Defining division by zero [duplicate]

I have looked through some of the previous questions posted on this topic, and I think mine is different.

Is there a flaw in defining division by zero? For example, define

$\frac{a}{0} = \infty_a$

it would seem like things work now, for example,

$\frac{a/0}{b/0}=\frac{\infty_a}{\infty_b}=a/b$.

What could go wrong with this idea, or more specifically, is it defined in some branch of mathematics?

Solution 1:

Since $0+0=0$, we have $a=0\cdot \infty_a=(0+0)\cdot\infty_a=0\cdot\infty_a+0\cdot\infty_a=a+a$. So $a=a+a$, and therefore $a=0$.

It certainly makes life simpler to have everything equal to $0$.

Solution 2:

I'm not going to consider $a=0$ (i.e. the quotient $\frac{0}{0}$) as we can construct functions in calculus limit initially tends to $\frac{0}{0}$ after some work turns out to be any real number we'd like. It may be the case that allowing $\infty_0$ could be useful, for example we have $0 = 0 \cdot \infty_0$, but I don't consider it here.

I noticed in André Nicolas' post, he showed that allowing division by zero ends up being trivial when you assume that the new elements you add obey the distributive law. So I conclude that if these new numbers are well-defined, then we can't use the distributive property with them.

What follows are a few thoughts in that regard.

Assume that $a \neq 0$ and define $\frac{a}{0} = \infty_a$ for some element $\infty_a$. Certainly $\infty_a$ is not a real number, so let us extend $\mathbb{R}$ to a new set $\mathbb{R}^{\dagger}$ which includes every $\infty_x$ for all $x \in \mathbb{R} \setminus \{0\}$.

What properties will this expanded set $\mathbb{R}^{\dagger}$ have? We decided above that it should not have the distributive property.

Well, we know that $\frac{a}{0} = \infty_a$ so perhaps $a = 0 \cdot \infty_a$. This strikes us as odd, because we know for any real number, multiplication by zero always yields zero. So we are at a crossroads. We can do one of two things:

(1): We can say "the new set $\mathbb{R}^{\dagger}$ must follows the rules of multiplying by zero" in which case we would derive $a=0$, which would be a contradiction (remember, we assumed $a \neq 0$ in the beginning). If we enforced this restriction, we would find our new set of numbers paradoxical and then throw them out.

(2): Allow this strange property of zero in this new set and accept all the consequences for its use.

Here is one consequence of (2):

Proposition: If $\mathbb{R}^{\dagger}$ is associative and commutative, then it contains only three elements.

Proof: Let $a, b \in \mathbb{R} \setminus \{0\}$. Now since we assumed (2), we know $a=0 \cdot \infty_a$ and $b = 0 \cdot \infty_b$, so we consider the product $ab = (0 \cdot \infty_a) (0 \cdot \infty_b)$.

We can write this product in two ways:

$$(i): (0 \cdot \infty_a) (0 \cdot \infty_b) = (a \cdot 0) \cdot \infty_b = 0 \cdot \infty_b = b, $$

but on the other hand

$$(ii): (0 \cdot \infty_a) (0 \cdot \infty_b) = \infty_a (b \cdot 0) = \infty_a \cdot 0 = a.$$

We conclude that $a=b$. So we have $\mathbb{R}^{\dagger} = \{0, a, \infty_a \}$.

Perhaps we should not assume $\mathbb{R}^{\dagger}$ is not commutative or not associative or not both, then...

Solution 3:

Not "flaw", rather, "could be defined differently". Introducing new definitions for division by 0, including new numbers, follows a traditional path trod by others. For general background, see Patrick Suppes' Introduction to Logic, Chapter 8, Sections 5 and 7, titled respectively The Problem of Division by Zero and Five Approaches to Division by Zero.[a] Specific examples defining division by 0 include Meadows[b], Wheels (Carlström, 2004), Wheels (Setzer, 1997), as well as the extended complex plane and its relatives that use one or more point(s) at infinity (e.g., the (affinely) extended real numbers), or Möbius transformations (i.e., the exact real arithmetic of Edalat and Potts).[c] Meadows, following Suppes' approach 3, is alone in not using a new definition.

