The Coverage Problem - thread 3

[Haydon Berrow]

I think we have agreed that the statement of the coverage problem is

There exists a complex number x such that x is not an algebraic integer but n*x is an algebraic integer for all integers n that aren't zero or a unit.

I'd like now to step through the construction in http://somemath.blogspot.co.uk/2015/11/non-polynomial-factorization-short.html and try to understand it if that's alright with you. It's bed time over here so this will be my last post of the day.

Before we can start, what are the definitions of

a) integer-like numbers.
b) primitive quadratic with integer coefficients from the complex numbers to the complex numbers

• What is a test for a number to be integer-like? and
• what does primitive mean in this context?
  

[JSH]

On Monday, November 13, 2017 at 4:14:04 PM UTC-5, Haydon Berrow wrote:

I think we have agreed that the statement of the coverage problem is

There exists a complex number x such that x is not an algebraic integer but n*x is an algebraic integer for all integers n that aren't zero or a unit.

I'd like now to step through the construction in http://somemath.blogspot.co.uk/2015/11/non-polynomial-factorization-short.html and try to understand it if that's alright with you. It's bed time over here so this will be my last post of the day.

Have a good night.

 

Before we can start, what are the definitions of

a) integer-like numbers.

Found an article on Wikipedia think covers. Is SO technical though:

https://en.wikipedia.org/wiki/Integral_element#Integral_closure

To me is like how 2+4i has 2 as a factor, but 1+3i is coprime to 3, with gaussian integers.

Lots like integers, but not rational.

 

b) primitive quadratic with integer coefficients from the complex numbers to the complex numbers

• What is a test for a number to be integer-like? and

See above. 

• what does primitive mean in this context?

Had to refresh a bit. Am thinking in THIS context was just to say quadratic was not a multiple of something.

Like all the coefficients have 2 as a factor or something.

Guess is covered with this article: https://en.wikipedia.org/wiki/Primitive_part_and_content

Seems WAY complicated to me though.

___JSH

[Haydon Berrow]

My last post seems to have been deleted. Was this by accident or deliberate?

I realise that you are not obliged by your rules to give me a reason

[JSH]

On Tuesday, November 14, 2017 at 12:49:57 PM UTC-5, Haydon Berrow wrote:

My last post seems to have been deleted. Was this by accident or deliberate?

I realise that you are not obliged by your rules to give me a reason

Yeah, you top posted, which is against the rules. Something about -- just testin

Sent an email to you, but guess you don't monitor email for this fake account, eh Penny Hassett?

You are now moderated on this group.

The rules are for a reason. Comments should go down here, like this one. And not at top of the post.

Readers find that distracting.

___JSH

[Haydon Berrow]

Oh, sorry, that was a mistake. I wanted to transfer a post from thread 2 to thread 3 because I had accidentally written to the wrong thread. I found I couldn't without writing new text so I wrote something random (actually "just testing" because I was testing whether I could copy if I added to the reply) and then forgot to delete it after I had written at the bottom. I agree, I very much prefer bottom posting. I'll try again.

[Haydon Berrow]

On Tuesday, November 14, 2017 at 10:56:33 AM UTC, JSH wrote:

On Tuesday, November 14, 2017 at 4:57:07 AM UTC-5, Haydon Berrow wrote:

We seem to be at an impasse. The opening sentence of http://somemath.blogspot.co.uk/2015/11/non-polynomial-factorization-short.html is

It can be shown that there must exist additional numbers besides algebraic integers, which are also integer-like numbers.

and the proof starts with

 ...  P(x) is a primitive quadratic with integer coefficients ....

If neither of us knows what integer-like describes then I don't see how I can understand a proof that a number satisfies the definition. Similarly, the proof

I explained to you. I merely gave reference to cover all the bases. Integer-like numbers behave like integers in key ways, like 4+2i has 2 as a factor, and can be coprime with other numbers like 1+3i is coprime to 3. I like using gaussian integers as easy to highlight.

Integer-like numbers are NOT rational numbers that otherwise behave like integers. They also are NOT in any way like fractions.

So 1/(1+3i) is NOT an integer-like numbers.

 

starts with a primitive quadratic so presumably this is needed but, again, if neither of us knows what it means then I don't see how I understand a proof that uses its properties.

It JUST means that there is no common factor to divide across the coefficients in this context.

Like: 175x² - 15x + 2 = 0 is primitive.

5x² - 15x + 20  = 0, is not. Understand now?

 

Can you see a way forward?

Apparently I needed to simplify with you. If I give you too much detail you not only claim to not understand, you decide I don't either!

Remarkable.

___JSH 

 

Ah, OK, so the definition of a primitive quadratic with integer coefficients is that there is no common factor to the coefficients. Another example would be x²+3x+2. I was foxed because you wrote "Am thinking in THIS context was just to say quadratic was not a multiple of something". It may be a difference in language because over here that would imply uncertainty.

