Sure, I remember the discussion in first-year algebra where the teacher showed positively that division by 0 was meaningless and had to be disallowed because if allowed, it presented all kinds of paradoxes and impossibilities (e.g., 3=4).
But I recall thinking at the time that it seemed to be a glaring *hole* in the system. How can we arbitrarily disallow some function? For instance, why not disallow multiplication by 17, or not allow an odd number to be added to an even number... that sort of thing?
As a sort of parallel example: Extracting the square root of -1 is obviously impossible and nonsensical. Nevertheless, there is an entire branch of mathematics devoted to it. Complex numbers (using "i" to represent the square root of -1) are useful in many real-world applications
Now why can't we do a similar thing with another "impossible" operation and value? Let's call "one divided by zero" (1/0) "p" and for instance, 7/0 is 7p. Having split off the special part of this as p (for "preposterous") number, can we now come up with a set of operations and a system of mathematics for making useful calculations?
I apologize that this may sound like it's related to some crackpot "theory of existence" but I assure you that I am not postulating anything -- I'm simply looking for a clearly-worded and understandable explanation of the situation. I am no mathematician -- I struggled but was able to bluff my way through a high school Trig class many years ago. So be gentle :-)
I'd also be interested in discussing related topics. For instance, if such a system has been (or could be) developed, what are (or would be) the implications?
Another oddball thought: What if instead of 0 we just substitute 0.0000001? For instance,
7/0.0000001
is a perfectly valid operation. It seems to me that it would be very nearly the same answer as 7/0, but just not as accurate. What's wrong with my thinking on that?
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"Another oddball thought: What if instead of 0 we just substitute 0.0000001? For instance, 7/0.0000001 "
this idea is covered here with Talyors theorem - http://en.wikipedia.org/wi
where you can express a a formula and whan happens when the varialbe (of that formula) tends towards zero.. ..(i.e. 0.000000000007 or what ever)
sorry wrong link try ...
http://en.wikipedia.org/wi
"But I recall thinking at the time that it seemed to be a glaring *hole* in the system. How can we arbitrarily disallow some function?"
"I'm simply looking for a clearly-worded and understandable explanation of the situation. "
It wasn't arbitrary to disallow zero and treat it differently. Zero is not an arbitrary choice like "17" would be. Zero is qualitatively different from any other number. In that way, it's the only number representing nothingness while all the other numbers represent "somethingness."
"where the teacher showed positively that division by 0 was meaningless and had to be disallowed because if allowed, it presented all kinds of paradoxes and impossibilities (e.g., 3=4)."
---
There you have it. "presented all kinds of paradoxes and impossibilities (e.g., 3=4)"
dividing by 17 does not presente all kinds of paradoxes and impossibilities.
" Having split off the special part of this as p (for "preposterous") number, can we now come up with a set of operations and a system of mathematics for making useful calculations?"
Can we....? So far NO. If you (or anybody else does) they will be celebrated by mathematicians everywhere. Some people have tried but most figure that the probability of success is so low that their time is better spent elsewhere.
you can define divide by zero to mean what you want, but the problem is that it might then invalidate important mathematical results. I'm sure mathematicians did experiment with defining 1/0 as some number, and the results were probably unsatisfactory.
An example of a definition that works well is that 1 is defined as NOT a prime number, despite dividing only by 1 or itself (the definition for any other natural number), but the usual definition of a prime number simply states that 1 is not a prime number. And this works, without that proviso in the prime number definition all sorts of nntheorems regarding prime numbers just fall over.
It's ultimately the philosophy of mathematics, a definition should be as simple as possible, but the maths still has to work and be satisfying.
>> all the other numbers represent "somethingness."
Just for the sake of argument: There is no other number that represents seventeen-ness either... Why make a special place for zero?
>> dividing by 17 does not presente all kinds of paradoxes and impossibilities
That's the same logic I heard in that classroom and it still fails to ring true (in my humble ears :-). If there is a system wherein a set of operations and values causes paradoxes, doesn't that invalidate the system? In most of science, a single exception is typically all that's needed to show at least a flaw, if not a fatal flaw, in the system.
ozo,
In one of your links I see "it doesn't make sense... like when you add apples to oranges..."
