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Plowing though .debate and a couple numbers caught my eye! (below)
In lugnet.off-topic.debate, Larry Pieniazek <lar@voyager.net> writes:
> [..]
> > Try expressing 22/7
> > as a decimal (I *won't* wait)
>
> Here you go: 3.1 <base 7> (since 22 is 31 in base 7, dividing by the
> base merely shifts the base point over one place) or
> 3.142857<bar>142857</bar> <base 10> Those are both valid expressions..
Larry, IMBW, but I think John might've meant "pi" when he said "22/7" --
at least, I know I've heard people accidentally refer to pi in that manner
before.
John,
pi = 3.14159265358979323846264338327950288...
22/7 = 3.142857142857...
22/7 is just a rough approximation to pi; it happens to come within ~.04% of
the actual value, so it's a pretty good one that's easy to remember.
Another cool one is 333/106, which happens to come within ~.0026% of the
actual value:
333/106 = 3.141509433962...
> > It is infinite, and therefore you can't,
> > because you can't express the infinite in finite terms. Period.
> > That's not circle logic. It's a fact. Facts compute, no? ;-)
>
> You're pretty confused here, and you're not going to get very far trying
> to use number theory. The fact that when 22/7 is expressed using base 10
> you get an infinite repeating sequence of digits:
> 3.142857<bar>142857</bar> is quite predictable using science, it's a
> clear derivation from the fundamental properties of numbering systems.
> Infinity is a well understood concept, and one that we can reason about
> and make predictions about. In fact 22/7 is an example of a point on a
> countable infinity. Contrast that with the irrational numbers, which are
> points in an uncountable infinity. But that doesn't make them
> un-understandable. It merely means we need different tools. [...]
John, Larry's right about the infinities thing in finite terms. The
terminology is a bit confusing though. When it's said that something is a
"countable infinity," the word "countable" here really means "enumerable."
Nothing in the universe[1] could actually _count_ (as in _finish_ counting)
an inifinite set, but you can construct a system of enumeration which
assigns ordinal counting numbers to each member of the set.
Whole number fractions like 1/4, 22/7, and 333/106 are part of a "countable"
infinity because the numerator and denominator can be arranged on a two-
dimensional lattice grid and enumerated from the lower-left corner zig-
zagging outward (or spiralling outward if you include negatives).
OTOH, irrational numbers like sqrt(2), e, and pi are not part of a countable
infinity because (it's been proven that) there is no way to enumerate the
irrational numbers.
So some infinities are bigger infinities than others. While there are an
infinite number of whole-number fractions, and an infinite number of
irrational numbers, you might say that there are "more" irrational numbers
than there are whole-number fractions because there's no way to enumerate
the irrational numbers. However, even though it might seem as though there
are more fractions than counting numbers, there are the same number of
counting numbers {0, 1, 2, 3, ...} as there are whole-number fractions,
because they can be paired one-to-one.
It's true that irrational numbers like sqrt(2) and pi go on forever and ever
in their decimal expansions, and that the entire universe working forever
and ever to compute the entire sequence would never complete. So does it
mean that they can't be fully understood? Well, no. They can still be
fully understood in many cases (not all cases). Many (not all) irrational
numbers can be expressed algorithmically in terms of rational numbers, and
some neat things come out of that. For instance, you can divide sqrt(8) by
sqrt(2) and the answer is _exactly_ 2, even if you can't ever express
sqrt(8) or sqrt(2) as decimal numbers. So you can actually manipulate
infinite numbers. Another way of looking at the same example is to say that
[sqrt(8) - sqrt(2)] ^ 2 = 2. That's pretty hard to verify via brute-force,
but pretty easy to verify with algebra.
Note that there are some irrational numbers which can't be expressed
algorithmically in terms of rational numbers. I forget if this subset is a
larger or a small infinity than the ones which can be, but it's a really
fascinating subset of irrationals which gets into random number territory.
--Todd
[1] Except God or mathematics.[2][3]
[2] Is God part of the Universe, or is he outside of it, or is God the
Universe itself?
[3] Is mathematics part of the Universe?
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