I'm still struggling with what integer-like means and I realise that it looks as if I am being difficult, but that's not my intention. I want to settle it before we go further because your explanation says that the construction will produce integer-like numbers and so it must end with "and these numbers are integer-like because .... and .... and ....". I can't find the post, I don't know why, but I think you said the numbers must satisfy key-properties such as being able to define coprimeness.

Question 1) is integer-like a property of a single number or a set of numbers?

Question 2) what is a complete list of the key properties?

To illustrate my confusion, consider the ring Z[π] which I believe to consist of all numbers that can be expressed as a finite sum of integer-multiples of powers of pi. ie all numbers of the form a₀ + a₁π + a₂π² + ... + a_n-1π^n-1 + a_nπⁿ for some integer n. Coprimeness can be defined in this ring, so is π integer-like or is the ring integer-like?

I'm going to have to look up what coprime means, I know that there are subtle problems because prime is not the same as irreducible.

[JSH]

On Wednesday, November 15, 2017 at 7:29:17 AM UTC-5, Haydon Berrow wrote:

On Tuesday, November 14, 2017 at 10:56:33 AM UTC, JSH wrote:

On Tuesday, November 14, 2017 at 4:57:07 AM UTC-5, Haydon Berrow wrote:

We seem to be at an impasse. The opening sentence of http://somemath.blogspot.co.uk/2015/11/non-polynomial-factorization-short.html is

It can be shown that there must exist additional numbers besides algebraic integers, which are also integer-like numbers.

and the proof starts with

 ...  P(x) is a primitive quadratic with integer coefficients ....

If neither of us knows what integer-like describes then I don't see how I can understand a proof that a number satisfies the definition. Similarly, the proof

I explained to you. I merely gave reference to cover all the bases. Integer-like numbers behave like integers in key ways, like 4+2i has 2 as a factor, and can be coprime with other numbers like 1+3i is coprime to 3. I like using gaussian integers as easy to highlight.

Integer-like numbers are NOT rational numbers that otherwise behave like integers. They also are NOT in any way like fractions.

So 1/(1+3i) is NOT an integer-like numbers.

 

starts with a primitive quadratic so presumably this is needed but, again, if neither of us knows what it means then I don't see how I understand a proof that uses its properties.

It JUST means that there is no common factor to divide across the coefficients in this context.

Like: 175x² - 15x + 2 = 0 is primitive.

5x² - 15x + 20  = 0, is not. Understand now?

 

Can you see a way forward?

Apparently I needed to simplify with you. If I give you too much detail you not only claim to not understand, you decide I don't either!

Remarkable.

___JSH 

 

Ah, OK, so the definition of a primitive quadratic with integer coefficients is that there is no common factor to the coefficients. Another example would be x²+3x+2. I was foxed because you wrote "Am thinking in THIS context was just to say quadratic was not a multiple of something". It may be a difference in language because over here that would imply uncertainty.

Yeah I was uncertain. But not that uncertain. Wasn't sure if meant that or also was about reducibility. Was reflecting back on something I used YEARS ago, for efficiency which makes more sense to people who ARE mathematicians. I am not. Eventually came back to me, which it was.

These are things I wrote either a couple of years ago, or even further back, and kept copying as keeps it simpler and maintains accuracy.

 

I'm still struggling with what integer-like means and I realise that it looks as if I am being difficult, but that's not my intention. I want to settle it before we go further because your explanation says that the construction will produce integer-like numbers and so it must end with "and these numbers are integer-like because .... and .... and ....". I can't find the post, I don't know why, but I think you said the numbers must satisfy key-properties such as being able to define coprimeness.

Yeah, early mathematicians struggled as well. With integers most of us have a good concept, right? Well Gauss, clever as he was, threw in a wrinkle with what we now call gaussian integers. Which I like to use when talking integer-like numbers.

My position is that struggling with trying to encompass them, mathematicians in the late 1800's screwed up. So yeah, can be hard to conceptualize.

Integer-like moves away from common human experience, where integers are natural. We literally call them, natural numbers.

 

Question 1) is integer-like a property of a single number or a set of numbers?

Would always be about a set to be meaningful.

 

Question 2) what is a complete list of the key properties?

Such a hard problem! Where my claim is that mathematicians failed, for over a century. Quick answer here? Ok.

The object ring is defined by two conditions, and includes all numbers such that these conditions are true:

1. 1 and -1 are the only rationals that are units in the ring.

2. Given a member m of the ring there must exist a non-zero member n such that mn is an integer, and if mn is not a factor of m, then n cannot be a unit in the ring.

http://somemath.blogspot.com/2005/03/object-ring.html

 

To illustrate my confusion, consider the ring Z[π] which I believe to consist of all numbers that can be expressed as a finite sum of integer-multiples of powers of pi. ie all numbers of the form a₀ + a₁π + a₂π² + ... + a_n-1π^n-1 + a_nπⁿ for some integer n. Coprimeness can be defined in this ring, so is π integer-like or is the ring integer-like?