Well, I say that when you add 5 apples to 3 oranges you have 8 pieces of fruit. It does make sense!
And in fact, they all just tell me over and over that the answer can't make sense... but that's the very point I'm asking about! How can we work with a system that contains operations that yield nonsensical results? We can't just sweep it under the rug! I might be important!
Think of the situation when you identify the location of a star near the sun during a total eclipse. It's not where it should be! Should we just say "Well measuring the location of that star is not allowed" ???
>>Extracting the square root of -1 is obviously impossible and nonsensical.
it used to be thought that the number -1 was nonsensical in medieval times, they managed to solve cubic equations without using negative or imaginary numbers, but they were using them in a round-about way. When they realised they were using -1 and it did something useful, they decided to allow it and I think only a few years later, sqrt(-1) became accepted.
If you've got a solution to a real equation that is itself real, then if you used imaginary numbers in determining that result, then clearly the imaginary numbers had purpose. and in the mathematical world they exist.
Imaginary numbers were just a superb discovery (yes a discovery not an invention).
By the way... Is 0/0 equal to 1? (as in 5/5, 4/4, ...1/1, etc.)
Or is it 0 (as in 0/5, 0/4, 0/1, etc.)?
Or is it "not allowed? If so, why not? Is this another special exception we need to make in our system of numbers? And if there are two exceptions, perhaps there are many others... How can we trust such a flawed system?
n^0 is 1.
What orifice did somebody pull that one out of? Raising something to the 0-th power is a nonsensical operation. Let's disallow it!
this is going no where - its a bit like trying to explain how the earth goes round the sun and not vice-versa to the flat-earth society.
or explain to a 5 year old that there is more space between the atoms of kitchen table than the atoms that make tha table up.
or trying to work out the length of the hypotenese of a right angled triangle without using square roots
What I'm trying to you need to study mathemtics to higher level to understand the answers.
imaginary numbers (or Complex numbers) make sense if you study geometry (see my last link)
some early societies did not have the concept of zero and they could not develop solutions to many simple (real life) problems. A definiton of division is that of "sharing the dominator equally between the divisor" - ie. 20/5 is 20 shared among 5 which means each one gets 4. using zero is un-informed.
0^n is only defined for n>0, that pretty much covers 5/0 not being defined because x/y = x * y^-1
also 5^0 seems weird until it's considered that 5^2 = 25, 5^1 = 5, 5^0=1, 5^-1=1/5
each time you divide by 5 because x^n-1 = x^n/x
>>How can we trust such a flawed system?
it's all being analysed with mathematical rigour, and 1/0 doesn't invalidate that rigour in my opinion. But could there be a giant flaw somewhere? I can't say for certain myself, your welcome to look for one.
The problem of dividing by zero lies in just exactly what zero is. It gets a bit involved by essentially the ordinal numbers are derived from the cardinals, which are basically abstractions related to sets of objects. Like Dan's sets of apples. The set which has cardinality four can be put into one-to-one correspondance with anyother set of cardinality four. Dans five apples and three pears producing eight is doing precisely that. The numbers involved are cardinals.
Now the cardinal number zero is the cardinality of that set which cannot be put into one-to-one correspondance with any other set. Simplyx because it has no members. It is VERY important for all the other numbers - to be unique and useful - that this correspondance works. For that zero must exist and it must NOT match anything else.
Thus the idea of taking a set (say Dan's five apples) and dividing into into sub-sets (ie: division amongst the ordinals) works because the sets produced (say 2 sets of three and four apples) are identifiable by their cardinality, since these sets are placed in one-to-one correspondance with sets of known cardinality. But trying to divide a set into two sets, one of which is supposed to be the null set just doesn't work, since it is not possible to identify the sets after division.
> 0^n is only defined for n>0
sometimes one may want ^ defined that way.
but others prefer a more useful definition of ^
http://www.faqs.org/faqs/s
ozo's bald assertion that 0^0 is undefined -- on the heels of reading that link in which it's very clear that top minds have debated the issue for centuries and are still in disagreement -- is interesting, indeed.
chilternPC assertion that I must be like a naive little kid who does not understand complicated things like atoms and hypotenuses is, simply put, a cop out. It's the teacher saying "You'll understand, little Danny, about these adult things things when you grow up." But I'm perfectly capable of understanding. I posed this question because every explanation I've heard usually boils down to, "Well, it is because it is" and I think it's fun to pick at scabs (and root assumptions).
