# Thread: why is "0" "number 1"?

1. Indeed, when you multiply any number by zero, zero says, "screw you, i rank as number 1, as zero".  2.

3. Oh, ok. I didn't know that. Sorry.  4. my aim was not to insult you  5. This question has been bugging me for some time, regarding 0 and 1.

When you multiply zero by ANY NUMBER, 0 is the result.

It is a law.

That law is associated to other laws regarding the way numbers interact with one another.

Those laws that define how numbers interact with one another reach equations we are able to prove the correctness of by the use of rulers.

The equations of a sphere for instance employs the use of mathematical laws which can be verified with rulers.

Is it possible that another way of using the number system exists?

Is it possible that the OTHER WAY of using the number system, whatever that other way is, can derive the same equations of a sphere?

If it were possible, does that mean that OTHER WAY of using the number system is as VALID as the current use of the number system through the employment of contemporary mathematical laws and axioms?

I could highlight to you what that other possible algorithm is, it's attached to my website, but first lets consider the possibility. I could also suggest that I have been able to derive the equations for a sphere, validating that new mathematical algorithm, but first I ask the question, "is it possible, is there some clause in mathematics saying it is NOT possible?"  6. Originally Posted by streamSystems
When you multiply zero by ANY NUMBER, 0 is the result.
Only when multiplication is an allowed operation, and where 0 is defined. Neither by any means always is, though.
Those laws that define how numbers interact with one another reach equations we are able to prove the correctness of by the use of rulers.
What do you mean? You can find the ratio of the circumference of a circle to its diameter using rulers? I don't think so, but it's true that it's pi, anyway.

Is it possible that another way of using the number system exists?

Is it possible that the OTHER WAY of using the number system, whatever that other way is, can derive the same equations of a sphere?
Well, the sphere is a sphere, it must have an equation, all we require is it's the correct one, using any convention we choose. But, no, there need be no THE number system, as you call it. Our assignment of abstract entities called numbers to collections of other real (or abstract) entities is in some sense arbitrary. We could equally well call the cardinality of my dogs "bling" if we agree that we can place them in on-to-one correspondence with some set of which "bling" is a member, and where "blong" < "bling" and "blang > bling. So what? 1, 2, 3, .. is easier.

If it were possible, does that mean that OTHER WAY of using the number system is as VALID as the current use of the number system through the employment of contemporary mathematical laws and axioms?
Yes (and no), by the above. Numbers are a abstraction for sure; if you want to be technical, any set whose elements can counted, measured or calculated via a well-defined operation is isomorphic to some "number system", which is usually real or or complex, but not always. But why bother to define 3<2<1<0, when the conventional way works just fine? I guess I don't understand the point of your ramblings.

Edit: Oh, you might want to look at this, found on my hard drive:

http://arxiv.org/abs/math.QA/9802029

Most of it you may not (or may?) understand, but the intro might interest you  7. Yes, indeed, there are a number of "ways" one can find the various equations of a circle and a sphere, through the employment of different forms of measurement using different types, even, of measuring sticks (aka rulers).

What if I were to suggest an algorithm that actually presents the equations for a circle and sphere that INCLUDES an equation, an explanation, of "pi", by the employment of a particular type of USE of the number system.

Let's say 0 = bling and 1 = blong. How about an algorithm that represents a particular way of associating bling to blong. For me, using 0 and 1 is much more practical, if not symbolic. It's a lot easier. The algorithm I propose basically represents the hidden LINK between 0 and 1 without the use of fractions or decimals between 0 and 1.

It represents a "new way" of using the number system...........a new type of axiom.

To summarise my ramblings, basically I have developed a new "equation", a new algorithm, using 0 and 1, applied to a construct of perception (and yeah, oh ye-ah, the difficult I have in explaining this new mathematical language, as though I have just landed on the planet), that has DERIVED, "de-rived" (is that a confusing word), the equations of a sphere, TOGETHER WITH deriving what APPEARS to be the mechanics-operation of an atom. I thought it would be an interesting point of topic.

I caution anyone though, if this, this rambling, is too much, the book will take you under..........it is a rambling upon a rambling upon a rambling, etc etc etc. If I am not making sense, don't read it........that's the simplest thing I can say.

BUT, if anyone IS interested in such a mathematical algorithm, take a look.  8. Originally Posted by streamSystems
This question has been bugging me for some time, regarding 0 and 1.

When you multiply zero by ANY NUMBER, 0 is the result.

It is a law.
It is? That is news to me, i've never seen 0a = 0 as a ring axiom!  9. I forget the last time I was able to judge someone by their comment that was only so long as to be able to fart or burp longer than the statement. I am talking about my own statement here. I was trying to initiate conversation. I was seeing who was awake, and what your response would be. Combining numbers and letters, um, I wasn't referring to that technique. But then again, if I could have rambled on a bit more I would have explained that. Still, I enjoy any response.  10. Numbers and letters? What are you on about man?

