# Laws of Zero

This page is based on an essay by Shadowjack, and may be read in its original form at this permanent link. Here, comment is interspersed; any user may comment constructively, the goal being education. Please sign comments. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

For the Wikipedia article, see w:Division by zero. The Wikipedia article is in the form of an essay, written without inline references; it represents "common knowledge," which is how many Wikipedia articles were originally written. —Abd (discusscontribs) 14:54, 30 December 2013 (UTC)

Laws of Zero

This article states the laws of zero as groups zi, zii, ziii, ziv, zv and in each of the groups are contained four units designated as ' " "' "". If that is not what is clear on reading this article then please be sure to check the "view history" file in order to find the original article.

The link at the top of the page points to the article as last edited by Shadowjack. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

The particular matter that has prompted this note is that students in different schools are prohibited from dividing by zero. The error is to confuse can not with must not, and the difference between rules and orders.

"Prohibited" is a social phenomenon, not a mathematical one. Division by zero is well−defined in most systems, particularly through approach to zero as a limit. Division of zero by another integer is also well−defined. What is undefined, usually stated as "indeterminate" is division of zero by zero, because N/D, with D −> 0 and N finite and not zero, approaches infinity, whereas 0/N remains zero for all values of N other than zero. As we will see, Shadowjack appears to rely on the latter relation, and thus declares that 0/0 equals 0.
Students are "prohibited" from avoiding division by zero because the result can be indeterminate; if it is treated as determinate, certain major errors can result. Shadowjack does not consider this problem.
We may define division from multiplication. If A * B = C, then C / A = B and C / B = A. The quotient of C / A can be defined as that number which when multiplied by A equals C. We may need to expand this definition slightly to handle division by zero. That is, there may be more than one number which satisfies the relation. The answer to the problem may be a set of numbers, or infinity.
If we ground mathematics in the study of existent reality, there is no zero outside of counting, and there is no infinity except as a pure concept. "Infinite" and "infinitesimal" are not measurable. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

In this theory, rules are a close approximation to actual laws built in to the construction either of nature or alternatively of the nature of the system. Orders are commands that are applied for some political reason, that need have no reference to nature nor to truth.

What Shadowjack states here is a general epistemological or ontological principle. The problem I see is that, while setting aside the "political," he substitutes his own ideas as "nature" or "truth." He is not alone in this! It is a fundamental epistemological error to confuse interpretation with reality. Shadowjack has not, so far, grounded his discussion in nature and reality, but simply manipulates ideas in ways that seem logical to him. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

The logical necessity of the sign labels and the system of mathematics is that we show division by zero, even if we are not exactly sure what that calculation gives. It is easily possible that division by zero gives a not obvious result similar to Q ! or e.

He has not stated this. Q is not defined here. e is a constant, an irrational number, which, however, is generally defined through approach to limit (the same as pi). Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

My own preferred answer is to say that division by zero either gives the result zero or alternatively does nothing and returns the original object number. Either answer is satisfactory to the logical necessity of showing the calculation.

There is no natural observation which involves division by zero. The problem arises entirely in language. Shadowjack, here, acknowledges that there is more than one "answer," "satisfactory" to logical necessities. The necessity is invented. From what I have stated above, we can see that a general answer to 0 / 0 is that it returns, not the "original object number" (I do not yet find Shadowjack's meaning clear), but the set of all numbers, unlimited (so infinity is, effectively, included.) That is, any answer from the set of all numbers satisfies the underlying multiplication, because 0 * N = 0. If N is infinite, the check is impossible to verify. We are, again, outside of the natural reality, we are looking at pure thought only, invented.
Does Shadowjack's answer, differing from my proposal, remain logically consistent? I don't know as I write this. I am, with most of us, a student. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

The only wrong answer in this system that is not by its nature obviously incorrect requires some explanation.

"Wrong" is undefined. This may seem picky, but if we are going to propose that most people are "wrong," we'd better be clear about what we mean. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

That is to make the usual mistake of people who touch on infinity mathematics to think that calculations by zero may reflect calculations by infinity, leading to the compounded mistake that zero equals infinity.