Introducing a new type of number as you suggest follows Suppes' fourth approach, the one he says is "most consonant with ordinary mathematical practice". Like you, I've thought about introducing unique quotients, but there are reasons it's not normally done with 0 - the arithmetic becomes very limited. To avoid at least some of these limitations, I've played around with going a step further. I've replaced 0 with a redefined zero from which unique quotients arise.

A couple of things that I've done so far to define a different number of nothing:

Used a different notion of nothing. Placeholder zeros provide a guide. They indicate the absence of some particular things, as, for example, 0 does the other nine digits. So a replacement zero could represent the absence of the real numbers or some other set of numbers. The importance of the idea of "absence" allows for "presence" and I'll try to show the importance of this for unique quotients.
Used a different notation. The notation, while awkward, makes it simple to see how to divide according to the usual rules for division.

To accomplish 2, the new zero will be in two parts. One part will indicate "absence" and the other part will indicate, in this case, the reals. Division, and only division, will change the absent into the present. Otherwise, the new, "absent", zero functions just like 0. So

Part A. introduce a bar to indicate absence
Part B. use the reciprocal of the Roger Penrose / John A. Wheeler / John Wallis definition[d] for $\infty$

So the notation $/1/\infty $ or $ \overline{\frac{1}{\infty}} $ would read "the absence of the reciprocal of the reals".

An example of obtaining a(n) unique quotient.$$ \frac{4}{\overline{\frac{1}{\infty}}} = \infty_4 $$

Division inactivates the absence bar, thus making the absent present. Now we can apply the usual rules for complex fractions. The "revealed" reals then extend orthogonally from 4. Geometrically, this would be a line that extends out from the point at 4 on the real number line.

This zero does not have a reciprocal because $$ \overline{\frac{1}{\infty}} \times \infty_4 $$ still equals zero. Yet, by definition, $\infty_4 $ has a reciprocal - the "free" or "revealed" part of the new zero. $$ \infty_4 \times \frac{1}{\infty} = 4 $$ With repeated division a plane can be constructed, as well as other objects. Basic arithmetic operations seem to make it possible to construct $n$-real-dimension space. Speaking visually, with a redefined number zero it becomes possible to carry out arithmetic operations, not only with points on the real number line, but also with lines, planes, and other higher dimensional constructs.

A less undeveloped version of the somewhat simplified arithmetic introduced here that I've been playing around with may be found in my paper Replacing 0.

Footnotes.

[a] Sections 8.7 and 8.5 begin on pages 181 and 184 of the linked PDF, not the pages listed in the table of contents.

[b] ArXiv has papers by a number of mathematicians using "meadows" in the title. An overview of their research program and links to papers is here http://staff.science.uva.nl/~janb/FAM/topFAM.html

[c] Links to referenced authors can be found in the bibliography to the paper "Replacing 0" linked at the end of my answer above and on my user profile.

[d] Penrose defines $\infty$ as an array of the real numbers. John Wallis, the inventor of the infinity symbol, did this implicitly.

Solution 4:

What you want to look at is called nonstandard analysis. You don't divide by zero but by infintesimals. See http://en.wikipedia.org/wiki/Non-standard_analysis

Solution 5:

Presumably for non-zero $a$ you have $\frac{1}{\infty_a} = \frac{0}{a} = 0$, so this suggests for non-zero $a$ and $b$ that $\frac{1}{\infty_a} = \frac{1}{\infty_b}$ which in turn suggests ${\infty_a} = {\infty_b}$ and so perhaps $a=b$. More briefly $\frac{1}{\infty_a} = 0$ suggests $\infty_a = \frac{1}{0} = \infty_1$.

I doubt you intend this, so at some stage the distinctions you are trying to create or the manipulations you hope to preserve are lost.