[math-fun] real numbers as infinite decimals
Do any of you know of books or articles that construct the real numbers (and their arithmetic) in terms of their binary (or decimal) expansions? I know of just two: the article "The real numbers as a wreath product" by Faltin, Metropolis, Ross, and Rota; and Gowers' web-screed "Real numbers as infinite decimals", aka "What is so wrong with thinking of real numbers as infinite decimals?". But I suspect there are others. Do you know of anyone else who has done anything along these lines? or even anyone who has given a conscientious definition of what's really involved when you add two infinite decimals and perform carries? Note that for this kind of analysis, 0.999... is not the same as 1.000..., even though both represent the same real number. (Here's a fun fact I haven't seen anywhere: if you add two (positive) infinite decimals, the only way their sum can end in infinitely 0's is if both summands do. Have you seen this before?) Any references or suggestions you can offer will be appreciated! Jim Propp
Chapter 16 of Ken Ross' "Elementary Analysis: The Theory of Calculus" doesn't construct them, but does provide an elementary proof that there is a bijection between real numbers and decimal expansions (after the choice of 9-tail or 0-tail is made if necessary for a given real number). Scott On 2/11/2010 11:15 PM, James Propp wrote:
Do any of you know of books or articles that construct the real numbers (and their arithmetic) in terms of their binary (or decimal) expansions?
I know of just two: the article "The real numbers as a wreath product" by Faltin, Metropolis, Ross, and Rota; and Gowers' web-screed "Real numbers as infinite decimals", aka "What is so wrong with thinking of real numbers as infinite decimals?". But I suspect there are others.
Do you know of anyone else who has done anything along these lines? or even anyone who has given a conscientious definition of what's really involved when you add two infinite decimals and perform carries? Note that for this kind of analysis, 0.999... is not the same as 1.000..., even though both represent the same real number. (Here's a fun fact I haven't seen anywhere: if you add two (positive) infinite decimals, the only way their sum can end in infinitely 0's is if both summands do. Have you seen this before?)
Any references or suggestions you can offer will be appreciated!
Jim Propp
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I have a vivid memory of performing this construction as an exercise, I think when I was a senior in high school. I was motivated by annoyance at the Cauchy-sequence construction, and was certain that the formal-decimal-expansion approach would prove much more straightforward. I don't remember whether it was, in fact. On Fri, Feb 12, 2010 at 8:38 AM, Scott Beaver <sfbeaver@minetfiber.com>wrote:
Chapter 16 of Ken Ross' "Elementary Analysis: The Theory of Calculus" doesn't construct them, but does provide an elementary proof that there is a bijection between real numbers and decimal expansions (after the choice of 9-tail or 0-tail is made if necessary for a given real number).
Scott
On 2/11/2010 11:15 PM, James Propp wrote:
Do any of you know of books or articles that construct the real numbers (and their arithmetic) in terms of their binary (or decimal) expansions?
I know of just two: the article "The real numbers as a wreath product" by Faltin, Metropolis, Ross, and Rota; and Gowers' web-screed "Real numbers as infinite decimals", aka "What is so wrong with thinking of real numbers as infinite decimals?". But I suspect there are others.
Do you know of anyone else who has done anything along these lines? or even anyone who has given a conscientious definition of what's really involved when you add two infinite decimals and perform carries? Note that for this kind of analysis, 0.999... is not the same as 1.000..., even though both represent the same real number. (Here's a fun fact I haven't seen anywhere: if you add two (positive) infinite decimals, the only way their sum can end in infinitely 0's is if both summands do. Have you seen this before?)
Any references or suggestions you can offer will be appreciated!
Jim Propp
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On Fri, Feb 12, 2010 at 10:26 AM, Allan Wechsler <acwacw@gmail.com> wrote:
I have a vivid memory of performing this construction as an exercise, I think when I was a senior in high school. I was motivated by annoyance at the Cauchy-sequence construction, and was certain that the formal-decimal-expansion approach would prove much more straightforward. I don't remember whether it was, in fact.
In some sense, the difference is that the Cauchy-sequence construction defines an equivalence relation on Cauchy sequences, while the "infinite decimals" construction chooses a canonical representative for each equivalence class (except for rationals with certain denominators, where it ends up choosing two canonical representatives instead. In defining multiplication on infinite decimals, it's easy to define a sequence that represents the product, but you then have to find an equivalent canonical sequence. If this isn't significantly easier than first showing that the sequence is Cauchy and then showing that *every* Cauchy sequence is equivalent to a canonical sequence, then completing the infinite-decimals construction pretty much includes proving that the two constructions are equivalent. Or is there a cleverer way to define multiplication of infinite decimals that I'm missing? You can take the products of the finite decimal approximations, but I don't see an easy way to show that each digit is ultimately constant as you take this sequence. Andy
I regret that I do not remember the details; reconstructing them now would basically be working from scratch. Obviously the well-definedness of multiplication can be proven in the decimal construction; the only question is how ugly the proof is. The Cauchy construction has the burden of first constructing the rationals; the decimal construction avoids that step, but again, I'm not sure if it turns out to be worth it. On Fri, Feb 12, 2010 at 10:40 AM, Andy Latto <andy.latto@pobox.com> wrote:
On Fri, Feb 12, 2010 at 10:26 AM, Allan Wechsler <acwacw@gmail.com> wrote:
I have a vivid memory of performing this construction as an exercise, I think when I was a senior in high school. I was motivated by annoyance at the Cauchy-sequence construction, and was certain that the formal-decimal-expansion approach would prove much more straightforward. I don't remember whether it was, in fact.
