October 12th, 2017, 02:40 PM  #21 
Senior Member Joined: Sep 2016 From: USA Posts: 167 Thanks: 79 Math Focus: Dynamical systems, analytic function theory, numerics 
1. I haven't raised 0 to any negative power. This is simply notation for zero's inverse. You are assuming that it is possible to satisfy the equation $x\cdot 0 = u$ where $u$ is a unit. Note that EVERY element in a field other than zero is a unit. I have shown this is impossible in a field, hence why we don't allow 0 to have an inverse. 2. Please read this: https://en.wikipedia.org/wiki/Welldefined 3. I still don't understand what $z_1,z_2$ are. You can't define the product of 2 elements to depend on a 3rd. This makes multiplication no longer a binary operation. 
October 12th, 2017, 03:58 PM  #22  
Member Joined: Aug 2012 Posts: 51 Thanks: 0  Quote:
But in any case I am not claiming (x = 0^1) is the inverse of 0. I have stated to you very clearly on a previous post that 1 is the inverse of 0 and 0 in the inverse of 1. I can define the product of two elements to depend on a third. This is called relativity. It specifically makes "binary" multiplication relative.... If you have trouble understanding z1 and z2 consider the following then.. Let z1 be the multiplicative property of 0 Let z2 be the multiplicative identity property of 1 Let zero posses both properties Let only one property be used in any binary expression If both numbers are 0 then z1 must be used as default. A x 0(z1) = 0 0 x A(z1) = 0 A x 0(z2) = 1 0 x A(z2) = 1 no tables necessary....any better for you? we then extrapolate for division and then for multiplicative inverses....  
October 12th, 2017, 05:38 PM  #23 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,595 Thanks: 620 Math Focus: Wibbly wobbly timeywimey stuff. 
Conway, I think you are misunderstanding what SDK is driving at. He is mainly trying to tell you that your construction is not a field. Your take on this system is perfectly allowed so long as you realize that. Dan 
October 12th, 2017, 05:47 PM  #24  
Member Joined: Aug 2012 Posts: 51 Thanks: 0  Quote:
Perhaps. Perhaps not. What is the point of your reply? Would you care to regulate this idea to some "new" field like meadows....fine....as you said... "Your take on this system is perfectly allowed".... I have no other way to interpret that other than to say you therefore see some form of "validity" in this idea....  
October 12th, 2017, 07:26 PM  #25  
Math Team Joined: May 2013 From: The Astral plane Posts: 1,595 Thanks: 620 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Dan  
October 12th, 2017, 07:51 PM  #26  
Member Joined: Aug 2012 Posts: 51 Thanks: 0  Quote:
Well now....my my....this is a far cry from how you have treated me on other forums. As well as your original post on this thread.... "I notice that, though you certainly may ask questions on any forum, you haven't changed anything since that last long conversation on MHB. What new thing do you expect here?" Dan That said....THANK YOU....seriously. To other matters... I have never really intended to debate what sort of "construction" this is. I have only intended to validate the mathematical conclusions of the given axioms. This you have agreed has been done. Now...I do consider it a field...(maybe falsely so) 1. "The multiplicative property of 0" is a valid property in a field 2. "The multiplicative identity property of 1" is a valid property in a field 3. If a unique solution can be shown for the "use" of both of these properties regarding zero....then this "construct" is valid in a field. This doesn't really carry us all the way though does it. We must do the math....as SDK showed earlier with his equations. However his equations relied on the assumption that (A * 0 = A ) in all cases. My reply here being that I can at "will" change the properties of zero to fit the desired product in the equation. IF and ONLY IF the solutions are unique. So can you pose another equation showing contradiction.... The key here is to remember the expression (A * 0 ) is relative in product. Thank you P.S. Should you wish to discuss "applicability" I would love to. But perhaps that is better done in another thread. Also I remind you of a "list" of goals I posted on another forum. These goals being your "applicability". Last edited by Conway51; October 12th, 2017 at 07:58 PM.  
October 12th, 2017, 08:37 PM  #27  
Math Team Joined: May 2013 From: The Astral plane Posts: 1,595 Thanks: 620 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Quote:
The existence of a multiplicative inverse in the field means that, for any given element $\displaystyle a \neq 0$ of the field we have an element $\displaystyle a^{1}$ belonging to the field. You say that you have a multiplicative inverse for 0. According to the above axiom that can't happen in a field. What you have is a field plus one more condition...I have no idea what to call it. But no matter what the nomenclature that means you don't have a field. End of story. Dan Last edited by topsquark; October 12th, 2017 at 08:41 PM.  
October 12th, 2017, 08:54 PM  #28  
Member Joined: Aug 2012 Posts: 51 Thanks: 0  Quote:
"Let no "ordered pair" be represented by another further "ordered pair" See Dan I do try to learn. I DID change something. You clearly missed it because you didn't read it. You were only interested in trolling me. On your second point.... The existence of a multiplicative inverse in the field means that, for any given element a≠0 of the field we have an element a^1 belonging to the field. I agree...there however is not problem in the mathematics if you use zero as z1 instead of z2. I have shown this repeatedly. So we keep the stament and only add the element 0 has an inverse of 1 Set this aside for a moment. Then consider with me Dan...if the idea is valid "except" as a field...then reapply the ideas of space and value as z2 and z1. Then "field" or otherwise I could care less... What it does mean is that division by zero is then defined. Multiplication by zero is relative Multiplicative inverses for 0 and 1 Varying amounts of 0 As well as a new way in which to consider mathematics..... Field or otherwise...... Last edited by Conway51; October 12th, 2017 at 08:58 PM.  
October 13th, 2017, 12:32 AM  #29  
Math Team Joined: May 2013 From: The Astral plane Posts: 1,595 Thanks: 620 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
And it's good that you aren't calling it a field. I am good with that. SDK and others can worry about how your axioms work. As I've said before I don't know much about constructions with a multiplicative inverse for 0. Dan  
October 13th, 2017, 05:13 AM  #30 
Senior Member Joined: Jun 2015 From: England Posts: 676 Thanks: 194 
Conway, when you are proposing a new mathematical object/entity this doubling up stuff is counter productive. Have a look at the biography of Lord Hamilton. When he introduced a new mathematical entity that was a number on steroids so it was more than a number, more than a vector, he had the good grace to give it a new name and symbol so that no confusion would arise. He is also famous for carving the theorem onto a milestone in Dublin with his penknife as he thought of it. The quaternion. Others have done this when introducing a single element extension to the number system, but quaternions are a whole new system in themselves. In order to accommodate the new material, the framework of algebra had to change and be extended. This process has happened many times over the history of Mathematics. Someone who is a first rate modern mathematician who specialises in this stuff has written a really good, but rigorous, book at about your level, called Unknown Quantity about all this. His name is John Derbyshire. Read this  it would help you redevelop your ideas into a format that could (if it really is any good) be incorporated into algebra. It also explains stuff about Fields and that you are clearly unaware of. As matters stand at the moment you have a communications barrier with others to surmount, that is preventing you moving forwards. It is up to you to deal with this, others cannot do it for you, they can only offer help and advice. Last edited by skipjack; October 13th, 2017 at 07:15 AM. 

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