Examples of bad notation and its consequences

Question

An example of bad mathematical notation that comes in my mind and has caused complications throughout history is the notation for imaginary numbers. The original notation used to represent imaginary numbers was "$\sqrt{-1}$", where the square root symbol was used to indicate the square root of a negative number. However, this notation can be confusing and misleading, as the square roots of negative numbers cannot be represented as real numbers.

The Swiss mathematician Leonhard Euler, in the middle of the 18th century, proposed a new notation to represent imaginary numbers. He introduced the letter "$i$" to represent the square root of $-1$ (i.e. $i^2 = -1$). This notation simplified and clarified the representation of imaginary numbers, allowing a better understanding and manipulation of them in calculations and equations.

The original notation, using the square root symbol for imaginary numbers, led to misunderstanding and confusion in mathematics. For example, people might think that $\sqrt{-1}\cdot\sqrt{-1} = \sqrt{-1\cdot{-1}} = \sqrt1$, which is clearly incorrect when working with imaginary numbers. The introduction of the "$i$" notation for imaginary numbers helped solve these problems and allowed further development in the field of complex mathematics.

I am looking for other examples of bad notation and its bad consequences.

naturally bad and good in in the eye of the beholder but iam eagera to hear friends of this community about the topic an their examples and opinions and references where the subject is studied scholarly.

Answer 1

In my opinion, bad notations are often a confusing notations, having different possible meanings.

For example, the use of parenthesis for too many things can be confusing:

• argument of functions,
• priority in operations,
• open intervals (I do prefer the french notation $]a,b[)$
• couples, triples,...
• row matrices,
• cycles (permutations).

I have been very disturbed when teaching on polynomials. For example $Q(X-\alpha)$ usually stands for the composition $Q \circ (X-\alpha)$, whereas $(X-\alpha)Q$ is used for the product. Or when working on linear maps on $\mathbb{R}^n$, because we use the same notation when we multiply a vector $(x_1,\ldots,x_n)$ by a real number or when we apply a map to this vector.

Using superscripts that can be misunderstood as exponents is also problematic. In my opinion, the two-dimensional Euclidean unit sphere should be denoted by $\mathbb{S}_2$ and not $\mathbb{S}^2$.

In differential calculus, when I was a student, I was puzzled the first time the teacher computed the second differential of $g \circ f$ by differentiating $D(g \circ f) = (Dg \circ f) \circ Df$. When trying to understand the computation, I finally realized that the symbol $\circ$ used twice in the right-hand side does not have the same meaning at the two occurrences. The correct statement is $D(g \circ f)(x) = Dg(f(x)) \circ Df(x)$ for all $x$.

I now give two notations that have a lot of advantages, but cause occasionally troubles.

In probability theory, the intuitive notation $f(X)$ for $f \circ X$ may induce in error for example in the formula to compute the conditional expectation of a bounded function of two independent random variables $X$ and $Y$, namely $E[f(X,Y)|X] = g(X)$ where $g(x):=E[f(x,Y)]$. It is very tempting but false to write $g(X)=E[f(X,Y)]$.

Polynomials of matrices also lead to confusions, like the false proof of Cayley Hamilton theorem by replacing $X$ by $A$ in the equality $\chi_A(X) = \det(XI-A)$. Distinguishing the $0$ in $K[X]$ from the $0$ in $\mathcal{M}_n(K)$ is a simple way to convince students that this proof is false.

Last, I hate the notations like $$f(x) = g(x), \quad x \in \mathbb{R},$$ which may mean $f(x) = g(x)$ for some $x \in \mathbb{R}$, or $f(x) = g(x)$ for every $x \in \mathbb{R}$, depending on the circonstances.

A good way to prevent a lot of problems is to keep in mind:

• the definitions,
• in which sets the objects live,
• for which elements equalities or properties hold.

Answer 2

Suppose that $A$ is an oracle; then it is standard to write $\mathsf{P}^A$ for the complexity class $\mathsf{P}$ relativized to $A$. As I have mentioned elsewhere on MO, this is incredibly confusing notation. It can lead to the following spurious argument that has confused generations of students. Assume that $\mathsf{P}=\mathsf{NP}$. Then for all oracles $A$, $\mathsf{P}^A=\mathsf{NP}^A$. But by Baker–Gill–Solovay, we know that there exists an oracle $A$ such that $\mathsf{P}^A\ne \mathsf{NP}^A$. This is a contradiction. Hence $\mathsf{P}\ne \mathsf{NP}$.