BigRat's empty set discussion makes the most sense to me.
But the empty set is a subset of all sets. That brings me back to the idea of p numbers, where we split off the zero-anomaly portion of the operation and use a set of self-consistent operations on it. It just seems like that would be more elegant than saying (effectively) "Don't go there!"
I did not assert that 0^0 is undefined
I quoted http:#a23240951
and asserted that while ^ may sometimes be defined that way, there was another definition of ^ that is more generally useful.
Look at it from the side of multiplication :
5 * x = y
If y is given, you can always find one and only one x. However :
0 * x = y
If y is given, any x would be a solution (since there's only one possible value for y, namely 0).
So, 0 is definitely different from all other numbers.
Since you can define division as the inverse of multiplication, the use of 0 also poses a problem in division for that same reason : it doesn't make sense to divide by 0.
Just to play devil's advocate... what sense does it make to add or subtract 0? You end up with whatever you started with, so you might as well skip it altogether. Let's say that hereafter, addition by zero is not allowed... (just kidding, please don't explain it to me)
I believe that mathematicians do work with infinities, insofar as when they have two of them, they can be used to cancel each other out. Can't we do something similar with the result of a division by zero?
5 + (4/0 - 7/0)= 5
>> Let's say that hereafter, addition by zero is not allowed... (just kidding, please don't explain it to me)
At the risk of stating the obvious : addition of zero gives a defined result (granted, it doesn't change the original value, but it's still defined). Division by zero doesn't ...
>> Can't we do something similar with the result of a division by zero?
>>
>> 5 + (4/0 - 7/0)= 5
Suppose we do allow division by zero (as long as the "infinities cancel each other out"), and we work with the above, then, we could also simplify (4/0 - 7/0) to ((4 - 7)/0) to (-3/0), and we're stuck again. When introducing infinity as the result of a division by zero, you can make some operations work, but not all of them, you'll always find a discrepancy.
> (4/0 - 7/0)
At the risk of introducing more confusion, Quantum Chromodymaics does use calculations in which infinities are canceled, but you need to be very careful about how you do it.
4/0 and 7/0 also have meaning if you are working in the Real Projective Line
but (4/0 - 7/0) does not have a meaning in the Real Projective Line
DanRollins - are you in the banking industy? :-)
I think its the banking industry that has been mis-using Zero.
what I guess is they were using the unknown 'x' in thier equations and cancelling x out and dividing by it - only to find x=0
defining 'division by zero' as not allowed is one of the axioms or of mathematics.
(http://en.wikipedia.org/w
please see the following link: http://en.wikipedia.org/wi
under 'elementary arithmetic'
Happy Holidays...
Santa got me this great book for Christmas:
http://www.amazon.co.uk/Ma
excellent first chapter chapter on zero.
In math it is not about whether something is"allowed" or not, but rather whether it is defined or not.
In different branches of math man different things are defined as "addition" and "multiplication" (andsometimes what you'd cll addition is investigated as a multiplication and vice versa). However, one is stil quite restrictive in what one would want to call addition or multiplication. For example it makes no (or little? it may depend...) sense to call the line pasing through two distinct points their "sum" - simply becasue some basic rules (axioms or theorems depending on your point of view) do not hold, e.g. (a+b)+c = a+(b+c) for all possible a,b,c.
Addition in the realm of natural numbers is defined, well, quite naturally: If I have two apples and you have thre, then we together have five apples. Isn't it intrigueing that this is always true? Even if we replace aplles with pears? (Actually, it is not true if one of the apples is common property of us).
We use addition to define *subtraction* and declare that a-b be <i>the unique number x such that a = b+x holds, provided it exists</i>. Such a definition is based on *theorems* that one can *prove*, especially one needs the following:
If b, x and y are numbers such that b+x = b+y then necessarily x=y (so-callled left cancellation)
It is possible to *prove* this statement for natural numbers and this allows us to conclude that a number x such that b+x=a is *unique* (if it exists at all). Without this uniqueness, the definition of subtraction does not make sense. This may seem trivial, but be informed that some operations that mathematicians rightfully call "addition" (because they obey the rules mentioned above) do *not* allow left cancellation and hence one *cannot* define a corresponding sutraction operation:
Consider pairs (a,b) of natural numbers and define an addition (a,b) + (c,d) as follows:
If b=d, the result is (a+c,b).