No. Pi is never integer-like and anything WITH pi is blocked because pi is transcendental so mathematical operations with it, cannot make it.

And with my object ring above, pi was definitely on my mind when figuring that out. Good example though!

 

I'm going to have to look up what coprime means, I know that there are subtle problems because prime is not the same as irreducible.

Oh yeah, technical explanation for coprime is out there to me. With me just means do not share prime factors.

Like 2 and 3 are coprime, while 2 and 4 are not. I like it simple. The reason it gets more complicated is if:

x does not share factors with y, but y shares factors with x

But if x is coprime to y, then neither x shares factors with y, nor y shares factors with x.

See? With integers coprimeness is guaranteed if x does not share factors with y, and y is not a unit, but I like the word.

James Harris 

[Haydon Berrow]

Oh dear, are you saying that they are the same:

Definition: A number is integer-like if and only if it is in the Object Ring?

I struggled with your definition of the Object Ring for years without understanding it. My first problem is that sets are usually defined by saying something like "the set S is a set of things with <this property>". eg

• The set of algebraic integers is the set of numbers x such that x satisfies some monic polynomial with integer coefficients
• The set of numbers x such that x satisfies a monic quadratic with integer coefficients
• The set of numbers x such that x can be written as a sum of three integers
• The set of numbers x such that x can be written as a sum of four integers

and the property is about that element only. After that I then have to prove that the set is a ring. Sometimes it isn't obvious

Your definition isn't like that. Its decision on whether a number x is in it depends on the other numbers in it. I immediately ask myself if I will find numbers x and y such that if one is in it then the other can't be and vice-versa. If I were to write

• the set of integers such that if a is in it, then there is another element b such that a+b is a prime number and if a+b is of the form 4n+1 then b isn't the sum of 3 cubes
  

you would probably have to stop and think whether it is even well-defined. I admit it's not a good example but I can't think of a better

My second problem is that you just declare it to be a ring. I've never seen a follow-up in which you prove that if x and y are in it then so are x+y and x times y.

Musings off the top of my head

• are (1+i)/√2, (1+√5)/2, (1+√7)/2,  1/(1+√2)  integer-like and how would I prove it using this new definition? My first guess was that three of them are and one isn't, and then I remembered that integer-like is a property of sets of numbers and not individual numbers.
  
• why is it *the* object ring, why aren't there lots of rings with this property and are there any at all?
• why isn't pi in the object ring? If I were to take one object ring and append pi and 1/pi to it, why wouldn't I get another ring satisfying all the required properties?

You asked me why I was interested in this. I only want to understand the construction in http://somemath.blogspot.co.uk/2015/11/non-polynomial-factorization-short.html because I couldn't work out what was going on. I think that the Algebraic Numbers form a ring and so I think there must be an error in the calculation of the properties of these new numbers but I'm not interested in finding it and I'm not interested in persuading you that there is an error; just as you aren't interested in showing rotwang the reservations you had in the proof that the Algebraic Integers are a ring. I'll be upset if you say later on that I have a hidden agenda.

Do you think I can understand the construction without understanding the definition of the object ring?

[JSH]

On Thursday, November 16, 2017 at 12:01:44 PM UTC-5, Haydon Berrow wrote:

Oh dear, are you saying that they are the same:

Definition: A number is integer-like if and only if it is in the Object Ring?

I struggled with your definition of the Object Ring for years without understanding it. My first problem is that sets are usually defined by saying something like "the set S is a set of things with <this property>". eg

• The set of algebraic integers is the set of numbers x such that x satisfies some monic polynomial with integer coefficients
• The set of numbers x such that x satisfies a monic quadratic with integer coefficients
• The set of numbers x such that x can be written as a sum of three integers
• The set of numbers x such that x can be written as a sum of four integers

and the property is about that element only. After that I then have to prove that the set is a ring. Sometimes it isn't obvious

Your definition isn't like that. Its decision on whether a number x is in it depends on the other numbers in it. I immediately ask myself if I will find numbers x and y such that if one is in it then the other can't be and vice-versa. If I were to write

• the set of integers such that if a is in it, then there is another element b such that a+b is a prime number and if a+b is of the form 4n+1 then b isn't the sum of 3 cubes
  

you would probably have to stop and think whether it is even well-defined. I admit it's not a good example but I can't think of a better

Well, know that Gauss started things with what are now called gaussian integer: a+bi

THERE can just say 'a' and 'b' are integers, and can easily prove ring of gaussian integer works ok. And Gauss handled that.

However, when trying to move to encompass ALL kinds of numbers where can have key properties of integers, which are yet, not integers, gets really thorny.

What I managed to do was prove that a seeming fix, looking at roots of monic polynomials with integer coefficients, can be shown to fail to cover all such numbers with an easy proof relying now only on a generalized quadratic factorization. While I ALSO have a published paper, which relied on a separate argument relying on cubics.