In some sense, the difference is that the Cauchy-sequence construction defines an equivalence relation on Cauchy sequences, while the "infinite decimals" construction chooses a canonical representative for each equivalence class (except for rationals with certain denominators, where it ends up choosing two canonical representatives instead.
In defining multiplication on infinite decimals, it's easy to define a sequence that represents the product, but you then have to find an equivalent canonical sequence. If this isn't significantly easier than first showing that the sequence is Cauchy and then showing that *every* Cauchy sequence is equivalent to a canonical sequence, then completing the infinite-decimals construction pretty much includes proving that the two constructions are equivalent.
Or is there a cleverer way to define multiplication of infinite decimals that I'm missing? You can take the products of the finite decimal approximations, but I don't see an easy way to show that each digit is ultimately constant as you take this sequence.
Andy
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Wrt multiplication being well defined, the key point to establish is that an increasing sequence, with an upper bound, has a limit. This is annoyingly non-constructive, but repairing it means discussing "computable reals" and doing detailed error bounds. --- I'm curious about variations of the decimal->real construction. These variations are mostly about Addition, although Sub/Mul/Div and Compare (& Limit) can sometimes be carried over to the variation. The decimals are a map from Z -> Z10, with some side constraints. We can replace Z with N, and get integers; or play with the side constraints, and get N back. We can replace Z10 with Z2 &c; or interleave Z6 and Z10 and make clock time. We can build two-digit numbers (Z100) from single digits (Z10) by introducing a Carry-Function on the units digits: C(3,6)=0, while C(5,5)=1, with the domain being the tens digit group. The tens digit group might be Z6, while the units digit group is Z10. Can we use other groups besides Zn? Non-cyclic or non-abelian groups? Semigroups? How about more kinds of Carries? What are the rules for Carry functions? What do we need for Multiplication to work? We might replace the shelf-space backbone Z with other things: partial orders, or directed graphs with cycles. This would mean propagating carries to some other place than "the next position to the left". In building Z100 from Z10 & Z10, the carries out of the tens place are simply dropped. Simple modifications, like replacing Z10 with Z2, just regenerate the Reals. Replacing Z10 with i-1 gives Complexes. Changing the rules a little gives 10-adics. Each has a new feature, topology. Are any of the other variations interesting? Rich ----- Quoting Allan Wechsler <acwacw@gmail.com>:
I regret that I do not remember the details; reconstructing them now would basically be working from scratch. Obviously the well-definedness of multiplication can be proven in the decimal construction; the only question is how ugly the proof is.
The Cauchy construction has the burden of first constructing the rationals; the decimal construction avoids that step, but again, I'm not sure if it turns out to be worth it.
On Fri, Feb 12, 2010 at 10:40 AM, Andy Latto <andy.latto@pobox.com> wrote:
On Fri, Feb 12, 2010 at 10:26 AM, Allan Wechsler <acwacw@gmail.com> wrote:
I have a vivid memory of performing this construction as an exercise, I think when I was a senior in high school. I was motivated by annoyance at the Cauchy-sequence construction, and was certain that the formal-decimal-expansion approach would prove much more straightforward. I don't remember whether it was, in fact.
In some sense, the difference is that the Cauchy-sequence construction defines an equivalence relation on Cauchy sequences, while the "infinite decimals" construction chooses a canonical representative for each equivalence class (except for rationals with certain denominators, where it ends up choosing two canonical representatives instead.
In defining multiplication on infinite decimals, it's easy to define a sequence that represents the product, but you then have to find an equivalent canonical sequence. If this isn't significantly easier than first showing that the sequence is Cauchy and then showing that *every* Cauchy sequence is equivalent to a canonical sequence, then completing the infinite-decimals construction pretty much includes proving that the two constructions are equivalent.
Or is there a cleverer way to define multiplication of infinite decimals that I'm missing? You can take the products of the finite decimal approximations, but I don't see an easy way to show that each digit is ultimately constant as you take this sequence.
Andy
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Some items that might have relevance: 1. Redundant (non-unique) representations of numbers, a la Booth -- e.g, utilizing _three_ binary digits: +1, 0 -1 when encoding numbers. Thus, long series of "11111" can be represented by +1,0,0,0,0,-1 to avoid long series of carries. 2. One's complement arithmetic has always had the problem of two representations for zero. Gosper showed that the universe is twos-complement in Hakmem #154. I guess the bottom line is: where is it written that number representations must be unique? Non-unique representations seem to have more flexibility & perhaps more efficiency, so we're always going to have to deal with equivalence classes. Even IEEE floating point has signed zeros and signed infinities. At 11:15 PM 2/11/2010, James Propp wrote:
Do any of you know of books or articles that construct the real numbers (and their arithmetic) in terms of their binary (or decimal) expansions?
I know of just two: the article "The real numbers as a wreath product" by Faltin, Metropolis, Ross, and Rota; and Gowers' web-screed "Real numbers as infinite decimals", aka "What is so wrong with thinking of real numbers as infinite decimals?". But I suspect there are others.
Do you know of anyone else who has done anything along these lines? or even anyone who has given a conscientious definition of what's really involved when you add two infinite decimals and perform carries? Note that for this kind of analysis, 0.999... is not the same as 1.000..., even though both represent the same real number. (Here's a fun fact I haven't seen anywhere: if you add two (positive) infinite decimals, the only way their sum can end in infinitely 0's is if both summands do. Have you seen this before?)
Any references or suggestions you can offer will be appreciated!
Jim Propp
participants (6)
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James Propp -
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Scott Beaver