# 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?

• Here is a similar MO question: mathoverflow.net/questions/23478/… – PrimeNumber 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. – Chris Brooks 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. – PrimeNumber 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-Álvarez 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).

• @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.

• 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 $\text{Prob}(A) \text{Prob}(B)=1/4$.

I know of two misconceptions. One is that irrational numbers don't exist. The other is that 0.999... $$\neq$$ 1. I think the first one comes from being taught how to multiply and divide fractions. I think that some people since they learned how to do arithmetic on them, they thought the rational numbers were the only numbers. The other misconceptions probably comes from thinking of the real numbers as being constructed from the decimal notations. It turns out that the set of all real numbers with +, $$\times$$ and $$\leq$$ has been constructed and been proven to be a complete ordered field which is unique up to isomprphism. Neither the rational number system nor the system constructed from the decimal notations has that structure.

We can define a natural number as a finite ordinal number then construct the integers then construct the dyadic rationals, the numbers whose binary notation is terminating. Finally, take any subset of that set that's nonempty and its complement is also nonempty and for any member of the subset, all smaller members of the set are in the subset. Now, if the set has no maximal element and its complement has no minimal element, we can invent a number that's larger than all members of the set and smaller than all members of the complement. Now there's exactly one intuitive way to define +, $$\times$$, and $$\leq$$ on that set and when you do so, the result is a complete ordered field which is unique up to isomorphism. It can be shown that not all members of it can be gotten by dividing an integer by a nonzero integer.

That only says how to construct the real numbers but still doesn't say how each of them should be written. Since it's a complete ordered field. We can divide any integer by 10 to get a multiple of 0.1 and can again divide any of those by 10 to get a multiple of 0.01. Take the set of all numbers that can be gotten by dividing an integer by 10 as many times as you want. Since the set of all real numbers is a complete ordered field, if you take any subset of the set of all numbers that can be gotten by taking an integer and dividing it by 10 any number of times such that that set and its complement are nonempty and for each member of the subset, all smaller members of the set are in the subset, if the set has no maximal element and its complement has no minimal element, there is exactly one real number between them and otherwise, there is no real number between them. This gives an intuitive way of defining a notation for each decimal number but it ends up defining only one notation for 1 and forbids a notation of trailing 9's.

The notation can also be defined to have the meaning of an infinite sum. That's not constructing the number from a Cauchy sequence. It's using an easy to prove theorem that in the already constructed field, all Cauchy sequences of real numbers approach a real number and defining the notation to refer to the limit of the sum as you take more terms. It can be shown that according to this meaning, it's still true that all real numbers have a notation and they have two notations if and only if they have a terminating notation and one of their notations is exactly the same as their notation using the other definition of their notation.