There has been no error shown in either of these approaches, while I've had fierce arguments over the cubic approach even when simplified to a quadratic one which did on newsgroup sci.math years ago. And yeah, so many arguments, despite the published paper where yes, chief editor tried to delete out later, but that was a social response in my opinion. Have seen no evidence was anything else.

That others would jump up and down on an arbitrary decision as if it says anything about the math, is also just human beings being erroneously human. 

Back to my consideration of solutions, realized that rather than try to say what an integer-like number was, why not focus on excluding numbers that were not?

A simple mental shift!!!

Rather than try to say, let's figure out everything that is integer-like, why not EXCLUDE every number that is not?

That will leave a set of numbers that are.

Is SO clever, and DO say so myself, but guess it takes really wrapping your mind around some IMMENSE abstractions.

 

My second problem is that you just declare it to be a ring. I've never seen a follow-up in which you prove that if x and y are in it then so are x+y and x times y.

Musings off the top of my head

• are (1+i)/√2, (1+√5)/2, (1+√7)/2,  1/(1+√2)  integer-like and how would I prove it using this new definition? My first guess was that three of them are and one isn't, and then I remembered that integer-like is a property of sets of numbers and not individual numbers.
  

If you can find another number that multiplies times those numbers to get an integer, is first step. Understand?

 

• why is it *the* object ring, why aren't there lots of rings with this property and are there any at all?

Because concept is to EXCLUDE all other rings where numbers are NOT integer-like. Rather than try to list everything possible, decided to push out everything NOT integer-like. Is a mental shift in logical reasoning, which I think is rather clever. Even if I do say so myself about my own reasoning.

It's so brilliant actually.

 

• why isn't pi in the object ring? If I were to take one object ring and append pi and 1/pi to it, why wouldn't I get another ring satisfying all the required properties?

Can you give ANY number that multiplies times pi to get an integer, which does not violate the rules given? Answer is, no.

 

You asked me why I was interested in this. I only want to understand the construction in http://somemath.blogspot.co.uk/2015/11/non-polynomial-factorization-short.html because I couldn't work out what was going on. I think that the Algebraic Numbers form a ring and so I think there must be an error

Your belief is irrelevant. I detect an emotional attachment. You should rely on logic only.

If there is an error, as is only a quadratic argument, where?

Logically, if an error exists, you could note a step.

A mathematical proof begins with a truth and proceeds by logical steps to a conclusion which then MUST be true.

Begin at start of the argument: is it true? Proceed then, are steps logical?

If you follow those instructions, then you will be fine. Note point at which you JUMP OFF from following them, and you will see a point of emotion.

Your emotion? Is meaningless to the math. The math does not care.

 

in the calculation of the properties of these new numbers but I'm not interested in finding it and I'm not interested in persuading you that there is an error;

There is, or there is not. There is no persuading.

Belief is irrelevant.

The math is.

 

just as you aren't interested in showing rotwang the reservations you had in the proof that the Algebraic Integers are a ring. I'll be upset if you say later on that I have a hidden agenda.

Your emotion is irrelevant. 

Do you think I can understand the construction without understanding the definition of the object ring?

Yes. Follow instructions as given. And I can keep talking like the Borg from Star Trek fiction. Or like Yoda. Either one works for me!

Your emotion is irrelevant. Your feelings do not matter. The math does not care.

See? Is fun! But seriously, you have an opportunity: study where your FEELINGS intrude, and ask yourself, why.

Report back. Am curious.

James Harris 

[Haydon Berrow]

On Thursday, November 16, 2017 at 5:46:39 PM UTC, JSH wrote:

On Thursday, November 16, 2017 at 12:01:44 PM UTC-5, Haydon Berrow wrote:

The object ring is defined by two conditions, and includes all numbers such that these conditions are true:

1. 1 and -1 are the only rationals that are units in the ring.

2. Given a member m of the ring there must exist a non-zero member n such that mn is an integer, and if mn is not a factor of m, then n cannot be a unit in the ring.

Musings off the top of my head

• are (1+i)/√2, (1+√5)/2, (1+√7)/2,  1/(1+√2)  integer-like and how would I prove it using this new definition? My first guess was that three of them are and one isn't, and then I remembered that integer-like is a property of sets of numbers and not individual numbers.
  

If you can find another number that multiplies times those numbers to get an integer, is first step. Understand?

 

First step ...

• (1+i)/√2, yes. 2*√2/(1+i)
• (1+√5)/2, yes, 4/(1+√5),
• (1+√7)/2, yes, 10/(1+√7)
• 1/(1+√2), yes, 2(1+√2)

what's the next step?

 

 

• why isn't pi in the object ring? If I were to take one object ring and append pi and 1/pi to it, why wouldn't I get another ring satisfying all the required properties?

Can you give ANY number that multiplies times pi to get an integer, which does not violate the rules given? Answer is, no.

2/pi multiplies times pi to give 2, an integer. Is 2 a factor of pi? I don't know but I'm beginning to see the argument.

