🚀 go-pugleaf

In article <5gmqsn$bs8@milo.vcn.bc.ca>, Mark Gibson <mgibson@vcn.bc.ca>
writes
>Martin Ward (martin@ward-hull.demon.co.uk) wrote:
>: Pi day
>: 
>: 
>: *22 July* should be Pi Day.  22/7 = 3.142857...  is the good
>: approximation to pi known since the days of the ancient Greeks.
>: 
>A very clever and entertaining post. I agree that July 22 should be Pi
>Day. Incidentally it is also the Deast of St. Mary Magdalene, patron saint
>of prostitutes and "fallen" women in the Calendar of the Catholic Chu7rch.
>
>But as for 22/7 being "close" to pi, "close" is a pretty bague term.

Yes, but the point is that it is as close a rational approximation as we
can get with a small denominator.  In this case with denominator <= 12.

>Compared with the spiral nebula in Andromeda, Betelgeuse is pretty close
>to Earth but don't hold your breath waiting for some airline to announce
>weekend excursions there.
>
>3.14159265379 is a closer approximation to pi that I usually use. But that
>isn't very close compared with the approximation developed by the guy who
>used a computer to work it out to millions of sig figs. 
>
>The most fascinating thing about all irrational numbers is that no matter
>how hard we try we can never really get "close" to them. The number itself
>always recedes over the horizon away from our attempts at precision.
>That's why IMHO there really can't be any Pi day, e day, sqrt(2) day, etc.

We can get as close as we like to an irrational in the sense that if you
say get within a millionth of pi, I can produce a rational number that
close; and if you say a billionth I can get that close too; and so on to
any degree of accuracy you want.

What we cannot do is state an irrational *exactly* as a fraction or
decimal.
>
>Incidentelly I'm sure it must have ocurred to someone that there are
>DEGREES of irrationality among the irrational numbers. For axample some
>irrational numbers when subjected to a simple operation can be mapped to a
>rational number. e.g. sqrt(2)*sqrt(2) = 2. Whilst as far as I know e*e and
>pi*pi, e*pi, etc are still irrational. Since this ocurred to a
>mathematical dunce like me it can't be that important but I though it was
>interesting, so there.
>
>MSG
>---
>THEY'RE COMING TO TAKE ME AWAY, HA-HA
>THEY'RE COMING TO TAKE ME AWAY...

You are quite right about this, and it is an important insight.
Irrational numbers like sqrt(2) are said to be algebraic, because they
are the roots of an algebraic equation (with rational coefficients).  In
this case:

        x^2 - 2 = 0

Irrational numbers like e and pi which are not the root of any such
equation are said to be transcendental.  (It is rather hard to *prove*
that pi and e are transcendental).
-- 

Martin Ward

_________________________________________________________________________

"When *I* use a word", Humpty Dumpty said in rather a scornful tone, "it
 means just what I choose it to mean - neither more nor less".
"The question is", said Alice, "whether you *can* make words mean so many
 different things".
"The question is", said Humpty-Dumpty, "which is to be master - that's all".

"Through the Looking-Glass" - Lewis Carroll.
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