Here's the question you clicked on:
abdul_shabeer
Does a point have no dimension?
yes a point is dimensionless
0D = Dot 1D = Line 2D = Plane 3D = Solid 4D = Space-time 5D = ...idk :P
Then how does it make a line?
. <-- This is not a point, it's just a representation because a point has no height, no length, no thickness.
One dimension: A line, numbers, they have only a length;
2D: Shape, Length and width 3D: Solids, Length, width and thickness
What is the definition for a line?
4D Spacetime: 3 dimentional object travelling/moving through time and space.
Straight objects without depth and width.
Is it made of many points?
Um, you can see it that way :)
When a point has no dimension then how can a collection of points make a line?
A point has no length, depth, width, when you align then, you just created a length
take a point and send it in some direction through space the path that it traces out is a line the real number line is infinitely dense so now we are getting into some of the trickyness of infinity
|dw:1341410594698:dw|here is the real number line (still a line, consisting of infinitely many points) I can ask you to pick out the point 1
|dw:1341410679112:dw|and there it is I can also ask you to find numbers between numbers so if I ask you to find 1.5 it should be on the line
My doubt is when a point has zero dimension, then how can it make a line that has one dimension?
|dw:1341410742343:dw|similarly I can keep asking you to find points between the two we have already found 1.75 must be on the line 1.85 1.8 1.8243574687654365768.... etc. must all be on the line therefore there are infintiely many points on a line
between any two points there are infinitely many numbers, do you agree?
like between 1 and 2 are there not an infinity of numbers?
Though there are infinitely many points, each has zero length, zero breadth etc.
but you are trying to add an infinity of infinitely tiny lengths!\[\infty\cdot\frac1\infty=\text{undefined}\]you can't treat infinity like a number and perform addition on it; it is a concept, rather than a number.
it would require an infinity of points lined up in a row, and they still would have not length this may seem paradoxical, but it reflects the infinite density of number distribution on the real line, so it is not trivial
Wouldn't 0+0+0+0........ be zero?
no, because \(0\cdot\infty\) is undefined
again, I know this does not make instinctive sense infinity is a tricky topic that has driven many mathematicians mad!
When 0*infinity is undefined, then how does it produce a length?
because you don't have to build a line one point at a time; it's a mathematical construct that contains infinitely many points
if there were some finite number of points that you could put together to make a line, then the real number line would not be infinitely dense (i.e. some points would be missing from the line because there would have to be a limit on the number of times I can tell you to find the midpoint between two points)
|dw:1341411610187:dw|take any line segment; I want the midpoint (whatever it is)
|dw:1341411643038:dw|now I want the midpoint of the right half
|dw:1341411666632:dw|now again, the midpoint of the right half
|dw:1341411687412:dw|we could do this forever, right? that means there are infinitely many points on that line; I can always ask you to find the midpoint between any two points.
yet even though it is an infinity of points, they only add up to a finite line segment, so an infinity of things can add to something finite. Reference to Zeno's paradox.
What does "though it is an infinity of points, they only add up to a finite line segment" mean?
TuringTest just showed that there's an amount of point between two point, but when this infinite adds up, it gives something finite, something you can just count on your fingers.
How can infinite things add and give some finite value?
It is important to understand here that points and lines are simply geometric realizations of abstract concepts. As mentioned earlier, a . is not a point, nor is a drawing of a line actually a line. A point takes up no space, so it can't be shown, and similarly a line can't be shown because it has no height. Even a plane can't be shown really, because it has no depth. So, when you're asking these questions, it is best to avoid relying too heavily on your geometric interpretation of the concepts. Instead, it is better to look at them using the tools of analysis. TuringTest is referring to some relevant results of analysis, for example, the uncountability of the real line. Not only is the real line infinite in the number of points, it is uncountably infinite. This ends up implying that the number of points between 0 and 1 is actually the same as the number of points on the entire real line. So, to interpret this somewhat geometrically, you can zoom in as close as you would like to the real line and still be looking at the same number of points. Of course, this doesn't work like any real-world object. Another relevant way of looking at the real line is through the lens of topology. We can talk about the real line as a metric space, and we define the metric abstractly. We simply define how distance works. One sees by learning some introductory topology that the real line is actually just one way to look at the set of real numbers, and there are many others. So, this is another way to look at the problem.
in answer to how an infinity of things can add up to something finite do you know what an infinite series is @abdul_shabeer ?
The sum of all terms of the geometric sequence that halves each time would be the perfect example :D
\[\sum_{n=1}^\infty(\frac12)^n=\frac12+\frac14+\frac18+...=1\]when you study them this may make a little more sense the one above can be said to represent Zeno's paradox
In general, we don't have an infinity of things simply "adding up" to finite value. We have an infinity of things approaching a limit that has finite value.
Hey guys - I'm at work at the moment but spotted this discussion and remembered a thread I saw that might help explain this to to you @abdul_shabeer : http://mathforum.org/library/drmath/view/55297.html Hope it helps - back to work now... :)
|dw:1341412717556:dw|