Re: [math-fun] when and why was it agreed that 0.999...=1?
From: "Huddleston, Scott"
David Wilson asks:
Anyway, I was wondering, does 0.99999... = 1 in the surreal numbers?
That depends on how you map infinite decimals into surreals. For sets of numbers A and B with a<b whenever a in A and b in B, surreal "A | B" is the "simplest" number between A and B. The surreals that have finite sets of reals for A and B are dyadic rationals (including integers). ... A better (IMO) mapping of infinite decimals to surreals is .3333... = {.3, .33, .333, .3333, ...} | {.3+.1, .33+.01, .333+.001, ...}
Let a be 0.33, in A, and let b be 0.3+0.1, in B. a<b does not hold. So "|" is not defined. Phil
On Friday 16 November 2012 11:56:39 Phil Carmody wrote:
From: "Huddleston, Scott" ...
A better (IMO) mapping of infinite decimals to surreals is .3333... = {.3, .33, .333, .3333, ...} | {.3+.1, .33+.01, .333+.001, ...}
Let a be 0.33, in A, and let b be 0.3+0.1, in B. a<b does not hold. So "|" is not defined.
[A] 0.33 < 0.4 = 0.3+0.1 [B]. -- g
I'm sure Senator Huddleston meant .3333... = {.3, .33, .333, .3333, ...} | {.31, .331, .3331, ... } On Fri, Nov 16, 2012 at 2:43 PM, Gareth McCaughan < gareth.mccaughan@pobox.com> wrote:
On Friday 16 November 2012 11:56:39 Phil Carmody wrote:
From: "Huddleston, Scott" ...
A better (IMO) mapping of infinite decimals to surreals is .3333... = {.3, .33, .333, .3333, ...} | {.3+.1, .33+.01, .333+.001, ...}
Let a be 0.33, in A, and let b be 0.3+0.1, in B. a<b does not hold. So "|" is not defined.
[A] 0.33 < 0.4 = 0.3+0.1 [B].
-- g
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GAH! I committed a similar mistake -- they are easy to make. And I misread Senator Carmody's critique, which I now believe to be wrong. On top of that, I'm misattributing everything in sight. Please be kind and ignore me. On Fri, Nov 16, 2012 at 2:48 PM, Allan Wechsler <acwacw@gmail.com> wrote:
I'm sure Senator Huddleston meant
.3333... = {.3, .33, .333, .3333, ...} | {.31, .331, .3331, ... }
On Fri, Nov 16, 2012 at 2:43 PM, Gareth McCaughan < gareth.mccaughan@pobox.com> wrote:
On Friday 16 November 2012 11:56:39 Phil Carmody wrote:
From: "Huddleston, Scott" ...
A better (IMO) mapping of infinite decimals to surreals is .3333... = {.3, .33, .333, .3333, ...} | {.3+.1, .33+.01, .333+.001, ...}
Let a be 0.33, in A, and let b be 0.3+0.1, in B. a<b does not hold. So "|" is not defined.
[A] 0.33 < 0.4 = 0.3+0.1 [B].
-- g
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participants (3)
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Allan Wechsler -
Gareth McCaughan -
Phil Carmody