Constructing the Real Numbers through Infinite Decimal Expansions

[Skrew]

I was looking at the construction of the real number system.

I know dedekind cuts can be used(completely worthless in terms of understanding I think) and Cauchy sequences can be used (I wish my analysis book used them) but I would like to see a construction based on infinite decimal expansions.

I believe one exists as mentioned on wikipedia and my analysis book which does a really minimal job of it, but I can't find any information which goes into detail on how multiplication and division would be defined, etc.

Could you define it as a sequence of rational numbers in a decimal expansion and do something similar to the Cauchy sequence construction?

When ever I try to study analysis(self study, have not had a course yet) I always get bogged down into these details because I find them so interesting.

 

[micromass]

Yes, I think this is in fact possible. But such a construction would be very technical and ugly, which is probably why nobody ever defines it that way.

However, I really think it'll be a great exercise to work such a thing out for yourself. But you'll run into problem immediately: for example, how would you define the equality of two real numbers??

And I agree with you that Dedekind cuts aren't that pedagogical. However, Dedekind cuts are still used because they offer an advantage which the cauchy definition doesn't do. Namely: it is very easy to generalize Dedekind cuts to more general objects. For example, a completion of lattices works by Dedekind cuts, and the construction of the surreal numbers is by Dedekind cuts...

 

[disregardthat]

One problem with decimal expansion is how to define multiplication. Remember that multiplication is not given, that is a property of real numbers as defined, so you will have to give a formula for the product of two general real decimal expansions. It is not as easy as defining the product of two power series...

I favor the Cauchy-sequence definition for real numbers. Cauchy-sequences are also used to complete rings in algebra.

 

[lavinia]

I wonder if the reals can be defined purely algebraically - say as a maximal ordered field that contains the integers? One would appeal to the Hausdorff maximal principal and then prove uniqueness? Not even sure if this statement is true.

 

[disregardthat]

I wonder if the reals can be defined purely algebraically - say as a maximal ordered field that contains the integers? One would appeal to the Hausdorff maximal principal and then prove uniqueness? Not even sure if this statement is true.

There are many ordered fields containing the integers and the reals. E.g. non-standard reals. How do you define "algebraically", by the way?

 

[Hurkyl]

but I can't find any information which goes into detail on how multiplication and division would be defined, etc.

The elementary school multiplication and long division algorithms should suffice, with only minor tweaking.

My recollection is once you figure out the trick to turn the addition and subtraction algorithms into a well-defined operation on right-infinite sequences of digits, the rest is straightforward.

 

[Hurkyl]

I wonder if the reals can be defined purely algebraically - say as a maximal ordered field that contains the integers? One would appeal to the Hausdorff maximal principal and then prove uniqueness? Not even sure if this statement is true.

I suspect that the field of real numbers is maximum amongst ordered archimedean fields.


As an aside, you may be interested to hear about "real closed fields".



As another aside, the definitions via Dedekind cuts and via Cauchy sequences generally disagree in intuitionistic logic -- in some sense the Dedekind definition is the "right" one.

 

[micromass]

There are many ordered fields containing the integers and the reals. E.g. non-standard reals. How do you define "algebraically", by the way?

Well, yeah, but you need to impose some axioms on the ordered field. For example, we should want the ordered field to be Archimedean. But it will be quite tricky to do this construction.

Furthermore, it would involve the axiom of choice, which would be very undesirable. But lavinia proposed a very interesting idea. If I'm bored, then I will try to work something like that out

 

[Skrew]

The elementary school multiplication and long division algorithms should suffice, with only minor tweaking.

My recollection is once you figure out the trick to turn the addition and subtraction algorithms into a well-defined operation on right-infinite sequences of digits, the rest is straightforward.

But it would be defined as a limit in terms of an infinite number of operations, is that OK?

 

[micromass]

As another aside, the definitions via Dedekind cuts and via Cauchy sequences generally disagree in intuitionistic logic -- in some sense the Dedekind definition is the "right" one.

That's really interesting to hear. You've got any kind of reference on that?

 

[Hurkyl]

But it would be defined as a limit in terms of an infinite number of operations, is that OK?