If b>d, the result is (a,b).
If b<d, the result is (c,d).
You are invited to check that the follwoing properties hold
Associativity: (x + y) + z = x + (y+z)
Commutativity: x + y = y + x
Neutral element: There is an object n such that x+y = x for all x
However, we do not have left cancellation: (1,0) + (0,1) = (0,1) = (0,0) + (0,1)
(Exercise: Define a multiplication by (a,b) * (c,d) := (a*c,b+d). Is this multiplication associative? Commutative? Does it have a neutral element? Does the distributive law hold, i.e. (x+y)*z = x*z+y*z for all x, y, z?
Even though the definition of subtraction makes sense for natural numbers, you will have noticed that subtraction is only a *partial* operation insofar as we cannot (within the set of natural numbers!) determine 3-5 or 4-6.
The solution to this problem, as you'll know, lies in extending the number system from natural numbers to integers. The point is, that
- the natural numbers are a subset of the integers
- one can define addition and multiplication of integers in such a way that one obtains the "old" result if one adds/multiplies two integers which happen to be natural numbers
- the basic rules mentioned above which hold for addition/multiplication of naturals still hold
- the equation a+x=b now has a (unique) solution for arbitrary natural numbers a,b (in fact for arbitrary integers a,b).
It turns out that the formerly undefined expressions 3-5 and 4-6 are (necessarily!) the same integer, namely the one written as "-2".
In the same manner, division is only partially defined, i.e. a/b is the uniue (natural or whatever we talk about) number x such that b*x=a.
In the theory of natural numbers, 3/2 is not defined. This corresponds to the fact that you cannot distribute three sheep equally among two shepherds. We extend to the field of rational numbers to avoid the problem, later we extend to real numbers and complex numbers - but in all these cases, division remains a partial operaion because a/0 is not defined for any a. It turns out that you cannot eat the cake (define division by zero) and keep it (i.e such that the basic rules still hold):
If 0*x=1, we find by applying these basic rules (esp. the distibutive law) that 2 = 1+1 = 0*x+0*x = (0+0)*x = 0*x = 1 - a contradiction. Thus you are perfectly allowed to *define* 1/0 (and define it as whatever known value or symbolic name you want) but are henceforth not allwoed to make unrestricted use of the distributive law - this is very unpractical, so unpractical indeed that it is best to leave 1/0 undefined.
>>But the empty set is a subset of all sets.
You've missed my point entirely. The set whose cardinal is zero (ioften called trhe empty set) is a set which CANNOT be put into correspondance with any other set. It cannot be directly a subset of every set, since there is no way to ascertain if the subset actually contains elements from the main set. Thus [2,3] can be a subset of [1,2,3,4,5,6,7] but [] cannot be a TRUE proper subset.
In order to delove into the realm of arithmetic we'd need to define ordinals, which requires relationships on sets and their elements, and then ordered relationships on those relationships are finally types and characteristics of sets. This is mathematically a lot of work - it has indeed been performed to put mathematics on a firm ground - but is far too much for a simple discussion. Therefore I'd like to take an alternative approach.
If we take a set of, say, six elements and split it into two sets, we'd get threre elements in each set. That is terribly intuitive. But take the other path, namely split the set into sets containing three elements. How about sets containing two elements? How about sets containing only one element? How about sets containing any number of elements you'd like. We'
d first have two sets, then three sets, then six sets and then how many? Is the old adage "two goes into six three times REMAINDER none" true, or is it "Two goes into six three times remainder none and none and none and none and none"? It looks as if with division there can be no "remainder none".
Look at it the other way round. Try splitting the empty set by 2. Do you get two empty sets? Since neither of these sets are equivalent to any other set, are they equivalent to themselves? The answer is undefined, since equivalence requires entities (elements) with which one can manipulate.