Let's see if he shows that. The concept of zero and that of infinity are linked, but, obviously, zero is not infinity. Rather, if we have N / Z, with any positive N, the quotient grows without limit as Z approaches 0. Hence we readily say that N / 0 is infinite.
The problem, of course, is what happens as N approaches zero. N / N is 1, regardless of the value of N, we easily think. That is an error, because if N equals 0, the quotient is *undefined*. It could be any number and satisfy the relation. I'll repeat, this problem arises only in language, the language of mathematics. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

Calculations by zero and calculations by infinity do not give equivalent answers, although they must give coherent non-contradictory answers.

What I think may be the correct line into the infinity mathematics is to hypothesise that zero divided by zero approximates infinity. The problem is that a/a=1 and 1/1=1 but we know that 0/0 =/= 1. So we say that 0/0=0, even though we cannot justify the arbitrary change in rules.

But if 0/0 = ∞ that would make sense in regard to the requirement to use a higher order language to explain a calculation that we cannot prove at the level of the given mathematical signlabels. And if we then show the set equivalence we obtain ɸ < ɸ => ∞.

Now that may actually work because we are saying if the empty set contains the empty set then infinity.

This is not a coherent sentence. He is referring to what he does not explain. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

Since that could be a necessary consequence of the empty set containing the empty set.

There are two meanings of zero. Shadowjack may be confusing them, here. Zero is the count of elements contained in the empty set. Zero is also a quantity on a continuum. The former is an integer, a count; the latter is a number, a value. Counted quantity exists as a basic distinction in language; it depends on the definition of objects to be counted, and "object" is, again, a creation of mind. It's quite a basic one! Objects are abstractions, resulting from the distinguishing of A and not−A. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

Furthermore, if 0/0=/=1 and 0/0=/=0 and 0/0=∞, then we prove 0=/=∞ and 1=/=∞, which we would like to do.

Sure! What fun! I'm not sure what Shadowjack's notation means here. If 0 / 0 = {the set of all numbers}, then to give a specific value is to equate the set of all numbers with a particular number, which will then lead to obvious contradictions. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

To help oneself understand the calculations intuitively the following approach is useful: That is, to specify the difference between nothing, unity and multitude.

I'm going to assert that there is no difference. Why? "Because it's fun," and because doing so can lead to some very interesting insights. I'd never claim that this was "true." Merely useful. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

Given one can intuitively suppose that nothing, unity and multitude must each be different to the other then specific understandings are coherent and non-contradictory. We can describe the existential operator infinity is an unlimited multitude for some kind of quantity, which is different to the universal operator infinity which is unlimited multitude of all of some kind of quantity. It means we are able to think about different kinds of infinity as infinity is applied to different measures of quantity.

That's done. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

One appropriate correct definition of infinity is: Bounded unlimited multitude.

"Bounded" is undefined, though he tries: Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

We say bounded, because particular. We say unlimited because if existant then without end. We say multitude because not nothing or unity.

How particular? If it is infinite and existent, yes, without end. But it is then everything, and it is then a unity, because if anything is outside of it, it is limited. It is itself. And it is precisely and exactly like nothing.
Now, there is a way out of this mess. The number of real numbers contained in the interval 0 to 1 is equal to the number of real numbers in any interval, that can be shown through one to one correspondence. So we can define a limited but unbounded set. It is limited by the bounds, but the count of members is unlimited.
Once again, we can see how the problem arises entirely in language. There is no real experiment that we can perform that produces an infinite count, or that measures infinitesimal values. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

In terms of minutae please note that our standard idea of a/a=1 only holds if a is a positive integer. If a is a negative integer or zero then all of the argumentation as to division of negatives by negatives or division by zero must be applied to a.