There is no law that 0 times any number must be 0 - that was my point. You claimed there was a law. I am merely pointing out that this claim is in fact incorrect.  11. WELL, that's my point as well. Do you know what I am alluding to though?  12. Ya, well, let's not get too heated chaps. In any so-called algebraic set, we have a non-exclusive choice of algebraic operations. Let's think of the two most familiar to us: + and ×. (There are many others, let's not go there).

For any algebraic set, we insist on something called the "identity under the operation". If, pro temp we call this guy e, this means that a·e = e·a = a (where the · is a generic operation), and obviously where an algebraic set admits of the operation +, we say that e = 0, and where the operation is × we say that e = 1. This is merely convention, we could call them e<sub>+</sub> and e<sub>×</sub>, but that would be a pain. No problems here I hope.

But where, say, the operation × is not defined, then neither is 0 × a, and therefore 0·a could be anything, nothing or just outright silly.

In the case that the operation is +, we very, very loosely say that "1 is the number (cardinality) of the singleton {a} such that the cardinality of {a} + {a} is 1 + 1 = 2", but this requires our set S to be countable (in one-to-one-correspondence with N; not all sets are), and in any case is really an isomorphism N → S.

If you really want to get into this, we can, it's deeper than you might suspect. Anyway, that reminds me; I have long wanted to show Cantor's Theorem, which is just too cool for words. Maybe later.  13. Originally Posted by Guitarist
... Anyway, that reminds me; I have long wanted to show Cantor's Theorem, which is just too cool for words. Maybe later.
I'd like to see it. Plus... it might be a good way to hijack this stupid thread....

Cheers  14. Cantor as in |X| < |2<sup>X</sup>| or Cantor-Bernstein as in |X| ≤ |Y| and |Y| ≤ |X| => |X| = |Y| guitarist?  15. Originally Posted by river_rat
Cantor as in |X| < |2<sup>X</sup>| or Cantor-Bernstein as in |X| ≤ |Y| and |Y| ≤ |X| => |X| = |Y| guitarist?
Still composing, and cooking dinner, too. But, ya, the former, but gimme time. If I can finish tonight UK, I will.  16. I have long wanted to show Cantor's Theorem, which is just too cool for words
'Too cool for words', Is that a mathematicians joke ?  17. Happy to have this thread become derailed, believe me: not what I expected.

Anyway, is anyone interested in a proof of a theory for the equations for a circle and sphere that uses a square-rule to derive the equations. It basically contradicts Ferdinand Lindemann's assertion that it is not possible to use a square rule. It presents a theory, a structure of logic, for a transcendental number. It's on my website (see www button below), the download (DOWNLOAD NOW button), 5MB. I could present it here, but, wow, not enough space.  18. Yeah, sorry for the delay, the following proved rather more difficult to explain both succinctly and clearly than I at first thought. As you see, I failed in my first objective, sorry. Let me start with Cantor's Theorem:

For any set S, the cardinality of S is strictly less than the cardinality of its powerset P(S).

This may need some explaining. If you don't need it, you can skip this preamble:

Don't worry too much for now what the powerset P(S) really is(we can return to it later, if you want), just think of it as all possible subsets of S.

The cardinality of a set is simply its number of elements. Some people, river_rat included by the look of it, use |S| for the cardinality of S. For reasons I don't need to get into, I don't like this so much, and will use cardS.

Now using all available fingers and toes, and remembering that S and Ø are always subsets of S, for some suitably small cardS you can easily convince yourself that cardP(S) = 2<sup>cardS</sup>. The question now is, is this always true? To prove this, we only need to prove the case where cardS = ∞ (use the simple-minded definition of ∞ for now). (By the way, the assumption that an infinite set has a powerset is not entirely uncontraversial, but we'll assume it has).

Next thing you need to know is this. A set is said to be countable if its elements can be placed in one-to-one correspondence with the elements of a subset of the "counting" numbers N. As N is always a subset of N, and as N is infinite, we have the quaint expression "countably infinite" (aren't mathematicians loves?). We'll assume here that S is countably infinite.

The one-to-one correspondence I referred to above means that whatever we can prove for N will by true for any countable set, so we want to prove that cardP(N) > cardN.

Proof: We proceed by contradiction, that is we start by assuming otherwise.

We start by assuming we can make a countable list L of all possible subsets of N (we don't care about the ordering of L).

By assuming that L is a countable list, we are assuming we can offer a unique index (with order) from N to each s<sub>n</sub> in L, like I just did. We start by forming the subset D (for diagonal) under the condition that

* the number n is in D iff n is in s<sub>n</sub>.