Answer 3

Elements of groups (or monoids, semi-groups, non-commutative algebras) are composed from left to right, functions, maps, functors from right to left. Things get very confusing when working for example with actions of the symmetric group (it is even worse when using the cycle notation). I think the Polish notation (arguments first) is a brillant idea but it is probably too late now for imposing it.

Answer 4

1) Ornstein–Uhlenbeck process

Consider the SDE for the Ornstein–Uhlenbeck process:

$$X_t:=X_0+\int_{h=0}^{h=t}\theta(\mu- X_h)dh + \int_{h=0}^{h=t}\sigma dW_h.$$

Solution can be found by applying Itô's lemma to the function $F(X_t,t) := X_t e^{\theta t}$, which leads to:

$$X_te^{\theta t}=X_0+\int_{h=0}^{h=t}\left(e^{\theta h}\theta\mu\right)dh+\int_{h=0}^{h=t}\left(e^{\theta h} \sigma\right)dW_h.$$

The final step is then to isolate $X_t$ on the LHS by dividing through by $e^{\theta t}$:

$$X_t=X_0e^{-\theta t}+\int_{h=0}^{h=t}\left(e^{\theta(h-t)}\theta\mu\right)dh+\int_{h=0}^{h=t}\sigma e^{\theta(h-t)} dW_h.$$

The short-hand notation for the Orsntein–Uhlenbeck process leads to confusion:

$$dX_t:=\theta(\mu- X_t)dt + \sigma dW_t.$$

Applying Itô's lemma and continuing in the short-hand notation leads to:

$$d(X_te^{\theta t})=\left(e^{\theta t}\theta\mu\right)dt+\left(e^{\theta t} \sigma\right)dW_t.$$

Using the short-hand notation, I have seen students mistakenly cancel out the terms $e^{\theta t}$ that would normally be written as $e^{\theta h}$ in the long-hand notation inside the integrals.

2) Geometric Brownian Motion

Even the very well-known SDE for the Geometric Brownian Motion written in short-hand notation leads to confusion:

$$dS_t=\mu S_tdt+\sigma S_tdW_t$$

Again, the terms on the RHS would normally be written as $S_h$ inside an integral, which makes it obvious that the terms cannot be "taken out" of the integral until they are integrated.

In the short-hand notation, I have seen far too often attempts to divide through by $S_t$ and write:

$$\frac{dS_t}{S_t}=\mu dt+\sigma dW_t$$

The "next step" would then be to "integrate both side" (with respect to what variable?) and write:

$$\ln(S_t)-\ln(S_0) = \mu t+\sigma W_t$$

Which is obviously wrong (the solution is actually $\ln(S_t)-\ln(S_0) = \mu t - 0.5 \sigma^2 t + \sigma W_t$, again using Ito's lemma).

In conclusion, I would argue that the short-hand notation for SDEs is rather unfortunate, particularly for new students to the field. I would encourage anyone new to stochastic calculus to use the long-hand notation until they become very comfortable with the subject.

Answer 5

Littlewood, J. E., A mathematician’s miscellany, London: Methuen & Co. VII, 136 p. 18 diagrams. (1953). ZBL0051.00101.

In §12 , Littlewood shows two proofs of the same thing. First a "beginners' proof" and then a "civilised proof".

Answer 6

Any notation that has become indelibly ambiguous, like (Does it mean the set of nonnegative, or just positive, integers?), has become bad notation. Not necessarily through any fault of its own.

Another example in my opinion is the backslash, which was introduced (as far as I know) by Hu to mean set subtraction: X \ Y = {x ∊ X | x ∉ Y}, apparently to avoid overloading the minus sign.

But the minus sign used to mean set subtraction seems to have caused no confusion that I know of. While the backslash can also be used for quotienting a structure Y on the right by a substructure X on the left.

Answer 7

I have long felt that the convention that writes the numerator above the denominator in fractions is the wrong way round. The consequences don't bother mathematicians but sometimes cause beginners to stumble.

When children first encounter a fraction like $3/5$ they think "divide something in $5$ parts and take $3$ of them". In that description you see the $5$ before the $3$. Writing "$3/5$" counterintuitively names the number you take before telling the reader how many parts there are.

When adding fractions, you must deal first with the denominators to find a common one. Only then do you think about the numerators. When you teach that to schoolchildren you require them to read from bottom to top.

In calculus, $dy$ is (roughly speaking) the change in $y$ caused by the change $dx$ in $x$. That sentence talks about the cause before the effect. Not how causality works.

