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## How much is zero divided by zero?

A not-too-serious (but under-the-hood slightly serious) post about very simple mathematics. And it gave me an excuse to try out WordPress.com’s math support.

So, what is the result of $\frac{0}{0}$?

### No value?

The first thing you learn at school is that you can’t divide by 0. After all, 8 divided by 4 loosely means: the outcome is the number of times 4 fits in 8. We then learn about remainders (7 divided by 4 is 1 and the remainder is 3) or non-integer results (7 divided by 4 is 1.75 or 1¾). But if I try to divide 8 by 0, I’m asking: how many times 0 makes 8? And as we know, anything multiplied by 0 is 0, so we are confronted with a contradiction:

$0 \times \mbox{something} = 8 \Rightarrow\\0=8$

This is a way the mathematician tells us that a calculation has no result. We learn: you cannot divide something by zero, it is not allowed as a mechanism in algebra.

### One?

But later in our education we are told the following:

We know that anything (except 0) divided by itself is 1, or as the mathematician writes it: $\frac{x}{x} = 1, \mbox{for }x \neq 0$. So $\frac{1000}{1000}=1$$\frac{100}{100}=1$$\frac{10}{10}=1$, etc. and going on $\frac{0.1}{0.1}=1$$\frac{0.01}{0.01}=1$$\frac{0.001}{0.001}=1$, and so forth. No matter how small we make x, as long as we do not make x 0, the answer is always 1.

We learn the mathematician’s shorthand for this: the limit: $\lim_{x \downarrow 0}\frac{x}{x} = 1$.

We can do the same from the other side of 0: $\frac{-0.1}{-0.1}=1$$\frac{-0.01}{-0.01}=1$$\frac{-0.001}{-0.001}=1$, and so forth, leading to: $\lim_{x \uparrow 0}\frac{x}{x} = 1$.

And combined we can say: $\lim_{x \rightarrow 0}\frac{x}{x} = 1$, the mathematician’s way of saying: if you calculate x divided by x and you make x approach 0 (without actually reaching 0), the result is always 1.

For most of us who were interested in this in the first place, this analysis burns itself into our mind. And we walk away with the feeling that — while it is illegal to divide by 0 — there is a special case: 0 divided by 0 is 1. But we have been tricked, because the answer was provided under the provision that it was not valid for x equal to 0.

So, ‘no value’ still?

### The result

Actually, no. There is, I think, a result. For that, we need to go back to essentials. If we say $\frac{a}{b} = c$, what do we mean? Well, we are looking for that c for which holds $c \times b = a$, right? Now suppose a is not 0 and b is, we are then looking for that c for which holds: $c \times 0 = a$, which leads to $0 = a$ but that is in contradiction with the assumption that a is not 0. This way, we have recreated the original contradiction above.

Bu what if a actually is 0? Then we are looking for that c for which holds: $c \times 0 = 0$. And that is true for every value of c. So, because $\mbox{anyvalue} \times 0 = 0$ we may conclude:

$\frac{0}{0} = \mbox{anyvalue}$. Or in a more mathematical way, because $(\forall c \in \mathbb{R}: c) \times 0 = 0$ we write: $\frac{0}{0} = \forall c \in \mathbb{R}$ or in short:

$\frac{0}{0} = \mathbb{R}$

### What does that mean?

Before discussing that, forget about the idea that what I am doing is illegal. At one time, the whole concept of 0 was considered to be illegal, satanic even. That is why we do not have the year 0, only the year +1 and -1. At some time negative numbers were illegal, the result of 2 minus 3 was undefined, you could after all not give away more than you had (given how bankers have black-magicked with debt, we should have kept it that way…). At some time, there was no answer to calculating the square root of a negative number, until someone just posed the number i as the ‘imaginary’ number that satisfies $i^2=-1$ and whole new avenues of calculation opened up. Here, I’m just taking algebra to its logical conclusion and arguing that we are short-changing ourselves.

If you take the function $f(x) = \frac{x}{x}$, you classically get this graph:

It has a value 1 everywhere, except for when x equals 0. At that point there is a hole in the graph. I’ve put a little hole in to represent that, even if in mathematical reality the ‘hole’ is infinitely small (and the line is infinitely thin). You get the idea.

Now what I do is add the answer for x equals 0 as follows:

Anyway, what does this mean? Well, just that the result of $f(x) = \frac{x}{x}$ is not single-valued for x equals 0. It just means that the result is not a ‘point’, it is a ‘line’. Instead of being dimensionless (which a point is), a line is one-dimensional. So, what I have done here is use 0 divided by zero to create a dimension. The dimensional value $\mathbb{R}$, represented by the vertical line “x=0” perfectly fills that ‘hole’ at (x=0,y=1).

To illustrate the effect of this approach, take for instance $f(x) = \frac{x^2}{x}$. What happens at x=0? Well, if $\frac{x^2}{x} = y$ then algebraically we are looking for that y for which is true that $x^2 = yx$ which is true for any y that is equal to x. The zero in the divisor doesn’t pose a problem anymore.