Now D is a subset of N and must be a member of L. Let's index this subset according to our rule for forming D, say s<sub>d</sub> and we have the mind-boggling conclusion that d is in D iff d is in s<sub>d</sub>. As this is D, by definition, we have the startling result

** d is in D iff d is in D.

I don't know about you, but I'm not about to write home to mother about this result. Ah, but wait. But first let me remind you we are assuming we have written an exhaustive list of the subsets of N, this will become important

Now, for any subset of N we can find its complement, that is, the subset whose elements are not in the first subset. Let's do this for D and say that C is the subset whose elements are NOT in s<sub>d</sub> i.e. D. This is definitely a subset of N and must there be a member of L, so let's give it an index, say c. Note by the definition of D, if C is its complement, c cannot be in D, and must therefore be in C
Then by * above

the number c is in C iff c is NOT in s<sub>c</sub>. But this is C! So

***c is in C iff c is NOT in C.

Crikey! This is clearly nuts. What did I do wrong? Well only by making the original assumption that P(N) is countable, which must be false. So N is countably infinite, P(N) is uncountably infinite, which a "bigger" number. Cantor's Theorem is proved. Isn't it pretty? I would rank this with Euclid'sprime number proof for simple elegance  19. Originally Posted by river_rat
Cantor-Bernstein as in |X| ≤ |Y| and |Y| ≤ |X| => |X| = |Y|
At last a bit of mathematics that looks perfectly logical to me at first glance, although I'm sure a proof would blow my mind. Anyway, it attracted my attention because although I'm no mathematician I have done some seismology which uses a fair bit of maths. One thing I learned was Fermat's principle, which says something about a ray path being both the minimum and the maximum distance between two points. This always confused me, how could something be both, perhaps there's some Cantor-Bernstein logic behind the statement? I dunno, stab in the dark really.  20. Originally Posted by billiards
One thing I learned was Fermat's principle, which says something about a ray path being both the minimum and the maximum distance between two points.
This is an example of a variational principle, which are extremizing functions. This means that, in first order terms, could supply a maximum, a minimum or a saddle point. Obviously the higher derivative distinguishes these alternatives but, as it is apparently usually clear from the physical nature of the problem which you are have, first order is normally sufficient. (Actually, it's on record here that I had to ask for help with this, so maybe a pinch of salt is required)

Anyway, I promised (threatened?) more on the powerset. This is the set formed from all possible subsets of a point-set. So, like, if X is a set of points {x, y, z}, then the powerset P(X) = {Ø, {x}, {y}, {z}, {xy}, {xz}, {yz}, {xyz}}. Note that the powerset of {x} = {Ø, {x}} and that of {x, y} is {Ø, {x}, {y}, {xy}}.

See a pattern? Yup, it looks like that, for a set with n elements, the powerset has 2<sup>n</sup> elements. Is this always true? For the finite case, the argument might go like this:

To each element x of X, the statement "x is in the subset A" is either true or false (note that it can never be both or neither) Set 0 = false, 1 = true, then we see that this is a map X → {0,1}, the set "2". Simple combinatorics tells us that for any string of 1's and 0's of length n, there are 2<sup>n</sup> possible strings.

So if the cardinality of X is n, the cardinality of P(X) is 2<sup>n</sup>, which is always greater than n. (By a slight, but quite benign, abuse of notation, the powerset of X is often written 2<sup>X</sup>). Cantor's problem was this; in general one would assume that 2<sup>∞</sup> = ∞, implying in this case cardX = cardP(X). We saw above this is false.  21. I know you want to hijack the thread and all, but...wouldn't it be better to start a new thread with the heading "Cantor's Theorem"?  22. Wrong. I didn't "want" to hijack the thread, it seemed to me relevant in context at the time. You disagree? Fine, no matter. More seriously, if you don't see the relevance, ask, or read a book.  23. I'm very curious about Cantor's theory, too. If you can explain it in layman's terms (for someone who only made it through 2nd semester calculus), that would be great.  24. The version of cantor's theorem Guitarist showed roughly says that you cannot list all the real numbers, even if i give you the luxury of an infinite list to write them all on. No matter how you write the list you will always miss at least one real number.  25. Originally Posted by Retromingent
I'm very curious about Cantor's theory, too. If you can explain it in layman's terms (for someone who only made it through 2nd semester calculus), that would be great.
Let me try. Have you ever seen one of those newspaper puzzles: how may words can you make from, say, "there". The rules are, in forming your new word, you may not change the order of letters as they appear in the old word, and for each letter in the old word you can use it only once in your new word.

What do you get? " there", "the", "thee", "he", "her", "here", "ere" etc. There may be others. Let's now relax the rules somewhat and say that you may have any word that has meaning in any language on any planet in the universe. Then it's safe to assume, let's say, that there is some planet where "t", "h", "th" etc are words. Let's also assume that, somewhere, " " means something.

Then all the words you can form from "there" are legit, and, taken together are called the powerset of "there" So let's see. Under these rules, a word of length 1 has 2 solutions in this game, a word of length 2 has 4, a word of length 3 has 8 and so on. Well, look at that a mo. 2 is 2 once, 4 is 2 twice, 8 is 4 twice, so we think we see a pattern; for any word of length n, we think it is safe to assume we can form words of length 2 × 2 × .. n times. A little math shows that this is 2<sup>n</sup>

But "safe to assume" doesn't satisfy mathematicians (maybe they should get out more?) - proof is what they crave. A sure proof would be that, if it can be proved this relation still holds where n = ∞, the math men can go to bet content.

This is the content of Cantor's theorem. But.... in my opinion, it's not the result so much that is neat, but the method of proof, which is, if fact, a generalization of a proof he had earlier used, now called Cantor's Diagonal Proof, that the set of real numbers is "bigger" than the set of "counting" (natural) numbers.  Bookmarks
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