To define an infinite decimal, it is enough to define each digit. If each digit is the result of an algorithm that terminates, then that certainly counts as a definition.


Now, I doubt things will be so simple -- it's already not that simple for addition. However, there is only one case where the left-to-right version of elementary school addition doesn't work, and it's easy to treat it specially, arriving at a definition of addition in all cases.


Scratch that, it strikes me there is another algorithm; we could indeed define each digit as a pointwise limit, and there would be no special cases.



Aside: if you weren't interested in defining things, but in writing a computer program to do computations, there would be difficulties. Addition is simply not a computable function on strings of decimal digits. This approach to defining the reals would not be a useful one in "constructive analysis".

 

[Skrew]

To define an infinite decimal, it is enough to define each digit. If each digit is the result of an algorithm that terminates, then that certainly counts as a definition.


Now, I doubt things will be so simple -- it's already not that simple for addition. However, there is only one case where the left-to-right version of elementary school addition doesn't work, and it's easy to treat it specially, arriving at a definition of addition in all cases.


Scratch that, it strikes me there is another algorithm; we could indeed define each digit as a pointwise limit, and there would be no special cases.



Aside: if you weren't interested in defining things, but in writing a computer program to do computations, there would be difficulties. Addition is simply not a computable function on strings of decimal digits. This approach to defining the reals would not be a useful one in "constructive analysis".

Also what about defining it as the limit of rational decimals multiplied?

IE, x(n)*y(n), x and y rounded to then nth place.

 

[Hurkyl]

Also what about defining it as the limit of rational decimals multiplied?

IE, x(n)*y(n), x and y rounded to then nth place.

That's an idea, but what limit are you taking? The ordinary limit of calculus? Then you have to show the limit exists. Is it a digit-wise limit? Then you have to show that the digits eventually stop changing.

 

[mathwonk]

I did it this way in a high school class I taught some 20 years ago and I have notes. they are based on the appendix to Spivak's calculus book if you have access to that book. I will also try to make it accessible here if feasible. We had fun working it out.

see post #34 of this thread:

https://www.physicsforums.com/showthread.php?t=466777&page=3

 

[Skrew]

That's an idea, but what limit are you taking? The ordinary limit of calculus? Then you have to show the limit exists. Is it a digit-wise limit? Then you have to show that the digits eventually stop changing.

The limit would be in terms of the decimal place.

You would never be able to show that the digits stop changing but you could write a maximum "error" which would shrink to 0 as the n was made larger.

IE consider 2.678...*3.532...

2.6*3.5 < 2.678...*3.532... < 2.7*3.6

Your value could approximated as (2.6*3.5+3.6*2.7)/2 with max error +/- (2.7*3.6 - 2.6*3.5)/2,

 

[Robert1986]

Or, just do combinatorics and not worry about this stuff :)

Seriously, decimal expansion seems to be the wrong way to go. This is highly biased to whatever base you are working in. I'd be to scared that what I was doing would only work for base 10, first of all, but secondly, I have always disliked messing around with decimal expressions.


This particular part of mathematics really, really, really, bores me. Don't get me wrong, I'm glad that people like you are interested enough so that I may rest easily knowing that the reals are complete; it just don't interest me, personally. However, I prefer the Dedekind Cuts method.

 

[Liangpan]

Hope a preprint on arXiv could give you some help:
http://arxiv.org/abs/1101.1800

Abstract of this preprint:``In this paper we provide a complete approach to the real numbers via decimal representations. Construction of the real numbers by Dedekind cuts, Cauchy sequences of rational numbers, and the algebraic characterization of the real number system by the concept of complete ordered field are also well explained in the new setting"

I was looking at the construction of the real number system.

I know dedekind cuts can be used(completely worthless in terms of understanding I think) and Cauchy sequences can be used (I wish my analysis book used them) but I would like to see a construction based on infinite decimal expansions.

I believe one exists as mentioned on wikipedia and my analysis book which does a really minimal job of it, but I can't find any information which goes into detail on how multiplication and division would be defined, etc.

Could you define it as a sequence of rational numbers in a decimal expansion and do something similar to the Cauchy sequence construction?

When ever I try to study analysis(self study, have not had a course yet) I always get bogged down into these details because I find them so interesting.