I can't find the post in which you give some numbers that are integer-like (and so in the object ring). Can you give me 5 examples to chew on?

[JSH]

On Sunday, November 19, 2017 at 12:17:29 PM UTC-5, Haydon Berrow wrote:

On Thursday, November 16, 2017 at 5:46:39 PM UTC, JSH wrote:

On Thursday, November 16, 2017 at 12:01:44 PM UTC-5, Haydon Berrow wrote:

The object ring is defined by two conditions, and includes all numbers such that these conditions are true:

1. 1 and -1 are the only rationals that are units in the ring.

2. Given a member m of the ring there must exist a non-zero member n such that mn is an integer, and if mn is not a factor of m, then n cannot be a unit in the ring.

Musings off the top of my head

• are (1+i)/√2, (1+√5)/2, (1+√7)/2,  1/(1+√2)  integer-like and how would I prove it using this new definition? My first guess was that three of them are and one isn't, and then I remembered that integer-like is a property of sets of numbers and not individual numbers.
  

If you can find another number that multiplies times those numbers to get an integer, is first step. Understand?

 

First step ...

• (1+i)/√2, yes. 2*√2/(1+i)
• (1+√5)/2, yes, 4/(1+√5),
• (1+√7)/2, yes, 10/(1+√7)
• 1/(1+√2), yes, 2(1+√2)

Is the result of any operation on that number a number not allowed, like a fraction? 1/2 is not allowed. Do you know why? 

what's the next step?

 

 

• why isn't pi in the object ring? If I were to take one object ring and append pi and 1/pi to it, why wouldn't I get another ring satisfying all the required properties?

Can you give ANY number that multiplies times pi to get an integer, which does not violate the rules given? Answer is, no.

2/pi multiplies times pi to give 2, an integer. Is 2 a factor of pi? I don't know but I'm beginning to see the argument.

How do you construct pi? 

I can't find the post in which you give some numbers that are integer-like (and so in the object ring). Can you give me 5 examples to chew on?

I could. But choose to not do so.

___JSH

[Haydon Berrow]

On Monday, November 20, 2017 at 3:05:20 PM UTC, JSH wrote:

On Sunday, November 19, 2017 at 12:17:29 PM UTC-5, Haydon Berrow wrote:

On Thursday, November 16, 2017 at 5:46:39 PM UTC, JSH wrote:

On Thursday, November 16, 2017 at 12:01:44 PM UTC-5, Haydon Berrow wrote:

The object ring is defined by two conditions, and includes all numbers such that these conditions are true:

1. 1 and -1 are the only rationals that are units in the ring.

2. Given a member m of the ring there must exist a non-zero member n such that mn is an integer, and if mn is not a factor of m, then n cannot be a unit in the ring.

Musings off the top of my head

• are (1+i)/√2, (1+√5)/2, (1+√7)/2,  1/(1+√2)  integer-like and how would I prove it using this new definition? My first guess was that three of them are and one isn't, and then I remembered that integer-like is a property of sets of numbers and not individual numbers.
  

If you can find another number that multiplies times those numbers to get an integer, is first step. Understand?

 

First step ...

• (1+i)/√2, yes. 2*√2/(1+i)
• (1+√5)/2, yes, 4/(1+√5),
• (1+√7)/2, yes, 10/(1+√7)
• 1/(1+√2), yes, 2(1+√2)

Is the result of any operation on that number a number not allowed, like a fraction? 1/2 is not allowed. Do you know why? 

Yes, trivially, because, for example, I could multiply by  (1+√2)/10  and (1+√2)/10 * 2(1+√2) = 1/5 by the first half of your definition but I don't think this is what you mean.

what's the next step?

 

 

• why isn't pi in the object ring? If I were to take one object ring and append pi and 1/pi to it, why wouldn't I get another ring satisfying all the required properties?

Can you give ANY number that multiplies times pi to get an integer, which does not violate the rules given? Answer is, no.

2/pi multiplies times pi to give 2, an integer. Is 2 a factor of pi? I don't know but I'm beginning to see the argument.

How do you construct pi? 

I don't need to construct pi. We are working in a subset of the real numbers and I appended pi so it is already there

I can't find the post in which you give some numbers that are integer-like (and so in the object ring). Can you give me 5 examples to chew on?

I could. But choose to not do so.

That's unhelpful. I'll choose to abandon my effort to understand the Object Set and go on to try to understand that construction without understanding objects.

[JSH]

On Tuesday, November 21, 2017 at 5:11:29 AM UTC-5, Haydon Berrow wrote:

On Monday, November 20, 2017 at 3:05:20 PM UTC, JSH wrote:

On Sunday, November 19, 2017 at 12:17:29 PM UTC-5, Haydon Berrow wrote:

On Thursday, November 16, 2017 at 5:46:39 PM UTC, JSH wrote:

On Thursday, November 16, 2017 at 12:01:44 PM UTC-5, Haydon Berrow wrote:

The object ring is defined by two conditions, and includes all numbers such that these conditions are true:

1. 1 and -1 are the only rationals that are units in the ring.

