Basic non-obvious things to remember

We can only write $a_1 + \ldots + a_n$, without parenthesis, because of the associative property of the real numbers.

(Definition) Identity: given a set $S$ and an operation $\square$ on $S$, it's the element $e$ for which
$$\forall x \in S, x \square e = x$$

(Definition) Inverse:  given an operation $\square$ on set $S$, we say $\square$ has an inverse operation $^-$ if
$$\forall x \in S, x \square x^- = e$$

Commutative property is not a general rule: $$a + b = b + a, a - b \neq b - a$$

Division by 0 is undefined because division is defined in terms of multiplication: $\frac a b = a \cdot b^{-1}$, and $0^{-1}$ is undefined.

Given that  $a \cdot b = a \cdot c$, one could say that we can "cut" $a$, and say that $b = c$. Well, if $a = 0$, it's not true! But, when $a \neq 0$, it is, otherwise it's important to be aware of the fact that "cutting" $a$ is hiding this set of  equations:
\begin{array}{c} a \cdot b & = a \cdot c \\\\ (a^{-1})\cdot(a \cdot b) & = (a^{-1})\cdot(a \cdot c) \end{array}