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I have always found strange that in elementary analytic geometry points are defined by their names followed by their coordinates, for example:

"Find the distance between $A(5, -3)$ and $B(2, 1)$." (source)

Usually, mathematical objects are defined either by an equality or by a uniquely characterizing property. So one would expect points to be defined such as "Let $A = (0,1)$" (once the coordinate system is fixed) or "Let $A$ be the point of coordinates $x= 0$ and $y=0$," but not "$A(0,1)$".

Furthermore, usually a letter followed by some numbers between parentheses denotes the evaluation of a function at certain points, so "$A(0,1)$" could be misunderstood for "The value of the function $A$ at point $(0,1)$".

What is the historical, practical, or whatelse reason for this notation?

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  • $\begingroup$ This is routinely done in US high school and undergraduate sources, but is by no means universal. $\endgroup$ Feb 19 at 21:02
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    $\begingroup$ To relate the letter used on a diagram (and in the accompanying text) to the coordinates used in calculations. Putting the equality sign between them is technically incorrect, a pair of coordinates is not the same as a geometric point, and adds clutter with little point to it. Mathematical notations routinely omit nuances of the relationship when those are easily understood from context, that is the rule with shorthands generally. Confusion with the functional notation in analytic geometry contexts is highly unlikely because point letters are not function names. $\endgroup$
    – Conifold
    Feb 19 at 23:04
  • $\begingroup$ I'd certainly agree that this notation is in conflict with late 20th-century-and-beyond notation involving functions. It arose in the 19th century or earlier, so far as I can tell, when the "Euclidean geometry" tradition was still very large, but/and needed to be combined, operationally, with "analytic geometry". So this style of notation was telling the two different worlds' names for the point. Really, in that context, no necessity of an equality, if everyone understands. Just bad luck to run afoul of other notational conventions in more recent times. :) $\endgroup$ Feb 19 at 23:40
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    $\begingroup$ See also this post. $\endgroup$ Feb 21 at 8:50
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    $\begingroup$ In response to @Conifolds comment: in analytic geometry (and I assume this is done here, since coordinates are used) the plane is identified with $\mathbb{R}^2$, so a point is a pair of numbers and writing $A=(5,-3)$ is technically correct. It might be shorter to write $A(5,-3)$ but I agree with the OP that this is irritating in the context of established notations. $\endgroup$ Feb 22 at 7:58

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Well, in my humble opinion, a point in a coordinate system is understood in terms of an intersection of two lines (two lines intersect at one and only one point or never at all if they're parallel).

A point P (3, 4) is shorthand for the two lines x = 3 AND y = 4, the point of interest (P) being where these two orthogonal lines intersect. It's simpler to write this whole business down as P (3, 4).

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  • $\begingroup$ I think your explanation doesn't address the question which is about the notation itself, not about its meaning which is non-ambiguous. Besides, we don't need orthogonality here (coordinates have a meaning in a "slant" axes system) $\endgroup$ Mar 24 at 11:09

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