SravnaTestPrep wrote:
IanStewart wrote:
In any infinite probability question of this type, the probability of making any specific selection is always zero. So in the original
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To me this seems counterintuitive.
The theory that the distance between two points can be arbitrarily small defies common sense logic.
I think this is a fundamental problem beyond the level of even math.
I believe space does not exist as points. It is more as stretches and any two points in the stretch are not distinguishable and are one and the same. Thus i believe the reality is that, there are finite number of stretches and not infinite number of points.
One thing that's true of math (and of physics, for that matter) is that it doesn't matter what one "believes" or finds "intuitive". Mathematics studies what you can logically deduce from a set of definitions and axioms, nothing more. So if you want to take as an axiom that there's a finite number of points on a line, or that some distances cannot be subdivided, you can do that -- then you get into the well-developed field of mathematics known as "Finite Geometry". That kind of geometry barely resembles what we see on the GMAT, which is Euclidean Geometry, and in Euclidean Geometry, lines are continuous and distances can be infinitely subdivided.
It's easy to see why the geometry we're all familiar with (GMAT geometry) becomes self-contradictory (and thus nonsensical) if you assume there is a smallest possible distance, or that points have nonzero length. I'll prove that below, but this is completely irrelevant to GMAT test takers:
- if you can draw a length of 1, you can make a 1x1 square
- you can draw the diagonal of that square, so you can make a length of √2
- you can then subdivide that diagonal into a length of 1 and a length of √2 - 1
- if you can do that, you can make any length of the form (√2)(c) + d, where c and d are integers. For example, to make a length of 4√2 - 3, you can divide the diagonal of a 4x4 square into a length of 3 and a length of 4√2 - 3
- if you think there is a length that cannot be further subdivided, you must think that there is a smallest length we can make in this way, so a smallest positive number that can be written (√2)(a) + b, where a and b are integers
- that number is between 0 and 1, so if we square it, it gets smaller. But if we square that number we get (√2)(2ab) + 2a^2 + b^2. That's another number in the form (√2)(c) + d, so it's another distance we can make, and it's smaller than (√2)(a) + b.
- So if we assume there is a smallest possible distance, we reach a contradiction -- we can make an even smaller distance. So there can be no such thing, in ordinary geometry, as a "smallest distance"; distances can be infinitely subdivided.
So if you pick a random point from a line -- say, for ease of illustration, from the numbers on the number line between 0 and 1 -- the probability you pick a specific point or number, say 0.565656.... is zero, because you're trying to pick one specific thing from an infinitude of possibilities.
I'd add that the reason "paradoxes" like Zeno's paradox or the Hilbert Hotel Paradox exist is because most people do not find the concept of infinity "intuitive". But if you were to use whatever conception of infinity I gather you do find intuitive, and develop mathematics from the postulate that some distances cannot be subdivided, then you'd have to discard all of calculus, advanced probability theory, advanced statistics, analytic number theory -- most of advanced math besides discrete math.
Different fields of geometry based on different postulates may exist and GMAT may test only one of them. That is not my point. But my point is the postulate that there can be arbitrarily small distance between any two points is fallacious. Zeno's paradox which you pointed out was resolved only by considering that there cannot be instant magnitudes.
I am talking about absolute truths and not about the "truth" of a particular field of math.