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Can two points be added?

The reason I ask is because when I think about it all I see is vector addition. I understand the difference between vectors and points. I know we used to talk about points on the number line as a kid, and we added numbers, but I can't assume equivalency.

It also feels strange to say "if I add this location with this other location I get a new location."

What is the answer, and what is the root of my confusion?

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    $\begingroup$ What, in your opinion, is the difference between a vector and a point? $\endgroup$ – user137731 Oct 14 '15 at 0:33
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    $\begingroup$ you can add points on an elliptic curve $\endgroup$ – Will Jagy Oct 14 '15 at 0:46
  • $\begingroup$ You might find the section Vectors of this answer helpful. Read through it and then ask yourself, "what is the difference between a tuple and a point?" $\endgroup$ – user137731 Oct 14 '15 at 0:52
  • $\begingroup$ Here's part of my view on this: an ordered $n$-tuple of real numbers is sometimes called a "point", and sometimes called a "vector". The two terms suggest two different ways of thinking about and visualizing an $n$-tuple, and each term suggests certain things we might be doing with the $n$-tuple. When we add together two $n$-tuples, we are thinking about the $n$-tuples as vectors, not as points. (Except perhaps in some advanced areas of math such as elliptic curves or something I'm not familiar with.) $\endgroup$ – littleO Oct 14 '15 at 1:36
  • $\begingroup$ $n$-tuples can just be considered the algebraic representations of points. Technically a set of points by itself has no inheritly defined vector addition or scalar multiplication. But, assigning to each point a $n$-tuple, and taking advantage of the obvious way of adding and scaling them, allows us to talk about points -- which are inheritly geometric objects -- algebraically. $\endgroup$ – user137731 Oct 14 '15 at 1:58
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It also feels strange to say "if I add this location with this other location I get a new location."

What is the answer, and what is the root of my confusion?

You don't add locations, you add displacements.

Even as a kid you would not be taught scalar addition as adding points on the number line, but adding the distance moved between points.   Start at the origin ($0$), move distance $x$, then move distance $y$ from there and the point you arrive at has the value of the sum of $x$ and $y$.

That's basically adding 1-dimensional displacement vectors.


Are you sure that you understand the distinction between points and vectors?

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  • $\begingroup$ Yes, I believe I understand the difference. So, main post tells me "You can't add points." Eg: Point1 + Point2 just doesn't happen. $\endgroup$ – Plotnus Oct 14 '15 at 0:56
  • $\begingroup$ @Plotnus as an algebraist, your words hurt me. Surely you can add anything you want, no? As long as you define what "add" means! $\endgroup$ – Matt Samuel Oct 14 '15 at 1:22
  • $\begingroup$ @MattSamuel I don't know the answer to that. The part of me in in abstract land says yes, but then the physics part of me says no because 1second + 1meter = 1second + 1meter. So, because of the units no further simplification is possible. So I suppose the question can Point1+Point2 be expressed as a single point would have been the better way to express my question. So, Matt can you add anything you want without ending up in a tautology? What does your statement mean? $\endgroup$ – Plotnus Oct 14 '15 at 6:41
  • $\begingroup$ @plotnus it's true that some things do make sense to add normally and some don't. Points, though, at least in linear algebra, are exactly the same as vectors, which are exactly the same as $n$-tuples. Generally you add them coordinatewise. $\endgroup$ – Matt Samuel Oct 14 '15 at 10:54

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