I would like to ask whether some things are being taught wrong in some ways regarding math. For example, today I was studying about electrons and I found out that electrons do not "orbit" the nucleus like my teacher taught me but instead are present in various energy shells and do not have a fixed path. I am now trying to rebuild the image of my mind. So, I would like to avoid the same mistakes in math.
Doing secondary education now.
This is a good question. Here's a short list of things that are often taught "wrong" in high school.
You can't divide by $0$ - depending on the number system you're working in, this is wrong, look up the projectively extended real line to learn more.
Infinity is not a number - as above.
Negative numbers do not have square roots - depending on the number system you're working in, this is wrong, look up the complex numbers.
There's only one exponentiation function - not true. If $n$ is regarded as a natural number, then the expression $a^n$ makes sense for $a \in \mathbb{R}$. If it's regarded as an integer, we need $a \in \mathbb{R}_{\neq 0}$. If it's regarded as a real number, we need $a \in \mathbb{R}_{>0}$. The way I see it, these are all different exponential functions.
Pythagoras' theorem can be proved by geometric arguments - this is a half-truth, because in the standard approach, Pythagoras' theorem is either a definition (of distance) or obtained by a purely algebraic argument using inner products. Here's a recent question of mine that's relevant.
$f(x)$ is a function of $x$ - no! It's $f$ that's the function. The expression $f(x)$ represents an element of the codomain of $f$, so unless the codomain of $f$ contains functions, $f(x)$ is not a function.
Asymptotes are a clear, unambiguous concept - it seems to me that they're not.
The fundamental theorem of calculus always works - actually no, you need some assumptions on your functions. Check Wikipedia for the details.
The set of antiderivatives of $1/x$ is $\{x \mapsto \log(|x|) + A : A \in \mathbb{R}\}.$ This is untrue, because the arbitrary constant can be one thing for $x > 0$ and another thing for $x<0$.
$\sqrt{x^2} = x$ - this is a half-truth. Its true if $x \geq 0$ is assumed, but if it's not, the correct formula is $\sqrt{x^2} = |x|$. A related half-truth is that squaring and square root are inverse functions - this is a half truth. Its true on $\mathbb{R}_{\geq 0}$, but doesn't hold on the whole real line.
Related is the formula $(x^a)^b = x^{ab}.$ This is only a half-truth; look at the case $x = -1, a = 2, b=1/2$ to see what I mean. Its true if we assume $x>0$.
When you're learning about random variables, integration will show up, and you'll assume that the Riemann integral is the meaning of these expressions. But to do the theory properly, you have to use the Lebesgue integral.
The "fact" that $x/x=1$ is sometimes used in high school to simplify expressions. Unless you're working in the field of formal rational functions, you can't do that.
Axioms are statements that are "so evident or well-established, that [they're] accepted without controversy or question." This is not really true. Most of the things described as "axioms" in modern math are really conditions on the kinds of structures we're interested in studying. For example, the axioms of group theory are more like restrictions on what we consider to be a group rather than "self-evident truths."
Related to the above comment: Donlans Donlans in the comments points out that in high school, we usually take the real line as given. We just assume some axioms about it and go from there. But in fact, the real line can be constructed using set-theoretic techniques, and the "axioms" can be proved.
First, I want to comment on your statement that saying that electrons "orbit" the nucleus is "wrong" ... it may be wrong in some respects, but it is a useful simplification. It is a good first step into some area of interest, the details of which can later be revised.
And notice that with math, notions of 'right' vs 'wrong' are even harder to pin down. Is Euclidian geometry 'wrong' and non-Euclidian geometry 'right'? No, both are simply different worlds that we can study in isolation. And as far as their application to the real world goes, both turn out to have their uses.
Still, something analogous to the electrons "orbiting" the nucleus could be set theory: Naive set theory assumes that there is something like a Universal Set: a set of all things. Now, this is certainly an intuitive and helpful concept, and you can use it to prove all kinds of elementary results in set theory. However, it turns out that there is no Universal Set: that the idea of there being a Universal Set leads to a logical contradiction. So, this is something that more advanced students of set theory will be taught, and they will learn a more sophisticated axiomatization of set theory that hopefully avoids any such contradictions. And note I say 'hopefully', because we actually don't know if modern set theory is consistent: so maybe the large bulk of mathematics that can be based on it is 'wrong'!
The point is: beginning students are unlikely to ever hit upon the contradiction hidden in naive set theory. So, naive set theory is still very useful and perfectly ok to use in that context. ... it is certainly a lot easier as an introduction to set theory.
So, the answer to your question "Are there things being taught wrong" is yes and no: Yes, in naive set theory we teach something that is "wrong", but I would not say that it is 'wrong' to teach this 'wrong' thing. Sometimes to teach well, you have to initially teach things that are 'wrong' ... and from a pedagogical point of view, it is sometimes 'wrong' to initially teach things that are 'exactly right'. The truth can just be too much too handle for beginners.
Indeed, we often take this staggered approach to the very notions of right or wrong, . First, we are taught that there is a clear 'right' and 'wrong' ... and only later (when we hit secondary education) do we learn that things aren't as black and white as we had always thought.
As Knuth put it in The TeXbook:
Another noteworthy characteristic of this manual is that it doesn't always tell the ^{truth}. When certain concepts of \TeX\ are introduced informally, general rules will be stated; afterwards you will find that the rules aren't strictly true. In general, the later chapters contain more reliable information than the earlier ones do. The author feels that this technique of deliberate lying will actually make it easier for you to learn the ideas. Once you understand a simple but false rule, it will not be hard to supplement that rule with its exceptions.
Last sentence emphasised by me.
There are many ways in which various subjects are taught in a way that is not entirety accurate. The reasons for this may vary, but they are typically because
$1$. Our understanding of a subject matter is incomplete and/or improves over time, and older models of thought remain useful introductions to a topic, or
$2$. Subjects are deliberately explained in a way that obscures the finer details of a theory because such details would overwhelm or overcomplicate the learning process early on.
Often times, the "early" explanations of a topic are not incorrect; they simply apply to general cases of the phenomenon being studied and ignore minutia necessary for handling special cases. Or vice versa, the "early" explanation is one that is helpful for understanding special cases, but more detail is necessary to understand the general case, and this is taught later on.
A simple example would pertain to even roots of negative numbers. In grade school we learn that even roots of negative numbers is not possible. That is because the root is either negative or positive, and in either case the root multiplied by itself an even number of times is positive - not negative. Later on we learn this is only true for the real number system, but not so with imaginary numbers. This doesn't nullify our current understanding; it simply expands it.
Another example would be in the realm of physics. All physics courses begin with an introduction to classical Newtonian physics, and you actually develop an intuition for the physical universe that is not entirely accurate. If you continue down the rabbit hole with relativity and quantum mechanics, you realize classical physics is a "special case" of relativity, and then quantum mechanics destroys everything you thought you know about reality. In fact, unifying relativity and quantum mechanics is an open problem, which underscores that even our best understanding of the physical universe to date is incomplete and not entirely accurate.