That's strange: −a / −a = 1 as well. Division by negatives is not a problem, if division is defined as the inverse of multiplication. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)
This is another Shadowjack essay, it has been moved to [1] and blanked, the link is to the last unblanked version at this point. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

zi': 0 + 0 = 0

zi": 0 - 0 = 0

zi"': 0 . 0 = 0

zi"": 0 / 0 = 0

This is the only unconventional claim in this series of identities. It effectively would equate, in practice, zero with the set of all numbers, because the set of all numbers satisfies the relation stated. Zero does not uniquely satisfy it. I have the sense that it can be shown that this proposed identity can lead to preposterous results, but I don't intend to do that today. If I've seen an example, it was over fifty years ago. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

zii':0 . 1 = 0

zii": 1 . 0 = 0

zii"': 0 / 1 = 0

zii"": 1 / 0 = 0

Multiplying both sides of the identity by 0, and assuming that A / B * B = A (which can be the very definition of division), we end up with 1 = 0. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

ziii': 0 . a = 0

ziii": a . 0 = 0

ziii"': 0 / a = 0

ziii"": a / 0 = 0

Ditto. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

ziv': 0 . ~a = 0

ziv": ~a . 0 = 0

ziv"': 0 / ~a = 0

ziv"": ~a / 0 = 0

Again. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

zv': 1 - 1 = 0

zv": a - a = 0

zv"': ~a - ~a = 0

"Not A?". The meaning is unclear to me. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

zv"": a + ~a = 0

Let's say that the meaning is very unclear. Shadowjack has other pages where he gives logical notation, perhaps those might clear this up, but it looks incorrect. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

If the question is :"Why state the given laws of zero ?"

The problem is if not stated then we suppose that all in the whole world agree with the same not stated laws of zero even though we have no possible way of knowing what all in the whole world agree with.

There can certainly be a value in the study of what is routinely assumed without careful distinctions. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

An existing mistake is the earlier conditional statement that:

"What any one person knows, every person knows."

Only if we allow "every person" to "know" through the one person. Where was this "earlier statement"? It is not in this essay. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

And "If one person knows, then everyone knows."

What can be said without violating ordinary common sense is that "what one person knows is known by the set of all people." Notice that a possible error arises in equating "the set of all persons" with "everyone," in the latter, it is implied that each member of the set of all persons knows the matter. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

And "If everyone does not know, then no one knows."

That's just plain ridiculous. Some people may think that what they do not know is not known by anyone, but only in certain narrow areas where they imagine their knowledge is comprehensive. It's ridiculous even there, because it assumes a form of omniscience. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

Which then means that everyone knows the laws of thought.

This would be a strange meaning for "knows." Everyone *experiences* what may be called the laws of thought, that is, everyone who thinks. But experience is not identical to knowledge. We may have knowledge without experience and experience without knowledge. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

But everyone does not know the laws of thought.

Indeed. Some people, faced with declarations from those who have studied and practiced "laws of thought," deny them. That denial is itself a product of the "laws of thought," as I would use the concept. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

And as stated it is not necessary that all intelligent decision makers must agree with each of the individual laws as stated. Using the laws of zero in this document we can provide examples.

Sensible. However, Shadowjack could be considering agreement/disagreement, as if agreement and disagreement are "decisions," i..e, choices, that must be made. Underneath Shadowjack's explanations and examples in this document is his way of thinking. he is following his own "laws of thought," those that apply to him. Or is he? Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

0 + 0 = 0 Zero plus zero is zero.

0 - 0 = 0 Zero minus zero is zero.

0 . 0 = 0 Zero multiplied by zero is zero.

0 / 0 = 0 Zero divided by zero is zero.

Again, the single idiosyncratic term.

These laws define zero. And the nature of zero as defined by these laws is the identity of zero.

"Identity" is an invented concept. Zero is defined without that extra identity, and works quite well without that definition. Shadowjack is here collapsing a definition based on four identities with a single indentity, that of "zero," as if "zero" were one thing and one thing only. Yet, as we have mentioned, "zero" is a concept, invented in language, not a measurable. The only zero that we experience is the zero of absence. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

The question raised by the other laws is whether the zero that results from the calculation is the exact same identity as the zero of the laws that define zero.

