Exploring Dividing by Zero in Extended Real Numbers: A Search for Consistency

[Phrak]

I really don't know which section this should be posted in.

I spent some time yesterday dividing by zero in an extended system of real numbers. I haven't found any inconsistancies--yet.

Rather than reinvent the wheel, is there some formal system that I might read about?

 

[Hurkyl]

It's possible that you quite literally reinvented the wheel.

However, the projective real numbers are much more commonly used.

You probably did not rediscover the extended real numbers, because division by zero is still undefined there.

It's possible that you simply reinvented the polynomial, and just gave the variable a funny name (like "1/0").

 

[Phrak]

It's possible that you quite literally reinvented the wheel.

However, the projective real numbers are much more commonly used.

You probably did not rediscover the extended real numbers, because division by zero is still undefined there.

It's possible that you simply reinvented the polynomial, and just gave the variable a funny name (like "1/0").

Wonderful. I'll be looking those over.

I'm not so much interested in dividing by zero per se, but having a consistent pair of bijective maps. The first should map the reals to elements of infinities, and the second map reals to elements of infintesimals.

Other nice properties would be good to have as well, but that's primarily the objective.

 

[Hurkyl]

The first should map the reals to elements of infinities, and the second map reals to elements of infintesimals.

I don't understand what you're trying to say here.

 

[Cantab Morgan]

You might find the hyperreal numbers interesting. These got a bit of play back in the seventies when a mathematician named Robinson wrote a book about them. You might try googling for that, I think the text is online somewhere. The claim is that something very much like hyperreals are closer to what Newton had in mind when he developed fluxions. But calculus was put on a surer footing with reals, and the hyperreals idea (I don't think it was called that in those days) apparently didn't lead to more interesting results.

There was also some near nonsense that made some press a year or two ago, when a British school teacher introduced some kind of new notation for dividing by zero. I can't remember the details, but the hype was quite overblown for what was done.

 

[Dragonfall]

Let's not forget the surreals.

 

[Phrak]

This is almost an embarassment of riches. I had no idea there were so many systems that modify or extend the real numbers. There are also the hyperreal numbers.