Does a point have a size?

Question

I have just started my geometry class last week. The first things I learned about were the point, line, and plane. According to my teacher the point has no size. None at all. Is this true, or is it just unimaginably small? I don't see how points can make up lines and planes if they have no size. Its like 0+0+0 and so on. It equals 0. How can they exist without size?

I understand the existence of a point. Exact location in space and all that. However, the planes and lines being made of these things if they don't have a size is what I find baffling.

Oh, I know how to use these things in math. I don't have a problem with them in math. It's just when applying them to life.

I think I understand now. Thanks to all who answered. I will have to try to explain it to my friends who came up with similar questions.

Answer 1

Unlike all the uninformed speculation, I have an actual answer: your teacher's definition a modern derivation from Euclid's Elements, the first and most fundamental compilation of geometric thought. A point is defined by Euclid as follows:

"A point is that which has no part."

See: http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html#defs
http://en.wikipedia.org/wiki/Euclid%27s_Elements#First_principles

This means that in elementary geometry, a point has NO measure. It is not measure 0, its measure does not exist. There is a difference between a length of 0 and a length that does not exist (0 itself does, of course, exist).

Think of it this way - a line segment has a "length" measurement. A rectangle has a "length" and a "width." A box shape has "length" "width" and "depth." But what does a point have? Nothing. It's not that it has 0 length, but rather that it has nothing to measure.

In modern day mathematics (which your teacher is not using of course), a point is something else entirely. Its definition has been formalized and made more sensible.

In topology (a vast extension of analytic geometry), a point is simply an element of a set. You can think of a point on the real line as an element of the set of real numbers. A point in a coordinate plane is an element of a similar set. See:
http://en.wikipedia.org/wiki/Point_%28geometry%29#Points_in_topology
http://en.wikipedia.org/wiki/Geometry#Topology_and_geometry

In modern geometry, a point is NOT defined. Rather than define what a point is, the term "point" is considered to be an undefined term. For more on undefined terms and modern geometry, see:
http://en.wikipedia.org/wiki/Axiomatic_system#Properties
http://en.wikipedia.org/wiki/Neutral_geometry
http://en.wikipedia.org/wiki/Geometry#Contemporary_Euclidean_geometry
http://en.wikipedia.org/wiki/Geometry#Geometry_beyond_Euclid
http://en.wikipedia.org/wiki/Geometry#Modern_geometry

Answer 2

It is true. The problem is that you are trying to think of a point as an object (which is understandable, but wrong). A point is just an EXACT spot in space, so exact that there is no number small enough to represent it. If it had a size, length, area, or any other such measurements, it wouldn't actually be a point at all, it would be a line or a plane or a sphere etc. The dot on the paper may have a size, but that dot is just an estimate, just a representation of a specific spot in space

Answer 3

If the borders of three countries come together at a point, can you measure the area or thickness of that intersection? Can you measure the width of the line that seperates light from shadow? Can you measure the thickness of an acre?

The answer is no because talking about these dimensions doesn't make sense. A point has no dimension, a line 1, and plane 2. To make a claim about a dimension that they don't have is jibberish.

Answer 4

Jack,

What a great question to ask, and you show you've thought about it with the 0+0+0 comment. Let me try to give you a one or two sentence answer. I hope it makes some sense to you.

Geometry is a subject that, although it has many many many applications to the real world from engineering to architecture, is in the classroom a mental exercise, not something actual and physical.

In Geometry, to avoid conflicts in other theorems, we say that a point has no size, but it does have POSITION. It tells us where it's at, but it has no area. Difficult to grasp, but remember we're playing a mental game here, not something truly physical, although it's uses in the physical world are amazing.

I hope that helped. Never quit questioning!!

Answer 5

I'm not a math whiz or anything, but I can understand how a point would have no size because it is only the beginning place for other things. If there were an actual mark made to indicate a point, it would be the beginning of a line or plane or other such things. The 'point' is where all things start, but not the 'thing' itself.

Answer 6

Points are merely a defined place in space. They have no size, because they are dimension 0, because you can't measure anything. A line, has one dimension that you can measure (length), a square, 2 dimensions (length and width), a cube...etc.
In short, you can't measure anything on a point. And, as the others above have said, you cannot measure it in any way.

Answer 7

points are the abstract idea of location, "here" or "there" with all other detail omitted. don't make a big issue out of it. beginners always make things harder than they are. points, lines, planes -- these are UNDEFINED terms. if you don't already know what they mean, you can't talk geometry. similarly, if you don't already know what "Jesus Christ, Son of God, Saviour" means, you can't really talk Christianity. the meaning of point, line, and plane lies in the relationships, in how they're used. 2 intersecting lines have 1 point in common. 2 points have 1 line in common. 2 intersecting planes have 1 line in common. and so on.

Answer 8

It has no size. If you put a dot on a road map, the dot and the map only represent a place on the road.. There is no actual dot on the road.Therefore dot has no size, unless you actually place a dot on the road..

Answer 9

Is there a size to this point?

If there is someway to measure it, then there is a size to it.