Answer 8

• $\newcommand\E{\mathsf E}\newcommand\P{\mathsf P}\newcommand\Eb{\mathbb E}\newcommand\Pb{\mathbb P}\newcommand\R{\mathbb R}\newcommand\C{\mathbb C}$ChatGPT gives a few examples of bad notation used more or less long ago, including (i) Roman numerals; (ii) the original notation for logarithms, "which used geometric figures and decimal numbers"; (iii) the use of x for both multiplication and variables.

I think everyone would indeed agree that Roman numerals are inconvenient for arithmetical or other numerical operations.

I have seen the use of (cursive!) $x$ for multiplication of real numbers (!) even on MO! — Something like $axb$, to denote $ab$.

I cannot vouch for the original notation for logarithms. (Added later: On further questioning, ChatGPT said that its previous claim that the original notation for logarithms "used geometric figures and decimal numbers" was incorrect.)

• However, I think the commonly used notation, $\log$, for $\ln$ is bad. Indeed, $\ln$ is more (and completely) specific and shorter than $\log$. So, I don't see a good reason to use $\log$ for $\ln$. (I know that this suggestion may excite some passions.)

• Another example of commonly used bad notation is $\Pb$ and $\Eb$ to denote the probability and the expectation. It is better to use $\P$ and $\E$, or simply $P$ and $E$, leaving the blackboard-bold font for $\R$ and $\C$, and the like.

• Also, the standard convention used to be to write something like $\E X$ and $\E XY$, without any brackets or parentheses — with apparent understanding that $\E$ is a linear (and integral) operator, and we still commonly write $Tx$ rather than $T(x)$ if $T$ is a linear operator. Nowadays, people mostly write, I guess under the influence of computer science, $\E(X)$ and $\E(XY)$, or $\E[X]$ and $\E[XY]$, which in some cases makes formulas hard to read, with the necessity of going through all those parentheses or brackets. (I understand that this remark may excite some passions, too.)

• Alas, it is becoming more and more common (or maybe even "cool") to use the same symbol to denote a random variable and any of its values. This can clearly create confusion.

• Another complaint: the rather common use of $f(x)$ to denote a function $f$. In one post on MO, I even saw something like $\langle f(x),g(x)\rangle$ to denote $\langle f,g\rangle$. (Of course, $\langle f(x),g(x)\rangle$ will make sense only if the values of the functions $f$ and $g$ are in an inner-product space.)

• Yet another complaint is the common use of something like "for all $0\le x\le1$", which I find impossible to read aloud — in place of "for all $x\in[0,1]$" or "for all $x$ such that $0\le x\le1$".

---

Rejoinder: I expected passionate reactions to some of the items above.

Any use of ChatGPT seems to continue to excite some strong passions. In this case, a user (say CF) wrote in a comment: "It seems the irony that the generated list is numbered with Roman numerals is lost on ChatGPT." However, it should have been clear from the first three lines of my post that the lower-case Roman numerals (in parentheses) are mine. I also tried to explain this in my response 10 minutes after this comment by CF. There has been no response from CF to my repeated requests to deal appropriately with this counterfactual comment (which has garnered 10 upvotes).

Another user wrote in a comment (which has garnered 8 upvotes): "I tried to ask Chat GPT some group theory questions yesterday. Absolute garbage, wrong answers." It is not quite clear what this has to do with my post. Yet another user wrote in a comment: "anytime [ChatGPT] gets a fact right that’s a happy accident." -- In this case, ChatGPT immediately gave 5 suggestions, of which at least 2 were good (I think); is this a bad score for an immediate response? Anyhow, I think what should be judged foremost is, not the tools used, but the quality of the post itself -- which is the eventual product.

My complaint about phrases like "for all $0\le x\le1$" has also met some rather passionate opposition. The same user CF suggested that there was no problem "to change [the] grammar" to read the similar phrase "for all $1\le k\le n$" as "for all $k$ between $1$ and $n$". My response to this was that I cannot see any compelling reason for us to have our own grammar rules here given such perfectly grammatical alternatives as "for all $x\in[0,1]$", "for all $k\in[n]$", and "for all $k\in\{1,\dots,n\}$" (more details on this can be found in my comments, especially the most recent ones as of today, 23 April 2023).

Overall, my answer has had 8 upvotes (thank you!) and 12 downvotes. The overall score, -4, is disappointing. However, I am happy that some users have found my post useful, to some degree. Thanks to everyone who read this post.