And we must say definitely not. Without any doubt.

That's remarkable. The first three identities define a different zero than the fourth. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

It requires we explain the concept of flavours used in hyperdimensional physics. The suggestion being that by applying a process of multiplying by one or a or not a we change the spin of zero to generate a different flavour.

What was it that Feynman said? Something along the lines of "if you cannot explain a thing in language a child can understand, you don't understand it."

But because the minuteness of the difference does not offer itself on our measuring system we show each zero as exactly same.

He is explaining away his lack of clarity. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

## Uniqueness of zero

There can only be one zero. For if z had the same properties, we would have

• z + 0 = z
• z + 0 = 0

Yes, okay. If z had the same properties as 0 then that would mean z = 0.

The properties of zero are as follows:

Zero is that entity which added to itself gives itself, subtracted from itself gives itself, multiplied by itself gives itself and divided by itself gives itself. If z has the same properties as zero then z = 0 and if z does not have the same properties as zero then z =/= 0.

Shadowjack is working from the assumption that his set of four identities consistently define zero. He is not doing what scientists do: attempt to disprove their hypotheses. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

If z does equal zero then z + 0 = z and z + 0 = 0 which only means we now know what the unknown variable z is equal to.

The law of identity states that two different entity are the same if in any function where one is used the other can be used to obtain the exact same result. So z = 0 if and only if the two sign labels can be exchanged in any area where one is used without altering the outcome of the calculation.

However this is useful if zero comes in different flavours. Because rather than saying the term (a - a) is the type of zero we are using in this instance instead we can say zv" = (a - a) and to obtain that particular zero we use the sign label zv".

Furthermore zv"' = (~a - ~a), ziii"' = ( 0/a ) etc.

The reason we would do such is to distinguish between the spin applied to zero in any instance.

Shadowjack has not explained his "varieties of zero." He has marked his examples with his special notation, but I have not seen that he has explained it. These are expressed as apostrophes, 1 through 4 of them. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

To clarify the reasoning behind this notion it is useful to refer to some theory of Kant. Since my explanation of some of the theory of Kant is distorted in order to explain a particular matter it is not to suggest that Kant said what I am about to state. Merely that what I am about to state is derived from study of what Kant said.

The particular useful understanding is to distinguish between concept, idea and notion.

Great. What would be dangerous is to assume that the concepts so distinguished are "true." In general, this kind of work is useful or not useful, and, of course, utility depends on context. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

Concept, idea and notion can be considered similar in structure to element, atom and molecule.

Or to fruit, seeds, and apples. Clear? Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

Elements are the basic unit matter that do not reduce any further in the field to which they are relevant. Atoms are a unitary building block of different sorts depending on the specific elements involved in what quantity. Molecules are the first complex made from the combination of different atoms.

Okay. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

In the same way can we understand concept, idea and notion.

We could, perhaps. Whether we can or not depends on the foundation. There is no sign, so far, that Shadowjack is effectively communicating with anyone. Even if we can understand this essay, it is not terribly likely that we will. Now, let's see if I can prove myself wrong. If anyone cares to assist, welcome. This is a wiki. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

Where concepts are the similar to elements, and in this context can be understood as the specific individual sign labels: a , ~a, 0, 1, +, -, etc. As such the individual sign labels meet the definition of elements since they are a basic unit matter that do not reduce any further in the field to which they are relevant.

Where ideas are similar to atoms, and in this context can be understood as the specific individual calculation of zero. So therefore, zero minus zero equals zero is an idea that involves the synthesis of the concepts zero, minus and equals. We know that the idea of zero minus zero equals zero is a different idea to the idea that zero plus zero equals zero because the concepts of the second idea are not the same.

Bringing in analogies can clarify, and it can confuse. It will only clarify under certain conditions. Otherwise it is just adding more words to a pile of uncomprehended words. My own training suggests that we suspend the chatter, that we recognize the operation of the mind behind all this thought and these words, and then see what appears. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

Therefore we then know that 0 - 0 = 0 and 0 + 0 = 0 are two different ideas, definitely not one idea.