The point I'm trying to make is that zero is not a number. We make it look like a number and we build it into various rules as if it were a number, but ultimately it is not a number and that manens there are operations which just don't make sense with it.
gleaned from my new book:
zero or "0" is a symbol that means nothingness, Its used as a place holder in our decimal number system - so one could distinguish between 34 and 304 - This might be compared to the introduction of the 'comma' into language - both help with the reading the right meaning, but just as the comma comes with a set of rules for use - there are rules for using zero.
one rule is to say 7/0 is undefined. it is not permissable to get any sense from the operation of dividing a nonzero number by zero and so we simply do not allow this operation. in a similiar way it is not permissible to place a comma in the mid,dle of a word without descending into NONSENSE.
to recap:
what use is zero? - we could not do witrhout 0. The progress of science has depended on it. We talk about zero degrees longitude, zero degrees on a temperature scale and likewise zero gravity. On the number line, 0 is the 'number' that separates the positive numbers from the negatives in the decimal system, zero serves as a placeholder which enables us to use both huge numbers and microscopic figures.
Someone mentioned division as a sharing mechanism earlier.
If you ran a lottery and one week three people matched the lucky number, you would divide the prize pool by three and each of the winners can receive that amount. If the next week there were no people who matched the number it could be true to say that you divide the prize fund by zero and gave the result to each of the (zero) winners, not an undefined amount, you definitely gave nothing away that week.
dividing can also be defined by subtraction - 20 / 5 can be defined as keep subtracting 5 from 20 until there is nothing or less than 5 remaining. in this example 5 can be taken away 4 times, which is the answer. Using this method 20/ 0 means as subtract 'nothing' from 20 until less than 'nothing' remains - in this case it clearly breaks down.
>it clearly breaks down
Not really clear. You changed the definition from 'nothing or less than nothing remaining' to 'less than nothing remaining'. With your initial definition all you are asking with 20/0 is to continue to remove nothing until nothing remains. It isn't clearly breaking down, just clearly never actually going to complete the function so all you have to do is count or calculate how many times it will repeat.
The lottery example I offered could be a possible use of the 'p' number, it certainly seems to offer a division by zero that is then later multiplied by zero to get the correct answer instead of an undefined.
share = prize_fund / number_of_winners
total_payout = share * number_of_winners
so for zero winners: total_payout = (pf / 0) * 0.
Yes I can see that multiplying by zero may take precedence over the division, but if you placed this formula into a spreadsheet it would work for any values except 0. In this case it would throw an error which would need to be dealt with before the multplication was attempted. So the precedence of the multiply-by-zero has been altered for this special case.
don't agree that zero isn't a number, not in terms of the intergers and real numbers. Zero is definately a member of the real numbers.
You can't do 5/0 or 0/0, but that's a limitaion of the divide operation, a number doesn't have to be able to be used as a divisor though to be a number, you also can't do arccos(-3), but -3 is still a real number.
>limitaion of the divide operation
I'm thinking that this is the whole of the problem. Addition is the opposite of subtraction. Multiplication is a 'trick' to speed up repeated additions and division is a trick to speed up repeated subtractions. Although multiplication and division appear to be opposites, it is not necessary that they are. As deighton has said, there is a limitation of the divide operation. The divide-by-zero problem has been rather brushed under the carpet as otherwise the operation is fairly easy to understand and to use. What we need perhaps is a better way to do a repeated subtraction that returns the count of the iteration and any remainder, one that is the proper reverse of multiplication.
Thank you RobinD. Yes, that's my point.
Most of the comments here have been (as I expected) proofs and examples of why the result of division by zero must be undefined. But I already knew that. Everybody does -- it is taught in grammar school. Just like the algorithm for multiplication is taught. It's a basic rule of the system.
But IMHO, the system just seems unfinished. A jagged edge on an otherwise, clean smooth set of operations. And it seems that nearly everybody is willing to accept it -- yes, sweep it under the rug -- and move on.
Sure, we can make it an axiom; Division is an operation that can be used on any two numbers and any number can be the demoninator except zero. Fine!
In geometry, we know that two parallel lines never cross. It's axiomatic. But map the plane onto a sphere, and guess what?
Isn't some sort of "remapping" available here? Some way of looking at this situation that doesn't leave us with this special (seemingly, arbitrary) exception sticking out like a sore thumb and needing to be swept aside?
