(June 16, 2020 at 6:17 am)Grandizer Wrote: (June 16, 2020 at 5:20 am)Rahn127 Wrote: Pi= Area of a circle / the radius of that circle ^2
That fraction will never be a whole number.
Yeah, but you're answering a different question from what I'm pondering. Why is the answer so "messy" (irrational) rather than "clean" (rational)? Whole numbers are not the only numbers that are rational anyway.
There are a couple of questions in this one.
The first is what happens in nature. And there, all measurements are rational numbers (in fact, almost all have finite decimal expansions). The mathematics is used as a *model* that has a certain level of accuracy. And, for any given level of accuracy, there are both rational and irrational numbers that meet that level of accuracy. The ones we use are convenient and give, often, several significant digits, but it is impossible to tell is anything 'in nature' is rational or irrational because there are *always* error bars.
The second question is, perhaps, deeper and goes back to Pythagoras. Pythagoras believed, essentially, that all ratios could be described as ratios of whole numbers. It was therefore a deep shock when it was the Pythagorean relation for right triangles that gave one of the first example of an 'irrational ratio'. In fact, the ratio between the diagonal of a square and a side *cannot* be described as the ratio of two whole numbers. This fact challenged the Pythagorean philosophy to its core.
It is a fact about the natural numbers that no square of a natural number is twice the square of a different natural number (unless both are zero). This is usually translated to say that the square root of two is irrational, but it is actually a property of the natural numbers and shows there is no solution of x^2 =2 in the rational numbers.
So, if we *want* to have things like the square root of 2 we need to extend the number system. And, in fact, it is possible to do so. The usual way is to 'fill in the holes' of the rational numbers. There are a variety of ways of doing this formally, but the end result is a *much* larger number system that we call the real numbers. Note: no decimals were harmed in this construction. The particular way of writing numbers in decimal notation is irrelevant to this construction.
And, it is possible to prove that the real number system is *unique* in being a good number system with an order where there are 'no holes'.
It is this extension that makes calculus possible. In particular, integration requires the holes to be filled in to even define the integral (as a limit).
So, the question is why, if you define a real number by some functional equation, or by some limit process, you would find it likely to NOT 'fall into a hole' (in other words, get an irrational real number). If anything, I would find it much, much more fascinating if the numbers that come up turned out to be rational for even moderately interesting situations (real, even x^2 =2 leads to an irrational number).
And, since algebraic numbers are defined in terms of polynomials, one we start thinking about *non-polynomial functions', I don't think it is too surprising that we see non-algebraic numbers (transcendentals) coming up naturally and most of the time.
The number e is defined to make certain calculus constructions involving exponents nice. And since an exponential function is NOT a polynomial, it isn't too surprising, I think, that transcendental numbers arise.
The same is true for trig functions (related to complex exponentials or to areas, which usually lead to transcendentals).
TL;DR; It is more surprising, in many ways, when rather arbitrary real numbers turn out to be algebraic or rational. If you aren't dealing with lines (for rational numbers) or polynomials (for algebraic numbers), you simply don't expect the numbers to be those that arise from lines or polynomials.
(June 16, 2020 at 7:14 am)Grandizer Wrote: Like Rahn, you are answering a different question: a question of how to prove some number is irrational or not a whole number or whatever. I am asking about some deep principle underlying the "messiness" of numbers that seem to be non-arbitrary in terms of its uses and practical values but nevertheless have seemingly arbitrary and random decimal digits.
If we start from the natural numbers, we can add and multiply, but cannot subtract or divide.
We extend to integers to be able to subtract. We then extend to the rational numbers to be able to divide.
But, when we start to divide, do we expect the answer to a generic question to be an integer? Or do we need to go to the 'messy' rational numbers that that point? Recall that the ancient Greeks never got to the place of having rational *numbers*: they talked about ratios, but the identification of ratios with a new type of number happened considerably later.
After rational numbers, we can deal with pretty much anything involving lines. But, once we step away from lines (which may involve division) and start looking at even simply polynomials, the roots are no longer rational. The square root of two is an example.
So, we extend the number system again to include roots of polynomials. This gives us the system of algebraic numbers. This is enough to work with all polynomial equations.
but, if we look at a typical polynomial equation, do we expect to see answers that are rational? of course not! the whole point of the extension was that rational numbers can't deal with roots of polynomials. We *expect* the 'messy' algebraic numbers when dealing with polynomial equations and not just the rational ones.
But now we look at exponential functions. We can compute, algebraically, such exponents as 2^(2/3) or 5^(1/137) and these are perfectly good algebraic numbers. But, finding a root of something like 2^x =3 is impossible for x an algebraic number.
So, we 'complete' the algebraic numbers (which turns out to the equivalent to completing the rational numbers) to allow for the 'holes in the order' to be filled. In essence, we allow for least upper bounds to be found, allowing the solution of equations like 2^x =3.
And, in fact, at this leap, we find we get all sorts of ability that roots (of, say, any continuous function which is negative somewhere and positive somewhere else) for many different sorts of functions.
Also, this is the natural place to be able to do calculus. Things like limits and derivatives, and integrals ALL involve, in the end analysis, finding least upper bounds. So their answers are not *expected* to be algebraic. We extended to the real numbers *because* the solutions of things like this are not algebraic.
TL;DR; The reason we extend out number system at each stage is to allow for more to be computed, proved, and investigated. We don't expect the problems that lead to an extension to be solvable at previous levels of the extension process.
Also, decimals have nothing to do with most of this. They are *one* way to describe the numbers at each stage, but far from the only way to do so. For example, 1/3 does NOT have a finite decimal expansion even though it is rational. Other methods, like continued fractions, can provide insights into the structures here, but are more difficult to use for computation.
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