Which also makes sense if $\frac{0}{0} = \mathbb{R}$ as I stated before. After all: $f(x) = \frac{x^2}{x} = x\frac{x}{x} \Rightarrow f(0) = o\frac{0}{0} = 0\mathbb{R} = 0$. Anything multiplied by 0 is 0, right? During the algebraic transformation we created and squashed an extra dimension.

It also fits with the ‘limit’ approach. $\lim_{x \rightarrow 0} \frac{x^2}{x}$ converges to 0 quickly. Just look at the series $\frac{0.1^2}{0.1}=0.1$$\frac{0.01^2}{0.01}=0.01$$\frac{0.001^2}{0.001}=0.001$, and so forth.

And the graph for $f(x) = \frac{x^2}{x}$? It thus simply looks like the graph for $f(x) = x$, without any ‘hole’:

In calculus (the ‘limit’ way of handling this, the original one with the ‘hole’ in the graph for $\frac{0}{0}$), French Mathematician l’Hôpital created a rule for the limit of something that approaches zero divided by zero (and infinity divided by infinity):

$lim_{x\rightarrow a} \frac{f(x)}{g(x)} = \frac{0}{0} \Rightarrow lim_{x\rightarrow a}\frac{f(x)}{g(x)} = lim_{x\rightarrow a}\frac{f'(x)}{g'(x)}$

In other words, if you end up in a zero-divided-by-zero situation on your limit, you take the limit of the derivative (for those who don’t know what a derivative is, it is the direction of a graph at each point). If that again would deliver zero divided by zero, you take yet another derivative, etc. Note, for a good description of how l’Hôpital’s rule works (which has a wider application), see Paul Dawkin’s page. This is a way to make a determinate answer in cases where that is possible. It is compatible with the algebraic approach above, except for the actual case of $\lim_{x\rightarrow 0} \frac{x}{x}$. There l’Hôpital’s analytic mechanism gives 1 as the answer, and my algebraic method says the answer is $\mathbb{R}$, or ‘all’, which is opposed to current mathematical propriety. See ‘degenerate state’ from physics below.

### Afterword

In 1908, L.E.J. Brouwer wrote his famous article “Over de onbetrouwbaarheid der loogische principes”. There he argued, all of mathematics had been based on three mechanisms:

1. Causality: (if statement A leads to statement B and statement B leads to statement C, then if statement A is true, statement C is true)
2. Negation: (if statement A leads to statement B and statement not-B, then statement A is false)
3. Law of the Excluded Middle (LEM): If statement not-A leads to a contradiction, then statement A is true)

Brouwer argued that number 3 led to unacceptable outcomes and that there really was no reason to accept number 3 and so started intuitionist/constructivist mathematics.

I personally can read these as follows:

1. A mathematical statement has reliable values
2. A mathematical statement cannot have more than a single value (has maximally a single value)
3. A mathematical statement always has a value (has minimally a single value)

Brouwer dropped 3 (and thus effectively introduced the dimension of time in mathematics, but that is another story). I suggest it is possible to drop 2 as well. The only thing that is needed for this is to accept that we can do algebra not only with point-values (dimensionless), but also with line-values (one dimension), or with values in even more dimensions. That is like accepting i, negative numbers, or 0. We’ve done such a thing before, haven’t we?

PS. Physicists are well acquainted with one statement having multiple values. For mathematicians $x=8 \wedge x=0$ is a contradiction, for physicists it is a fact of life. They call that a ‘degenerate state’. By the way: for the physicists among us who are hunting for extra dimensions: take a close look at your formulas where you get zero divided by zero. Extra dimensions may lurk there… 😉

PPS. Apologies to all real mathematicians for loosely formulating. And the math support of wordpress.com? It is poor, this thing with the images is a kludge.

PPPS. Wasn’t this fun?

### 6 Responses

1. […] How much is zero divided by zero? | Gerben Wierda’s Blog […]

2. […] mirror. Fixing a defect is like creating a feature. Subtracting -1 is like adding +1 (I always love a little math for fun — link to an article about calculating zero divided by zero). Fixing technical debt is like […]

3. The 0/0 teaser – like all inderterminate forms – is a nice playground for some math fun. It even teased L’Hopital sufficiently to write a rule on it.

Would be nice if WordPress has decent support for Latex.

• I’ve added a bit about l’Hôpital’s rule to the story. Thanks.

4. […] Fixing a defect is like creating a feature. Subtracting -1 is like adding +1 (I always love a little math for fun — link to an article about calculating zero divided by zero). Fixing technical debt is like […]

5. […] mirror. Fixing a defect is like creating a feature. Subtracting -1 is like adding +1 (I always love a little math for fun — link to an article about calculating zero divided by zero). Fixing technical debt is like […]