They are different and they are the same. "Definitely" could betray a rigidity of thought, if it is an insistence on one possibility out of many. Is it?

When we combine the four different ideas:

0 + 0 = 0 Zero plus zero is zero.

0 - 0 = 0 Zero minus zero is zero.

0 . 0 = 0 Zero multiplied by zero is zero.

0 / 0 = 0 Zero divided by zero is zero.

He is being repetitious, perhaps. He has already established the four identities he is working with. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

Then we obtain a notion.

Where notions are similar to molecules and in this context can be understood as the synthesis of several coherent ideas involving related concepts.

The building from concepts to ideas to notions is called synthetic. The reduction from notions to ideas to concepts is called analytic.

And the conflation of different topics into a single essay is called "writing walls of text." No matter how interesting these excursions may be, in my training, it is called "drooling." Mea culpa, often. I am leaving off my commentary at this point, today. Abd (discusscontribs) 20:55, 29 December 2013 (UTC)

Translating the laws of zero into the language of sets provided in -> http://en.wikiversity.org/wiki/Boolean_algebra we obtain:

Shadowjack had edited that article to take it outside what a top level project educational resource properly would be. [2] Here is a link to the last version as edited by him.

zi': ɸ ^ ɸ => ɸ

zi": ɸ ^ ~ɸ => Q

zi"': ɸ v ɸ => ɸ

zi"": ɸ < ɸ => ∞

zii': ɸ v 1 => ɸ

zii": 1 v ɸ => ɸ

zii"': ɸ < 1 => ɸ

zii"": 1 < ɸ => ɸ

ziii': ɸ v a => ɸ

ziii": a v ɸ => ɸ

ziii"': ɸ < a => ɸ

ziii"": a < ɸ => ɸ

ziv': ɸ v ~a => ɸ

ziv": ~a v ɸ => ɸ

ziv"': ɸ < ~a => ɸ

ziv"": ~a < ɸ => ɸ

zv': 1 ^ ~1 => Q

zv": a ^ ~a => e

zv"': ~a ^ ~(~a) => !ɸ

zv"": a ^ ~a => e

And showing the equivalence relationship of shape not meaning between the two different languages we obtain:

zi': 0 + 0 = 0 <=> ɸ ^ ɸ => ɸ

zi": 0 - 0 = 0 <=> ɸ ^ ~ɸ => Q

zi"': 0 . 0 = 0 <=> ɸ v ɸ => ɸ

zi"": 0 / 0 = 0 <=> ɸ < ɸ => ∞

zii':0 . 1 = 0 <=> ɸ v 1 => ɸ

zii": 1 . 0 = 0 <=> 1 v ɸ => ɸ

zii"': 0 / 1 = 0 <=> ɸ < 1 => ɸ

zii"": 1 / 0 = 0 <=> 1 < ɸ => ɸ

ziii': 0 . a = 0 <=> ɸ v a => ɸ

ziii": a . 0 = 0 <=> a v ɸ => ɸ

ziii"': 0 / a = 0 <=> ɸ < a => ɸ

ziii"": a / 0 = 0 <=> a < ɸ => ɸ

ziv': 0 . ~a = 0 <=> ɸ v ~a => ɸ

ziv": ~a . 0 = 0 <=> ~a v ɸ => ɸ

ziv"': 0 / ~a = 0 <=> ɸ < ~a => ɸ

ziv"": ~a / 0 = 0 <=> ~a < ɸ => ɸ

zv': 1 - 1 = 0 <=> 1 ^ ~1 => Q

zv": a - a = 0 <=> a ^ ~a => e

zv"': ~a - ~a = 0 <=> ~a ^ ~(~a) => !ɸ

zv"": a + ~a = 0 <=> a ^ ~a => e

The person stood looking for the longest time at the thing he was looking for. A passer by asked if he could help. Oh, would you mind said the person. Can you see it? Its over there. Pointing with his finger at a location near some bushes.

