How can you have an infinite number of points in a finite area?

Question

I was told by my college professor that that there are an infinite number of points in any given area. A given area is finite. How can you have infinity in a finite area. I was told that a point has no size so there is infinity. If it has no size, It does not exist. The smallest thing known to man, and atom?? a photon?? has size. Some body please help my to logical pea brain.

Answer 1

This question actually has a simple answer without getting into math theory. Imagine that you have a $100 bill. This is a finite amount of money. Now imagine that you have 10 $10 bills. You have the same finite amount of money but the division is smaller. Now replace that with 100 $1 bills. Same amount but smaller division. You can replace the $1 bills with 1,000 dimes or 10,000 pennies. Each is the same finite amount of money but each is made up of smaller divisions. Now, we can arbitrarily divide the money even smaller into 1/10ths of a cent, 1/100ths, 1/1,000ths. We don't have to stop at any point; we can go to 1/millionths of a cent, 1/billionths, 1/trillionths. We can divide up our finite amount of money into any abitrarily small amounts. The only limit to how many divisions we can have is how small we make each division Since we have no limit to how small these units can be, the number of divisions is essentially infinite (uncountable) even though our amount of money doesn't change.

Answer 2

You are saying that the area is finite.

What is the meaning of finite area?

It has a finite number of units, unit being defined by you.

Say, the area is 5 square centimeter. We take a unit area as one centimeter square. Thus it has a finite area.

The same area can be told to be 500 square millimeter. Again it has a finite area.

Thus the area is finite as long as you define a unit area.

If you define a point whose area is so much, then we can say that the area contains a fixed number of points.

But we have agreed to the concept that a point cannot be defined by an area. Point has no definite area and it is decreasing to a size almost nearing to zero indefinitely. Mathematically its area is infinitesimal.

To calculate the number of points in an area, we divide the area by the area of a point.

The area of a point “a”, being infinitesimal, the limit (A/ a) tends to infinity when a tends to zero.

Similarly you can calculate the area by the product of the number of the points and the area of each point.

That is infinity x infinitesimal, we know this can lead to a finite value.

If we use the smallest particle as a point, then using the area of the point, we can count the number of particles in a given area.

But point is the smaller than the smallest particle. A particle has many points in it. A area of a point must tend to zero and it should not stop in the middle.

Answer 3

A "point" as you refer to it is a mathematical concept rather than a physical entity. For example you could drive only a finite number of nails in a finite area because the nails have a diameter and take up space. Two objects can not occupy the same space. However, a line and a point have zero diameters and therefore zero areas. If you divide zero area into an area of say one square unit, the result will be infinity. Remember that we can draw a line or make a point that has visible thickness using a pencil but the drawing merely represents the line or point that would otherwise be too thin to see.

Answer 4

Do this, trust me it will be worth it if you want to understand your question:

Draw an isosceles triangle in the middle of a piece of paper (one that looks like the one at the top of the page here: http://mathworld.wolfram.com/IsoscelesTriangle.html)
Now draw a line horizontally through the triangle, from one side to the other, so that you have a mini isosceles triangle inside the first one.
Now imagine that there is a long pendulum swinging from the top of the triangle down well below the base of the triangle.

Do you see that the string of the pendulum would touch only one point at a time on the base of the triangle and the base of the mini triangle, but it would touch all of the points on both lines as it rotated? So EACH point in the top line is connected with ONLY ONE point on the bottom line. (stop and think about this for a second) But doesn't the bottom line have more points because it is longer? No! Because they are matched one to one, point to point, they have the same number of points. This would not be possible if points had a size, right? So points have no size, so you can fit any number of them in a finite area. Hope that helps.

Answer 5

The basis of calculus is getting on good terms with this idea. Imagine, if you will, you have a needle. You will stick this needle somewhere in a huge field. Then you move the needle slightly over. You have a completely new point. Now imagine that your motor reflexes are perfect. You can now move the tip of this needle to any precision. You move it to the left by 1 atom. You move it to the left by 1 electron. You move it to the right by 1 quark. You move it up by one super string. etc etc. You then realize that you can continually reduce the size of your movements as many times as you want (because you have perfect movement). This is what math is all about. Since it is not limited by physical aspects, there is no reason why we should stop at the atomic scale, or even the sub-atomic scale. This is why there are an infinite number of points. In math, there is ALWAYS one more point you can do that is unique.

Answer 6

Since points have no dimension you can identify and match an infinite number of points on any given area.

.A point can be considered as the common intersection of directional lines(vectors).the point become what is called a coordinate or a frame of reference. Since direction can be divided into infinite number of angles so can its intersection which is a point..

Neverthe less the Universe does not live by points which are dimensionless. So what is concevable mathematically is not so in the real Universe.

What is called a singularity in the Big Bang is not really a point, but a volume of micro masses having a density..Therefore the Universe exists in terms of volume of both mass and massless. structures.
The Photon mathematical description is one of radiation power falling on a surface for one second. Main stream Physics has not given any size to Energy.And it considers the Photon masseless.

I really appreciate your logical reasoning.

Answer 7

I'm jealous because you defined (a finite) area.. your peabrain. By giving a definite size reference for your brain, you have created an area containing an infinite number of points!! And it's really close in relation to your head (aha! another defined area)!!! I'm in awe... truly amaxing!! It's all about the relationships man!!

Answer 8

Yeah, points do not exist. Only tiny bits so small, that from our perspective, there is no difference. But, since real "points" are finite in size, then in multitudinous aggregate they can form finite volumes (or apparent areas).

Answer 9

Well you can divide some finite are into half... divide that piece into half and divide that piece into half on and on and on.

Or think of integral. You take infinite amount slices to make accurate measurement in some curved surface.

Probably just ask him to explain.

Answer 10

well a point doesn't hav a size so there can be any number of points in a certain area. For example you might hav a coordinate plane... u hav a point at 1, then 1.1, then 1.2.... 1.01
it will go on forever... jus add decimal points