2. Given a member m of the ring there must exist a non-zero member n such that mn is an integer, and if mn is not a factor of m, then n cannot be a unit in the ring.

Musings off the top of my head

• are (1+i)/√2, (1+√5)/2, (1+√7)/2,  1/(1+√2)  integer-like and how would I prove it using this new definition? My first guess was that three of them are and one isn't, and then I remembered that integer-like is a property of sets of numbers and not individual numbers.
  

If you can find another number that multiplies times those numbers to get an integer, is first step. Understand?

 

First step ...

• (1+i)/√2, yes. 2*√2/(1+i)
• (1+√5)/2, yes, 4/(1+√5),
• (1+√7)/2, yes, 10/(1+√7)
• 1/(1+√2), yes, 2(1+√2)

Is the result of any operation on that number a number not allowed, like a fraction? 1/2 is not allowed. Do you know why? 

Yes, trivially, because, for example, I could multiply by  (1+√2)/10  and (1+√2)/10 * 2(1+√2) = 1/5 by the first half of your definition but I don't think this is what you mean.

Fractions are excluded. Like if 1/2 is in the ring, then 1/2(2) = 1, correct? Then are units in the ring. But only 1 or -1 are allowed to be rationals that are units in ring.

Idea is to exclude ANY numbers that violate the rules, versus try to include integer-like numbers.

Can see you're struggling with the concept.

Numbers that violate the rules? Or lead to a violations? Cannot be in the ring of objects.

Ring of objects works by exclusion. Flipping the problem on its head. As by excluding any numbers that break the rules, it automatically includes those that do not.

 

what's the next step?

 

 

• why isn't pi in the object ring? If I were to take one object ring and append pi and 1/pi to it, why wouldn't I get another ring satisfying all the required properties?

Can you give ANY number that multiplies times pi to get an integer, which does not violate the rules given? Answer is, no.

2/pi multiplies times pi to give 2, an integer. Is 2 a factor of pi? I don't know but I'm beginning to see the argument.

How do you construct pi? 

I don't need to construct pi. We are working in a subset of the real numbers and I appended pi so it is already there

Let me make up a number--bizarro. Now append bizarro to a ring! What is the number?

You need rules for construction. You say--pi. That is two letters. You think you know what pi is.

I ask you to construct it, and you say you appended it. Same can be said for, bizarro. 

I can't find the post in which you give some numbers that are integer-like (and so in the object ring). Can you give me 5 examples to chew on?

I could. But choose to not do so.

That's unhelpful. I'll choose to abandon my effort to understand the Object Set and go on to try to understand that construction without understanding objects.

Ok.

___JSH 

[Haydon Berrow]

On Tuesday, November 21, 2017 at 3:56:59 PM UTC, JSH wrote:

On Tuesday, November 21, 2017 at 5:11:29 AM UTC-5, Haydon Berrow wrote:

On Thursday, November 16, 2017 at 12:01:44 PM UTC-5, Haydon Berrow wrote:

The object ring is defined by two conditions, and includes all numbers such that these conditions are true:

1. 1 and -1 are the only rationals that are units in the ring.

2. Given a member m of the ring there must exist a non-zero member n such that mn is an integer, and if mn is not a factor of m, then n cannot be a unit in the ring.

1/(1+√2), yes, 2(1+√2)

Is the result of any operation on that number a number not allowed, like a fraction? 1/2 is not allowed. Do you know why? 

Yes, trivially, because, for example, I could multiply by  (1+√2)/10  and (1+√2)/10 * 2(1+√2) = 1/5 by the first half of your definition but I don't think this is what you mean.

I should have expressed that more carefully.

• I know that 1/5 is excluded by the part of the definition that says "1 and -1 are the only rationals that are units in the ring"
• Yes there is an operation that produces a fraction and it's trivially obvious. Just multiply by (1+√2)/10. This produces a fraction because (1+√2)/10 * 2(1+√2) = 1/5
  

I'm beginning to wonder if there are any numbers in the Object Set except for the integers

[JSH]

Progress! So then, why is sqrt(2) in it? Can you find ANY contradiction with its use? No.

The abstraction is actually not hard. For instance, we are humans. You can try to describe humans, correct? Or you can exclude other animals.

Which is easier? Shifting to another realm to help you ponder an abstraction that is logical. So please answer. Would you prefer to describe humans to find the set? Or exclude other animals? Or maybe something of both?

That is, the logic of the abstraction of exclusion to determine inclusion in a set is independent of numbers. It can be used WITH numbers, but numbers are just one area where such an abstraction works.

If you can grasp the abstraction then you can understand the object ring. If you cannot then you will not.

