Can a point, on a 2D graph, have dimensions (like length, width, height)? If not, are there any other situations in which a point can have dimensions? In the case of inequalities, when you graph them, an open circle indicates greater/less than, and a closed circle indicates greater/less than or equal to. Would the closed circle have dimensions? Would it have a slope (would it be included in the slope of a line that it is part of)?
First post so please forgive anything wrong on my part. Thanks!
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The way I think about this is the following. Points are 0-dimensional objects, they have no length, width or height. So, they have a dimension, it is 0. However, a bunch of points (in fact, infinitely many of them) stacked closely together can give a 1-dimensional object: a line or segment. For example, a graph in $\mathbb{R}^2$, like the circle you mentioned, is a 1-dimensional object (of course, only including its boundary), but it is the set of all 0-dimensional points $(x,y)$ such that $x^2+y^2 = r^2$ where $r$ is the radius. Now similarly, a bunch of 1-dimensional objects stacked closely together give a 2-dimensional object. For example, a bunch of straight parallel segments may depict a square. You can continue in this matter and think of 3-dimensional objects as a bunch of 2 dimensional objects put together, and 4-dimensional objects ... You get the point.
Of course, I am being extremely hand-wavy and non-rigorous, but this is the low-level intuition I have about why it makes sense that points are 0-dimensional.
If you are interested in this, you might enjoy reading "Flatland" by Edwin Abbott Abbott.
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A point is a location in space that has no dimensions. The interior of a circle is called a disk, which has area and is therefore 2-dimensional. The circle itself is a closed curve but cannot exist only occupying one dimension (otherwise it gets squished into a line) so I would say it is 2-dimensional.
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It is worth considering Euclid’s treatment of a point as a primitive term: a point is “that which has no part.” In other words, no width, length, etc.
Since a closed circle is just an “exaggerated” way of indicating a single point on a graph, a closed circle would not have any dimensions when considered on its own. However a closed circle when considered as part of a line would be a part of a 1 dimensional shape.
As for if we can find a slope at a single point, now we are getting into Calculus!
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Can a point, on a 2D graph, have dimensions (like length, width, height)?
We need to clarify what we mean by "2D graph", "point", "dimension" and even "length", "width", "height", although all these terms may be clear intuitively, there are many ways we can interprete all these terms in math.
For example, they can be captured by the notion of vector space $(V,+,\cdot)$ (over a field $K$). A point is an element of $v \in V$, the notion of dimension is a property of the vector space $V$ and is the number of elements of its base.
Of course the usual assumption when we hear "2d graph", and "point" is the vector space $(\mathbb{R}^2,+,\cdot)$ over the field $\mathbb{R}$. This vector space has dimension $2$. So in this case a point has the form $(x,y)$ with $x \in \mathbb{R}$ and $y \in \mathbb{R}$, does this point have "length", "width", "height"?, well, to answer this question we need to add another structure to our vector space called measure, in this case, the Lebesgue measure, which generalizes the notion of "length", "width", "height" for $\mathbb{R}^n$. Through the Lebesgue measure we could talk about the "size of a point" (dimension in the sense of "size"), that is, the measure of a point. The (Lebesgue) measure of any point in $\mathbb{R}^2$ is $0$, so in this case we can say the "size" of a point in $\mathbb{R}^2$ is $0$, so we can say it has no dimension (in the sense of size).
If not, are there any other situations in which a point can have dimensions?
Yes, as I said previously, there are many ways we can interprete the terms.
For example think of the vector field that takes point in the plane $\mathbb{R}^2$ and assigns it a vector in $\mathbb{R}^2$, in this case you have a "2D plane" whose points are vectors that have length (through their norm).
Image: six methods for visualizing the same 2D vector field (source)
In the case of inequalities, when you graph them, an open circle indicates greater/less than, and a closed circle indicates greater/less than or equal to. Would the closed circle have dimensions?
The circle you describe has a (Lebesgue) measure, different to zero, intuitively this means it has a "size", therefore a "dimension". Which is given by the well known $\pi r^2$.
Would it have a slope (would it be included in the slope of a line that it is part of)?
Here, I don't know what you mean by "slope".
In summary
The idea of "dimension" may be seen at least by two lenses.
In the linear algebra lenses, the dimension is the number of "coordinates" of your vector space, therefore is an attribute of the vector space and its a natural number or $\infty$ (for example $\mathbb{R}^n$ has dimentsion $n$).
In the Lebesgue measure lenses (measure theory), the dimension is the "measure" of a set $A$ living in another bigger set $B$ ($A \subset B$). And this measure can be interpreted as the "size" of the given set $A$. It is an attribute only for what we refer to "measurable sets" and its a real number or $\infty$.
Can a point, on a 2D graph, have dimensions (like length, width, height)? If not, are there any other situations in which a point can have dimensions?
In the case of inequalities, when you graph them, an open circle indicates greater/less than, and a closed circle indicates greater/less than or equal to. Would the closed circle have dimensions?
Would it have a slope (would it be included in the slope of a line that it is part of)?