Hold on, said the passer by, I need to get closer. Okay, said the person, if you see it, let me know. And by the way, we have to be very careful it does not get away. I know how to catch it, so if you see it just point it out to me and I will be ready. Sure, said the passer by.

They both stood, straining their eyes in concentration trying to see the thing they were looking for, which another person thought was quite interesting so they came over to see if they could find out what was going on.

Oh, we are looking for something over there, said the passer by. We think it is hiding in the bushes. I have very good eyesight said the third person. If it is hiding in the bushes I assure you it will not escape my attention. Yes, said the passer by, but don't startle it or it may escape. Apparently there is a very clever way of catching it, so when you see it, just let us know and we will do the rest. Okay, said the other person.

Before too long there was a small crowd of people all gathered around the general area looking for the thing they were looking for.

The original person who had been looking in that direction had moved a small distance away and was looking at the crowd of people looking for the thing.

What are they looking at? asked somebody to the person. I am not quite sure, said the person, but it must be important to get so much attention. I think someone said that if they see it they can capture it, but they are not sure what it looks like, nor the exact method for capturing it when they see it.

Well I am a doctor so if they need the trained mind to assist in the exact method of capturing the thing I am sure I can show them how to do it. I had better let them know.

So then the doctor went over and spoke to the crowd of people and explained that he would know the method of capturing the thing when they saw where it was. That is a relief said the original passer by. We knew we would know it when we saw it, but we had no idea how to capture it. So it is lucky that you are here.

Say no more said the doctor. Just point it out when you can and we will take it from there.

There it is, said one of the people in the crowd. I saw it move just then. Hold on, I will show you exactly where it was. Yes, but what did it look like, said another person. If we knew what it looked like, then we would be more likely to see it for ourselves. Maybe it was nothing said a different person.

Don't say it was nothing, said the person who had seen it move. I definitely saw it move I tell you. There it is said someone else. It does move doesn't it. That is how we will catch it. Maybe we can't see it unless it moves, but when it moves we can see it. And if we can see it we can capture it.

The doctor with his trained mind decided it would be easier to know it if they saw the thing if they could determine where specifically it was hiding. So he thought if they could outline exactly the area within which the thing was hiding they would be in a better position to see it. For this reason he obtained a long length of rope and extended it in a large circle around the general area that everyone was looking at. That is a good idea, said another person. It is definitely inside that circle.

Then to determine exactly where within the circle the thing was they placed four posts evenly spaced around the circumference of the circle and then tied two ropes diagonally across the diameter of the circle to make four quadrants. And then to make it really easy to work out which quadrant of the circle the thing was hiding in they connected each of the posts together by a rope to make a square.

Immediately the person who had seen it move spoke up and said that it was in the upper right quadrant. And a different person said no it isn't, they thought it was in the lower left quadrant. But the person who had seen it move said yes, that is where it was before it moved. But it moved from the lower left quadrant to the upper right quadrant.

So now everybody focused their attention on the upper right quadrant to make sure it didn't get away.

How big is it, asked someone who had just turned up. Well it is smaller than one quadrant of the circle, said another, but apart from that we are not sure. Well if we put a rope from each corner of the upper right quadrant diagonally to the opposite corner we can work out where in the upper right quadrant it is.

Only if you are sure, said the person who had seen it move.

Is it smaller than one of the quadrants of the upper right quadrant? asked one of the observers.

Hold on, let's jump on it, said one of the other others. What do you mean? said another.

Well, if it is smaller than one of the quadrants of the upper right quadrant, then if all of us jump on it at the same time one of us is bound to catch hold of it. The important thing is if you get hold of it don't let it go until the rest of us have been able to tie it down. We should all agree to jump on a different quadrant of the upper right quadrant and then no matter which bit it is hiding in it won't be able to escape.

Good idea, they all agreed. At the same time then, let's jump on it.