___JSH

[Haydon Berrow]

Just checking, but are you saying that

• 1/(1+√2) is not in the Object Set, but
• √2 is in the Object Set
  

[JSH]

Yeah. You're struggling with the concept: exclusion rules.

YOUR idea earlier can be given as: how do you include all numbers that are integer-like?

My position is--that's too hard. Why not ask: how do we EXCLUDE numbers that are not?

Here is a question for you: how do you know 1/2 is NOT an integer?

What if I ask you, why can't I just append 1/2 to the ring of integers and still have integers?

If you say, because humans say-so, you're lost. Human beings don't matter. I can say the moon is, whatever, so? Why mathematically is that the case?

What makes an integer, mathematically?

Here at least you can find reference. And yeah, you can also see examples of people appending 1/2.

How does that change things? When done, why can you NO LONGER say you're talking about integers without contradictions arising?

 

___JSH

[Haydon Berrow]

On Thursday, November 23, 2017 at 3:39:14 PM UTC, JSH wrote:

On Wednesday, November 22, 2017 at 2:30:05 PM UTC-5, Haydon Berrow wrote:

Just checking, but are you saying that

• 1/(1+√2) is not in the Object Set, but
• √2 is in the Object Set
  

Yeah. You're struggling with the concept: exclusion rules.

 

You are absolutely right, I'm struggling.

• I gave some numbers and asked how I determined if they were in the Object Set.
• You asked if I could do an operation to each of them to produce an integer.
• I said yes, here are numbers that you can multiply by to get an integer (and we don't know whether they, themselves, are in the Object Set)
• You said that showed they weren't in the set
• I can do the same with √2 and yet that doesn't show that √2 isn't in the set

I'm almost certain that there is a mistake somewhere and 1/(1+√2) is in the set after all

[JSH]

On Friday, November 24, 2017 at 4:20:21 AM UTC-5, Haydon Berrow wrote:

On Thursday, November 23, 2017 at 3:39:14 PM UTC, JSH wrote:

On Wednesday, November 22, 2017 at 2:30:05 PM UTC-5, Haydon Berrow wrote:

Just checking, but are you saying that

• 1/(1+√2) is not in the Object Set, but
• √2 is in the Object Set
  

Yeah. You're struggling with the concept: exclusion rules.

 

You are absolutely right, I'm struggling.

• I gave some numbers and asked how I determined if they were in the Object Set.
• You asked if I could do an operation to each of them to produce an integer.
• I said yes, here are numbers that you can multiply by to get an integer (and we don't know whether they, themselves, are in the Object Set)
• You said that showed they weren't in the set

Really? Are you sure? If so, was I right, or wrong?

• I can do the same with √2 and yet that doesn't show that √2 isn't in the set

I'm almost certain that there is a mistake somewhere and 1/(1+√2) is in the set after all

Can you give a reason for it not to be? Exclusion rules. Does it break any rules?

If it does NOT break any rules, and the set includes all numbers that do NOT, then, is it in the set or not?

And you did NOT answer any of my requests for you, so will answer one myself.

Like say you append 1/2 to the ring of integers, but have exclusion: only 1 and -1 are rationals that are units!

Contradiction as now 2 is a unit as 2(1/2) = 1. Therefore, you can rely on the exclusion.

___JSH

[Haydon Berrow]

On Friday, November 24, 2017 at 4:11:32 PM UTC, JSH wrote:

On Friday, November 24, 2017 at 4:20:21 AM UTC-5, Haydon Berrow wrote:

On Thursday, November 23, 2017 at 3:39:14 PM UTC, JSH wrote:

On Wednesday, November 22, 2017 at 2:30:05 PM UTC-5, Haydon Berrow wrote:

Just checking, but are you saying that

• 1/(1+√2) is not in the Object Set, but
• √2 is in the Object Set
  

Yeah. You're struggling with the concept: exclusion rules.

 

You are absolutely right, I'm struggling.

• I gave some numbers and asked how I determined if they were in the Object Set.
• You asked if I could do an operation to each of them to produce an integer.
• I said yes, here are numbers that you can multiply by to get an integer (and we don't know whether they, themselves, are in the Object Set)
• You said that showed they weren't in the set

Really? Are you sure? If so, was I right, or wrong?

Fair comment. I can't find why I thought this, I must have misunderstood somewhere. Here's the bit of the definition that applies

2. Given a member m of the ring there must exist a non-zero member n such that mn is an integer, and if mn is not a factor of m, then n cannot be a unit in the ring.

If you're willing I'll try with just three cases and return to the others later

case A) m= √2 , is it in the Object Set?

• does there exist a number n such that mn is an integer? Yes, n=√2 is an example
  
• does there exist a member n such that mn is an integer? I don't know because I don't if √2 is in the set
• is 2 a factor of √2? I don't know because I don't know if  1/√2 is in the set

case B) m= (1+√2), is it in the Object Set?

