Infinity, Pi, and Stranger Things

Date: 7/15/96 at 19:7:31
From: Anonymous
Subject: Infinity, Pi, and Stranger Things

The issue was December 1995 of Discover; the article is called 
"Infinity Plus One, and Other Surreal Numbers"

Martin Kruskal invented this set of numbers. Imagine there's a string 
attached to the top of your head, and you're dangling from it. Now 
imagine that your spine, instead of stopping at your skull and your 
coccyx, goes on forever along that string, extending above your head 
past the farthest star and below your feet through the Earth and 
beyond, never stopping. Now imagine that there's one small problem: 
somewhere along your spine, you have an itch you can't scratch.

"I'll scratch your back if you scratch mine," offers a friendly alien,
positioning its fingernail on the vertebra between your shoulder 
blades. "Just tell me where."

"Up," you say. The alien moves up a vertebra, but it hasn't yet 
reached the itch. "Up, up, up, up, up," you repeat, and each "up" 
inches the alien fingernail one vertebra higher. After 40 or so "up"s, 
you sense the alien is getting close, but at the 44th vertebra, you 
know it's gone too far. The itch is between vertebra 43 and 44. 
"Down," you say. The alien moves its fingernail down half a notch, to 
the point midway between 43 and 44. Unfortunately, it's gone past the 
itch again. "Up," you direct it. It goes up a quarter of a notch this 
time, to the midpoint between its former spot and vertebra 44, 
reaching the place 3/4 of the way between 43 and 44. "Right there," 
you say, and the alien scratches. Ah!

Kruskal's number set uses a string of ups and downs to get any real 
number. For example let's make the symbol for up a "1" and down a "!" 
So 5 would be 11111.  -3 would be! 

Kruskal introduces a number tree which starts at 0 which travels more 
and more to the right the more complex the string is. 1/2 would be 1!.  
-1/4 would be !11.  9 3/4 would be 1111111111!1. Now 1/3 would be 
1!(!1). Actually the parentheses are supposed to be a chord, which 
indicates that the segment is to be repeated "omega" times.

The surreals introduce two terms omega (w) and iota (i). "Omega" is 
the term which we _normally_ call "infinity". Just like Cantor's 
ordinal that describes the collection of all positive integers. This 
spot would be way up at the end of your infinitely long spine. Omega 
is the simplest surreal number larger than all real numbers.

Iota (i) is the simplest infinitesimal surreal, the simplest number 
larger than zero but less than all the positive reals. This would be 
an itch the barest smidgen above zero.

Later on the article says that the alien's back needs to be scratched. 
But the alien has infinite spines (omega spines), omega backs, omega 
bodies, omega populations of bodies, omega collections of omega 
populations, etc. To send you to a spot, say 1/3 of the way between 
the 2nd and 3rd vertebrae on the negative fifth spine on the eight 
back, the alien has but to tell you

((1))((1))((1))((1))((1))((1))((1))(!)(!)(!)(!)111!(!1)

where the extra set of paretheses are an extra "chord" symbol. Well, 
you get the idea?

There are an nested infinities, or an infinite amount of infinite 
amount of infinite amount of infinite amount of infinite...

It also says that every number is rational in the surreal system, even 
the square root of 2 -- just write it as  (sqrt2*w/w). (Remember w is 
the Greek symbol for omega).

There are 2 other terms: extravagance and intravagance. An 
extravagance is a positive number that can't be arrived at by 
performing finitely many algebraic, logarithmic, or exponential 
operations on earlier numbers. The earliest extravagance is omega. An 
intravagance is an enormously medium number, something very, very, 
very deeply embedded between two other numbers..something like Pi! :)

Anyway, enough copying out of the magazine... read the full article :)

Date: 7/15/96 at 19:12:53
From: Anonymous
Subject: Infinity, Pi, and Stranger Things

Hi Dr. Math,


The web is wonderful isn't it?  First I was using the local library
as a dial-up to use their Lynx Web browser, and I was searching 
through the phone directories. I looked up the one for Canada, a site 
called wys.net or something, which would be closed in a couple of 
days! I did a keyword search, looked through the keyword list and 
found "PI, HACKING, AND COMPUTERS" which had links to a HUGE hacker 
site list, other sites, and the "The Uselessness of Pi and its 
irrational friends" page.

I was ecstatic when I saw the page! It was 15 pages full of Pi links,
everything about Pi: pi poems, pi formulae, pi contests. It even has a 
link to a site from which you could d/l 500 million (500,000,000) 
places of Pi! In case you might want to check it out, the URL is:

  http://www.go2net.com/useless/useless/pi.html   

I found a link to this site from the uselessness of pi page. :)

Anyway, I was browsing through the elementary level questions about 
the concept of infinity, and I remembered that in a former issue of 
Discover magazine introduced this new set of numbers called the 
"surreals".

Date: 7/15/96 at 19:17:0
From: Anonymous
Subject: Infinity, Pi, and Stranger Things

I know that pi is a nonterminating decimal, but uhm if there was ever 
a "last digit" (I know this is a paradox, but who cares...), wouldn't 
the last decimal be a "0"?  Because say x = 1.2736. Since adding 
zeroes at the end won't change the value of x, x could = 1.27360. So 
the last decimal of x would be 0?

Date: 7/15/96 at 19:49:48
From: Doctor Sydney
Subject: Re: Infinity, Pi, and Stranger Things

Dear Robert, 

It sure sounds like you are having a good time surfing the net!  We're 
glad you found our site, and we hope you enjoy browsing  through the 
archives.  

I think you already know the answer to your question.  There are two 
kinds of real numbers -- those whose decimal expansions have a finite 
number of nonzero numbers, and those whose decimal expansions are 
nonterminating.  Numbers whose decimal expansions terminate are 
rational numbers, and numbers whose decimal expansions are 
nonterminating are sometimes rational (1/3 = .333...) and sometimes 
irrational.  The decimal expansion of pi is nonterminating since 
only rational numbers have terminating decimal expansions.  Thus, 
there really is no "last digit" in pi.  

When you are adding zeros to the end of a number like x=1.2736 it is 
like adding on 0/10^n, where the 0 is in the nth place to the right of 
the decimal. So, when we write x=1.2736 we are really using shorthand 
notation for the 1 + 2/10 + 7/100 + 3/1000 + 6/10,000.  We can add on 
as many 0/10^n's as we want, and we won't change the value of the 
number at all.

So, I hope this helps, and enjoy the surfing!

-Doctor Sydney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/