Answers Part I

So my previous post asked some questions. I want to give the billions of people who read this blog a few days to provide answers rather than just give them away, but I feel it is unfair to simply wait for days when the only (possibly) interesting part of that post was the questions. Therefore, I’ll provide an answer to the first question (deliberately ranked first to enable this) already.

“1) Imagine that you or a computer could pick a number at random from the interval [0,1]. What is the probability that this number will turn out to be a rational number?”

There are infinitely many rational numbers in this interval. The probability that a randomly picked number within this interval will be rational is 0.

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7 Responses to Answers Part I

  1. Jeff says:

    At what point do rational numbers become redundant? Meaning they can no longer be considered infinite? Pi becomes redundant, even if it seems like an ever repeating decimal (we can only make a perfect circle, unless nothing in nature or the universe is perfect – which for as far as anyone can tell, nothing is). And by redundant, I mean a decimal repeating to the point that it doesn’t matter how many decimal places it expands, it only matters to a few decimal places, unless you put a certain metric on it. What type of metric scale are you referring to or, better yet, does that metric even exist? =)

    Do metrics exist on a number line? I think not!

    • The rationals are always infinite. However, not all infinities are equal. Given the unit interval [0,1], it turns out that even an everywhere dense and infinite set like the rationals barely “fills out” the interval or “takes up space” in the interval. One way to approach this other than learning measure theory and real analysis is to consider a more intuitive comparison between a finite set (of numbers) and an infinite set like the integers or rationals. Imagine our finite set consisted of all the integers from 0 to 100. How much “bigger” than this is any infinite set? Well, if we add another 100, are we any closer to the “size” of an infinite set? No. Add a billion, same result. Googol? same result. No matter how many integers we add to our finite set, it is never closer to any infinite set.
      It turns out that there is something similar with infinite sets. The integers, rationals, and whole numbers are all infinite sets, but they are what we call “countably” infinite sets. We can “order” them in order to match them up in a one-to-one correspondence with the set of counting numbers. This is not true of the real numbers. There are more elements in the set of real numbers in the interval [0,1] than there are rational numbers. See example #2 here:

    • I should have mentioned: metrics are usually based on notions from the number line and definitely exist on the number line. “Metric” has a mathematically precise definition.

      • Jeff says:

        I may have been a little cynical (if not a little confused) when I responded, apologies, and thanks for the clarification.

        It brings up some good questions though. Like are all rational numbers based on some type of metric. Otherwise they must fall to an irrational counterpart?

        We can give numbers a value and numbers like 1,2,3,4 are considered whole numbers or natural numbers. At what point do they become unnatural, irrational or to push the boundaries a little, infinite. It tends to make me question the boundaries of infinities.
        I’m not mathematician and don’t claim to be (and you are smart so bar with me), but for example, if we were doing some type of theoretical stuff. We could say there are no metrics and these numbers have no values, they are just placeholders of infinities (or infinite possibilities). But if we were to give them some kind of value, we have to do the calculations based on the values we have assigned or given to them.

        It is kind like saying Bob weighs 128 lbs. We decide to do a little science experiment to see how many Big Macs Bob can eat before he needs his stomach pumped or dies from eating too much. At what point could we save his life and at what point will Bob die, 28 Big Macs, 28 – 1/32, 28 – 1/16, 28 – 1/8, 28 – ¼, 28 – ½, and so on. Because we all know Bob couldn’t eat an infinite amount of Big Macs, even if he would like to. =)