• does there exist a number n such that mn is an integer? Yes, n=2/(1+√2) is an example
  
• does there exist a member n such that mn is an integer? I don't know because I don't if 2/(1+√2) is in the set
• is 2 a factor of (1+√2) ? I don't know because I don't know if  (1+√2)/2 is in the set

case C) m= 1/(1+√2), is it in the Object Set?

• does there exist a number n such that mn is an integer? Yes, n=2(1+√2) is an example
  
• does there exist a member n such that mn is an integer? I don't know because I don't if 2(1+√2) is in the set
• is 2 a factor of 1/(1+√2) ? I don't know because I don't know if  1/(2(1+√2)) is in the set

[JSH]

(1+√2)(1-√2)  = -1

So is a unit. It multiplies times another number in the ring to give an integer. And -1 is a factor.

Passes all tests.

___JSH

[Haydon Berrow]

I see that (1+√2) is a unit but I don't see how it passes test 2. How do you know that (1-√2) is in the object set?

[JSH]

The object ring includes ALL numbers such that the rules are not broken. So any number is included, if is not excluded.

So is included as is not excluded. Now say, 1/2. Is excluded as 2(1/2) = 1, making 2 a rational unit. Its inclusion breaks a rule.

___JSH

[Haydon Berrow]

On Monday, November 27, 2017 at 4:47:40 PM UTC, JSH wrote:

The object ring includes ALL numbers such that the rules are not broken. So any number is included, if is not excluded.

Where does that definition come from. It's not the one I've been using:

From <https://web.archive.org/web/20160220152708/http://somemath.blogspot.co.uk/2005/03/object-ring.html>

The object ring is defined by two conditions, and includes all numbers such that these conditions are true:

1. 1 and -1 are the only rationals that are units in the ring.

2. Given a member m of the ring there must exist a non-zero member n such that mn is an integer, and if mn is not a factor of m, then n cannot be a unit in the ring.

I see that rule 1 fits your description and it removes all the rational fractions. Rule 2 is not what you just described. It says that if a number m is to be in the 'ring' then there must be another number n such that mn is an integer. It doesn't say anything along the lines "m is not in the ring if it fails a test", it says m is in the ring if it passes this test. Are you using some other definition that I don't know of?

[JSH]

Is right there, with that includes all numbers part. The object ring definition includes all numbers not excluded, by those rules.

For example 2 is included, as is not excluded by those rules. But 1/2 is excluded as 2(1/2) = 1, which would make 2 a unit in the ring.

___JSH 

[Haydon Berrow]

I'm still struggling. Rule 2 of the definition is

2. Given a member m of the ring there must exist a non-zero member n such that mn is an integer, and if mn is not a factor of m, then n cannot be a unit in the ring.

Suppose it had been

2a. Given a member m of the ring there must exist a non-zero number n such that mn is an integer, and if mn is not a factor of m, then n cannot be a unit in the ring.

would that have made a difference?

[JSH]

Yes. The object ring includes ALL numbers such that the rules apply. However, there are numbers which are NOT in the object ring, such as 1/2. So a member of the object ring is needed. Note that we can say 1/2, but in the context of the object ring, how do you construct it? Far as the object ring is concerned, 1/2 does not exist. I just use it to help explain.

___JSH

[Haydon Berrow]

You've used the phrase "constructing a number" before but I hoped I wouldn't have to understand it.

What does it mean to construct a number in the context of a ring? Are sqrt(2) and sqrt(3) constructible in the context of the object set?

[JSH]

You can construct a number using mathematically valid operations and operators. For example can add i to 1, to construct: 1+i.

___JSH 

[Haydon Berrow]

This is new.

• which operations are allowed?
• which numbers are allowed in the construction?
• what does "in the context of X" mean?
• why does this crop up with the object set? It isn't mentioned in the definition.

[JSH]

Ring operations. 

• which numbers are allowed in the construction?

Members of the ring. 

• what does "in the context of X" mean?

Huh? 

• why does this crop up with the object set? It isn't mentioned in the definition.

Ring operations are allowed. A construction involves using valid ring operations with members of the ring.

For example: 1 and i, construct to 1+i

Valid ring operation: addition

Members: 1 and i

___JSH 

[Haydon Berrow]

On Thursday, November 30, 2017 at 9:24:58 AM UTC, JSH wrote:

• which operations are allowed?

Ring operations

so, addition and multiplication. but not division and not square root

• which numbers are allowed in the construction?

Members of the ring. 

• what does "in the context of X" mean?

Huh? 

you wrote "Note that we can say 1/2, but in the context of the object ring, how do you construct it?" up there ^^^

• why does this crop up with the object set? It isn't mentioned in the definition.

Ring operations are allowed. A construction involves using valid ring operations with members of the ring.

Yes, but what has that to do with the definition of the object set? Construction and constructibility isn't mentioned in the definition but it seems to be important.

Neither sqrt(2) nor pi seem to be constructible by your explanation

[JSH]

You may never understand, ok?

I'm ok with that. 

___JSH

[Haydon Berrow]

OK