Infinity, Pi, and Stranger ThingsDate: 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/ |
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