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.

pedagogicallybad for mathematically inexperienced first-timers in complex numbers, but that's not enough to disqualify it. Arguably, it has certain shock value, which has turned it into a symbol on its own, subconsciously burried into every mathematician's mind: it hits you even before you digest the other parts of a formula. Many classical (and even newer) books on SVCs and Complex Geometry use it instead of $i$, which frees the latter to be used as an index, e.g. Donu Arapura's book, Griffiths and Harris' book etc. $\endgroup$thesquare root of $-1$", emphasis mine, because there is no such thing asthesquare root of $-1$ (unless you work in characteristic $2$). If you think that $\sqrt{-1}$ isasquare root of $-1$, then you should be untroubled by the suggestion that $\sqrt{-1}\cdot\sqrt{-1} = \sqrt{1}$: it says that the product of two square roots of $-1$ (which need not even be the same!) isasquare root of $1$, which is true. $\endgroup$write$\lim a_n$ until you have first proved that the limit exists. So an equation like $\lim a_n = 5$ may be neither true nor false but actually ill-defined. By contrast, using $\to$ as a relation, the formula $a_n \to 5$ at least has a definite truth value no matter what sequence $a_n$ is. $\endgroup$17more comments