What is a point?Date: 8/26/96 at 21:32:45 From: Anonymous Subject: What is a point? Define a point, please. Date: 8/27/96 at 22:15:6 From: Doctor Robert Subject: Re: What is a point? The word "point" is often left undefined in geometry texts. It is pretty easy for us to conceptualize a point, but it is quite difficult to define exactly. I would say that a point is an entity that has only one characteristic. That characterstic is its position. But then, you see, I haven't defined position, have I? Anyway, a point has no size, color, smell, feel, etc. It has only its position. A point is often represented by a smudge of chalk on a blackboard, or a smudge of pencil lead on a piece of paper. But, of course, those are only representations. A point can really only exist in our minds. I hope you're not more confused than when I started. -Doctor Robert, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 8/27/96 at 22:15:6 From: Doctor Ken Subject: Re: What is a point? Hi! The question you asked is actually quite a subtle one, and in a certain sense it makes a difference what kind of math you're doing. If you're doing Geometry, then no amount of physical torture will get me to define a point; it's an undefined term, and it's undefined on purpose. So this is really about math in general: what is an undefined term? Well, as you may know, math is about making definitions and proving theorems. For example, I can define a circle to be a set of points equidistant from a given point, and I can prove a theorem about that circle, such as "the perpendicular bisector of a chord of a circle will pass through the center of the circle." But in order to make these definitions and prove these theorems, I need to start from _something_. That's the role of the UNDEFINED TERMS and AXIOMS in math. When I defined the circle as a set of points, I used objects I already had (points) to define a new kind of object (the circle). So every time we define a new object we have to have some old object to base it on. If you think about the structure of math as a tree, there has to be something at the bottom of the tree, some objects that aren't defined. A point is one of these objects. It is undefined. It is just an object. In geometry, people usually think of points, lines, and planes as undefined objects (also known as undefined terms). So what are axioms? Well, they're statements that we don't have to prove (much as undefined terms are objects we don't have to define). A traditional example of an axiom in geometry is the statement "given any two points, there is one and only one line that passes through them." This statement is just accepted as true, so that we have a starting point, something we can use to prove theorems. It also does something else: it tells us something about points and lines. Any concepts we have in our heads about points and lines MUST satisfy this axiom. If we're thinking of points as bottles of beer and lines as telephone poles, then we have a problem, because I can show you a couple of bottles of beer that don't have a telephone pole connecting them. So that's how it goes in geometry. In other parts of math, for instance when you're using the coordinate axes, a point may not be an undefined term at all - it can just be a list of numbers, such as saying "consider the point (3,2)." For that, check out the answer from Dr. Chuck, which follows. -Doctor Ken, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 8/27/96 at 22:15:6 From: Doctor Chuck Subject: Re: What is a point? A "point" is an infinitely small entity at a specific location on a number line, plane, 3-D space, etc. When we talk about points, we are referring to one specific location. For example, along a number line the number "2" exists at just one point. I said that points are infinitely small because the point at '2' is different from the point at '2.000000001'. Here's a picture of a number line: The point 2 | \/ -infinity <==...---(-1)-----0-----1-----2-----3--... ==> infinity Okay, so this makes sense: if you want to distinguish one place along a number line, you "point" at it. You label that place with the corresponding number, and refer to it with that number. Now, how do you distinguish a location in 2-dimensional space (i.e. a sheet of paper)? Imagine that we have two number lines, one horizontal and the other vertical. We are "pointing" at a place "p": ... | 2 | | | | | 1 p | | | | | ...---(-1)-----0-----1-----2-----3--... | | | ... How do we describe where the point 'p' is? We can't just say p is at 2 because we don't know which number line that refers to - is it at 2 along the horizontal number line, or 1 along the vertical number line? To describe where 'p' is, you must talk about where it is both horizontally AND vertically. So, you can say: 'p is at 2 horizontally, and 1 vertically'. However, this is a mouthful to say. Because describing points in 2 dimensions is really useful, people have defined some conventions to make life easier. They call the horizontal number line the 'x-axis', and the vertical number line the 'y-axis'. The convention for talking about 2-dimension points is to write: ( position along x-axis , position along y-axis ). Therefore, 'p is at (2, 1)' 2-dimensional points can be described by any pair of numbers. For example, (4,5) (6.23432, 3.14...) and (-12, 4) are all points. Sometimes people want to describe a point in three dimensions. To do this, they need to use a triplet of numbers like (1, 2, -5) - do you see why? I hope this helps. -Doctor Chuck, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 8/2/97 at 12:54:51 From: James R Sheldon Subject: Re: What is a point I was reading your answers to this person's question, and I remembered seeing some stuff about points in a Geometry book: From The University of Chicago School Mathematics Project Geometry: 4 descriptions of a point: A point is a dot. A point is an exact location. A point is an ordered pair of numbers. A point is a node of a network. The dot one is used often in computers, the location and ordered pair are used in Geometry usually, and the node is used to solve problems like the Konigsberg Bridge Problem. I found the networks the most interesting since I had never seen those before. |
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