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

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In summary, the conversation discussed dividing by zero in an extended system of real numbers and the possibility of reinventing the wheel or discovering new systems such as the projective real numbers, extended real numbers, and hyperreal numbers. They also mentioned the need for consistent bijective maps and other properties. One person suggested looking into hyperreal numbers, which were popular in the seventies but did not lead to significant results. They also mentioned the hype around a British school teacher's new notation for dividing by zero. Another person brought up the surreals as another option, leading to the conclusion that there are numerous systems that modify or extend the real numbers.
  • #1
Phrak
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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?
 
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  • #2
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").
 
  • #3
Hurkyl said:
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.
 
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  • #4
Phrak said:
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.
 
  • #5
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.
 
  • #6
  • #7
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.
 

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

What is the concept of dividing by zero in extended real numbers?

Dividing by zero in extended real numbers is the act of attempting to find a quotient when one of the numbers involved is zero. In extended real numbers, there are two types of division by zero: positive infinity and negative infinity. Positive infinity is when the numerator is positive and the denominator is zero, while negative infinity is when the numerator is negative and the denominator is zero.

Why is dividing by zero in extended real numbers problematic?

Dividing by zero in extended real numbers is problematic because it results in an undefined or infinite value. In mathematics, division by zero is considered undefined because it goes against the fundamental principle of division, which is to find the number that, when multiplied by the denominator, gives the numerator. Furthermore, infinite values are not consistent with the rules of arithmetic and can lead to contradictions and inconsistencies in mathematical equations.

What methods have been used to explore dividing by zero in extended real numbers?

Several methods have been used to explore dividing by zero in extended real numbers. One approach is to use the concept of limits to analyze the behavior of a function as the denominator approaches zero. Another method is to use complex numbers, which allow for division by zero but introduce new challenges and complexities. Additionally, non-standard analysis has been used to explore infinite and infinitesimal values in mathematics.

What are the implications of dividing by zero in extended real numbers?

Dividing by zero in extended real numbers has significant implications in mathematics and other fields that use mathematical concepts, such as physics and engineering. It can lead to contradictions and inconsistencies in mathematical equations, making it challenging to solve problems and reach accurate conclusions. It also highlights the limitations of our current understanding of numbers and mathematical operations.

Is there a consistent way to explore dividing by zero in extended real numbers?

The search for consistency in dividing by zero in extended real numbers is an ongoing topic of research in mathematics. While some methods, such as non-standard analysis, have provided some insights and solutions, there is no universally accepted approach that resolves all the issues surrounding dividing by zero in extended real numbers. Ultimately, it highlights the complexity and mystery that still exists in our understanding of numbers and mathematical operations.

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