How do I express that a point $A$ has coordinates $(a, b)$, symbolically? $ $ Should I just write $A=(a, b)$?

Question

How do I express that a point $A$ has coordinates $(a, b)$, symbolically? $ $
Should I just write $A=(a, b)$, or there's a better way? $ $ Also, if $B$ has coordinates (c+1, d+2), and I know $A$ and $B$ have the same coordinates, can I write $(a, b)=(c+1, d+2)$?

Answer 1

The most common expression is $A:=(a,b)$ where the symbol "$:=$" means "defined as". Of course it is also common just write $A=(a,b)$.

Answer 2

I have seen the following notation; suppose $x\in X$ belongs to an abstract 2-dimensional space. By virtue of being two-dimensional, it has a basis $E$ of 2 elements, say $E = (e_1,e_2)$. Then there exists two numbers $a,b$ such that $x = aE_1 + bE_2$. We then define $$ [E, x]:= \begin{bmatrix}a\\b\end{bmatrix}$$ This notation makes it clear that there is a choice of basis.