Answers: Part II

Here I will go over all the answers to the questions posed in the post immediately prior to “Answers: Part I”, including the answer covered in “Answers: Part I”.

The first question contains much that is relevant to all the rest. There are a couple of points that I (deliberately) didn’t cover in much detail in part I of the answers.

1) It is important to understand that by “probability of picking a rational number” I do not mean the probability that one picks a specific rational number. The probability of picking any particular number from [0,1] is 0, but if one picks a number from this interval, the probability of picking a number is 1.
2) The rational numbers appear to have no gaps. The answer to question 2) “Between any two rational numbers there is another rational number. Does this mean that between any two rational numbers there are infinitely many rational numbers?” is “yes”. Logically, if there exists a single rational between ANY two rational numbers, then I can select two rational numbers R1 & R2 and find a rational number r. I continue to use R1, but I set R2 equal to r, giving me the same R1 but an R2 that is between R1 and the original R2. Then I select r between these new two rational numbers. I again keep R1 and set R2 equal to r. I do this again (find a new r, keep R1, and set R2 equal to my new r). I can continue this infinitely, showing that between my arbitrarily chosen initial R1 & R2, there are infinitely many rational numbers.
If there are infinitely many rational numbers between any two rational numbers, how can there be any “gaps”? If there are no gaps, where do the irrational numbers appear? It turns out that there are infinite sets which are uncountable. That is, one can’t find a 1-1 correspondence between e.g., the set of real numbers in the interval [0,1] and the ENTIRE set of rational numbers: there are more real numbers in that interval than there are rational numbers. This difference (at least partly) means that the countable set of rational numbers in the interval [0,1] barely counts. It is “negligible” or infinitesimal compared to the set of irrational numbers. We can compare this difference to the difference between a finite set and an infinite one. Let N be the set of whole numbers and Let S be the set of integers from 1 to 100. We can add another 100 integers to the set S, but we are no closer to the number of integers N. We can add a billion, a trillion, and so on, but we are still no closer. Likewise, no matter how many numbers we add to a countably infinite set, it will never get any closer to the size of a countably infinite set (this is a simplification, but not without reason). This is why the probability of picking a rational number from the interval [0,1] is equal to 0. It is like comparing the probability of picking 1 out of the set of all integers except 1 vs. the probability of picking any other of the infinitely many integers other than 1.

The rationals do contain a property (other than being “dense”) that the reals have and that other countably infinite sets don’t. Namely, there is no “next” rational number that we can find. To see this, start by trying to pick “next” (largest/greatest) rational number after 1. Maybe we start with 1.000000001. Immediately, we realize that we can extend the number of 0’s before that final one onto infinity and we will never leave the rational numbers OR reach the next rational number. In fact, as every rational number is either a terminating decimal or a repeating decimal (e.g., 1/3=.3333333…), we can show that there is no next largest/biggest rational number we can find. In the case of rational numbers that terminate (like 1), we can use the method we used above: simply extend 0’s infinitely in front of a final one: ….00001, ….00000001, …00000000001, etc. For repeating decimals like .3333…, simply take one of the numbers in the repeating sequence and add 1 to every decimal after it (e.g., .33333…. becomes .333344444; .123123123… becomes .12444444…). We can always push the point at which we add one to all following decimals back: .333344444… becomes .333334444…. to .33333344444… etc., and .1234444 becomes .12314444 to .12312444… etc. This relates back to the “dense” nature of the rationals.

We’ve already covered probability of any specific point or number in any interval or region of real numbers: they are all 0. But what about the probability of a set of points on the unit square that contains (0,0)? Well, if the probability is defined to be equal to the region, than we get probability 1 if we fill the entire unit square. But we can do this without using the point (0,0). So even though this point belongs to the “probability space” of the unit square, the probability of picking any set of points in this region can equal one but not include a point in that region.

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