      • It’s certainly true that with few (and very limited exceptions) we always rely on not only finite operations with finite sets but rational numbers too. However, one of the only reasons we can do this is by allowing for irrational numbers that “fill out” most of the real number line.
        For some “simple” examples (or fairly easily expressed examples, anyway), consider numbers like pi and e. So many discoveries in the sciences and in engineering require that these numbers have the property of being irrational. Also, to even allow for the possibility of irrational numbers (even just these two), we require knowing how to refer to these numbers in terms of e.g., metrics/distances (where exactly are they in the number line? If we can’t say this, how can we perform basic trigonometric operations involving the multiplication, division, addition, and subtraction of e.g., pi, because we can’t even locate values like 2*sqrt(2).
        A more problematic issue is that, while it is true that we rely on finite computations and rational numbers for any results in any experiment, measurement, etc., we require uncountably infinite sets for almost all probability models. Every single bell-shaped “probability” curve requires an uncountably infinite real-valued interval [0,1] that conforms to this idea that the rationals are a “tiny” infinity compared to the irrationals. Integration (Riemann, Lebesgue, Henstock-Kurzweil, Stieltjes, etc.) also requires the number line to have this structure/nature.
        Quantum mechanics, particle physics (and its foundation- QFT), statistical mechanics, and more all require that there exist uncountably infinite sets and that the real number line consist of mostly irrational numbers which are a larger infinity than the set of rationals. The list goes on.
        That said, it is absolutely true that some of the mathematics of infinity get more than a little theoretical and irrelevant for non-mathematicians. For example, a major problem for mathematicians was whether or not there existed a set larger than countably infinite sets but smaller than the smallest uncountably infinite sets (sets exactly as “large” as the reals). Also, because we provide a proof that the set of all subsets of any set (the “power set” of a set) MUST be “larger” than the set itself, the power set of every infinite set is “larger” than that set. This gives us an infinity of “sizes” of infinite sets. However, outside of very abstract set theory and similar areas of mathematics, this is of little relevance and I can’t think of any offhand examples of any practical relevance of “large” infinite sets much “bigger” than the set of reals. Also, as the rationals are infinitely dense, whenever we wish to use some irrational number for some model, computation, etc., we can use some rational number right next to it.

  2. Jeff says:

    Well that makes irrational numbers very interesting now and makes me wonder why Pi would seem to continue on infinitely, although since Pi is a mathematical constant, it would probably be more accurate to say it continues on indefinitely, rather than infinitely.

    It doesn’t increase in length, size, nor does it get smaller, it’s almost like it falls off the number line at some point, because it continues on indefinitely, in N-Dimensions. You can have different size circles and spheres. So it kind of goes back to probability and set theory, hey, back at full-circle!

    Ha.. Idk why but that is just fascinating now.

    An irrational constant! Just thinking about it is kind of mind boggling. Pi has been conquered! Take that Pi. =D

    • You are right. Continues infinitely isn’t a good description for pi. Rather, to represent the value of pi we require an infinite number of digits. One can say the decimals of pi are infinite (or continue indefinitely), but I agree that “continue infinitely” isn’t really accurate. I would add, though, that whether a number/value is a constant or not doesn’t really matter here. Every number is constant, but we reserve the label “constant” for particularly important values. In other words, “constant” refers more to the importance of a number/value than to its properties (or at least whether or not it e.g., requires an infinite amount of information and infinitely many digits/decimals as pi does).
      It’s also true that that, like any number/value, pi doesn’t increase or decrease. But neither do infinite sets. In fact, they fail to be “larger” in some rather interesting ways. The number line is 1-dimensional I’ll denote by R1 (the real numbers in 1-dimension). Even though this is a line, mathematically we call it a “space” (a 1D space, but a space nonetheless). Consider R10,000,000 (i.e., a real-valued ten million dimensional space). The “size” of this set of all elements in this ten million-dimensional space is exactly equal to the size of R1.
      I like the way you describe pi: “it’s almost like it falls of the number line at some point.” The problem that this property describes almost all numbers. Again, one way to see how this might be true (i.e., to get at the concept without formal proofs and the construction of the reals) is to consider that we can add/multiply to or subtract from/divide pi by any rational number we wish. 4*pi, for example, requires an infinite amount of information and infinitely decimals/digits to represent it. Same with any multiple, any value added to pi, etc.
      It’s also important to note that pi is a point in 1D (R1). If we consider the interval [3,4] which contains pi, we find that pi takes up an infinitesimally small “amount” of this 1D interval. Dimensions, both in physics and in mathematics, are like coordinates. The x,y-plane is 2D because every point requires a two values. For example, there exists a point in this plane (pi, pi) which is located near the point (3.14159265, 3.14159265). Both points exist in 2D space. For basically any N-dimensional space (Euclidean, Minkowskian, complex, etc.), the number of dimensions is determined by how many different numbers/values you need to locate a point (or vector) in that space. In 3D, for example, we need an x, y, & z coordinate for every point.
      I don’t know about conquering pi (I know I haven’t). I have too many problems with the darn number: it’s constantly irrational, an irrational constant, transcendental, and won’t go away. Pie, however, I can deal with (especially apple).

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