a function on the real numbers, much less differentiable, or noting that since $x$ isn't constant the third step is invalid. My question is, what are some other common mathematical misconceptions that people have, and how can they be combated?

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Here is a similar MO question: mathoverflow.net/questions/23478/… –  PEV Mar 26 '11 at 4:33
Is there such a thing as "community wiki" on this site (as on MO)? If so, I think this should be community wiki. –  Daenerys Naharis Mar 26 '11 at 4:35
@PEV: Thanks for the link! But I'm looking for more elementary misconceptions, i.e. ones for which most people who believe them will have difficulty understanding complicated formal arguments. –  Alex Becker Mar 26 '11 at 4:35
@Joseph: Yes there is. –  PEV Mar 26 '11 at 4:37
Honestly, that calculus thing just a mistake. The expression $\sum_{n=1}^xx$ simply does not make any sense. Such mistakes can be combated by insisting that students make sure that everything they write mean something. They can even extrapolate this advice to essentially all their other activities... –  Mariano Suárez-Alvarez Mar 26 '11 at 4:47

Way too many to count. Of course, one of the most common ones is:

All functions are linear. So, $(a+b)^2 = a^2+b^2$, $\sqrt{a+b} = \sqrt{a}+\sqrt{b}$, $\sin(a+b) = \sin(a)+\sin(b)$, etc.

All things can be sustituted: thus, since $\int\frac{1}{x}\,dx = \ln|x|+C$, then $\int\frac{1}{f(x)} \,dx = \ln|f(x)|+C$

At the bottom, all of these common misconceptions arise from not understanding that the symbols are supposed to have a meaning, and that the manipulations are not simply mindless rules. The first misconception comes from not understanding what the decimal expansion of a number means (it describes the coefficients of a series, and it represents the number that is the limit of the partial sums). The second comes form mindless manipulation (as does the "every function is linear" problem).

Others arise because the students are trying to memorize without understanding, and there's too much to memorize, often very similar to one another e.g., (you'll notice a theme, but that's because right now I'm teaching series, so these are fresh):

• The Divergence Test tells you that a series converges if the terms go to zero, and diverges if they don't.

• The Integral Test gives you the value of the series.

• In the limit comparison test, you have to see if the limit is greater than $1$ or smaller than $1$.

I'm not sure what is "the best way to combat them". After so many years, the Freshman Dream is alive and well, as are misconceptions about the nature of decimal expansions.

(I would say that your second example is not a "common misconception", but rather a fallacy that many people have a hard time figuring out and spotting; it's not like people actually think that $1=2$, whereas they do actually think that $1$ and $0.999999\ldots$ are different, or that you can talk about "an infinite number of $9$s, and then a $0$" in decimal expansions).

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@Arturo: I've talked to people who think that the $1 = 2$ example shows not that $1 =2$, but that calculus is "wrong" or "doesn't actually work". And on the subject of your second example, I think many such misconceptions arise from people not understanding one-way implications. –  Alex Becker Mar 26 '11 at 4:40
@Arturo: Editing to clarify that. –  Alex Becker Mar 26 '11 at 4:44
Arturo: that is a very good site you link to at the end of your answer, and I directed my calculus students this semester to look at it. But I do wonder how seriously the students paid attention to it. In my first lecture I emphasized that almost nothing except multiplication by a number is additive: (a+b)^2 is not..., 1/(a+b) is not..., sqrt(a+b) is not..., but 3*(a+b) is.... I can't tell if they really catch on. –  KCd Mar 26 '11 at 7:01
I've had students try to "divide by" the literal string of characters "sin"(or cos). So if you have sin(2x)= something, just divide by sin (and later, by 2...?). Yeah, not pretty! –  The Chaz 2.0 Mar 26 '11 at 8:07
@KCd: A friend of mine compared it to speaking a foreign language. We know the language, and cringe. The students say "I am a train station" (instead of 'How do I get to the train station?') and we explain carefully to them that they are not, in fact a train station. But the problem is they don't realize that they are actually saying something, they see it as shuffling symbols. so long as they keep saying the words without realizing that there is supposed to be meaning behind them, there's not much that can be done to correct them. –  Arturo Magidin Mar 26 '11 at 18:52

For an $m \times n$ matrix $$\sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij} = \sum_{j=1}^{n} \sum_{i=1}^{m} a_{ij}$$

Thus $$\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} a_{ij} = \sum_{j=1}^{\infty} \sum_{i=1}^{\infty} a_{ij}$$

Or

$$\int \int f(x,y) \ dx \ dy = \int \int f(x,y) \ dy \ dx$$ always.

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Yea, Fubini... not a single math student nor my TA could explain to me when exactly this can be applied. The answer was: "Constructing a counter-example takes 3 pages, so you are almost surely save" (pun indended, that was a stochastics class) –  Raphael Mar 26 '11 at 10:25
Three pages? $\text{sign}(y-x)$ for $x,y >0$ integrates to $+\infty$ one way and $-\infty$ the other. $a_{ii} = 1$, $a_{i (i+1)} = -1$, $a_{ij}=0$ otherwise sums to $1$ one way and $0$ the other. –  Douglas Zare Mar 26 '11 at 13:35
@Doug: I think I have font large enough to fit that on three pages :) –  The Chaz 2.0 Apr 29 '11 at 3:16

$\text{Prob}(A \cap B) = \text{Prob}(A) \text{Prob}(B)$

which only holds when $A$ and $B$ are independent. For example, if you roll a fair die, let $A$ be the event that you roll a number up to $3$, and let $B$ be the event that you roll an even number. $\text{Prob}(A \cap B) = 1/6$, while $\text{Prob}(A) \text{Prob}(B)=1/4$.

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