You can define operations on a Real Projective Line
http://en.wikipedia.org/wi
But then you no longer have field, and you lose some other properties.
There is no complete and perfect system
this was proved in 1931
http://mathworld.wolfram.c
http://en.wikipedia.org/wi
>> What would be the meaning of division by zero ?
Good question! It's one of the things I'm trying to find out.
When you examine a formal system for flaws, you would surely look first to the obvious exceptions. If an axiom about Boolean Logic stated:
When A is true and B is true, then the statement A AND B is also true, except on
Thursdays and alternate Wednesdays when the AND operation is not defined.
... then surely you would wonder about those exceptions.... wouldn't you?
=-=-=-=-=-=-=-=-=-=
ozo
There is some relevance to Gödel's incompleteness theorem. It shows that no system can be complete, but that's the case only when the system allows or includes self-referential statements. Most of the time, statements are not self-referential. For instance,
"This statement is false."
rarely comes up in real conversations. It's a special case.
But, that special case (it seems to me) is no different from the special case of division by zero. Why is that theorum so universally accepted and profound, when a pseudo-theorum like:
All mathematical operations are suspect because we arbitrarily disallow one
operation on one value. Thus, we cannot trust mathematics in general.
... is not taken seriously?
What Gödel showed was that any system interesting enough to include arithmetic will be able to model self-referential statements.
And Gödel did not say we cannot trust mathematics , just that it cannot do everything.
And there are ways of defining division by 0 and eliminating that particular special case as described above,
but that just moves the exceptions elsewhere.
> we arbitrarily disallow one operation on one value.
If this refers to division by 0, it is not completely arbitrary. It is because of the consequences of allowing it.
If is allowed, you will have to give up something else.
What one chooses to give up may depend on what is important for what you want to do using the system.
Dan:
This will add... ummm... "zero"... to the discussion, but I'm still going to say 'Thank you' for asking the question.
I also find something disturbing about this exception.
Unending repetitive subtraction of zero seems no more nonsensical to me than unending repetitive addition of '1' (or any other value).
BTW, one plus one doesn't always equal two when we discuss real-world examples. One liter of water plus one liter of alcohol doesn't result in two liters of liquid. And adding ten gallons of water to a vat containing ten gallons of sulphuric acid doesn't always give the same result as adding ten gallons of sulphuric acid to ten gallons of water.
Mathematics are extremely valuable languages because (and as long as) they closely match the "real world". They are supposed to be structured similarly to the real-world in order to be able to give descriptions that can be applied to real-world situations. That's where their predictive powers stand out.
I.e., we can manipulate the symbols and arrive at some predicted result that no one has ever seen before.
However, when we _assume_ that the structures are absolutely precise matches, we run into nasty surprises at irksome moments.
For division by zero, we see an instance where the symbols can be combined in a way that has no real-world correspondence. To me, that _indicates_ (without proving) that we don't quite have it perfect yet.
Mathematics as language -- English as language -- spoken English vs. written English --
There things you can 'say' in English that cannot be properly 'written' in English:
If I 'say' this -- "There are three 'to's in the English language: 'to', 'too' and 'two'" -- it can communicate the idea easily. But it has an obvious flaw when it's written because there is only one 'to' not three. Maybe I should write 'tu' as a kind of phonetic representation? But am I then writing it 'in English'? If so, then wouldn't there be four 'tu's in English: 'to', 'too', 'two and 'tu'? But then there would really only be one 'tu' not four...
Division by zero is an attempt to put some real-world process into symbolic form. What actual real-world operation is being described?
Tom
>> >> What would be the meaning of division by zero ?
>>
>> Good question! It's one of the things I'm trying to find out.
My point is that, if you don't need it (you don't know tha meaning of division by zero), why are you missing it ? Why are you pointing at it as being a flaw ?
The continuation of my point is that if for some reason you would need division by zero, you can use a different system that allows division by zero, or you could reason with limits, or ...
A number system is used for counting, performing useful calculations, etc. In general, division by zero is not a useful calculation (a bookkeeper wouldn't want to encounter it for example), so it makes sense that this number system disallows it. In fact, you could reason that it is a requirement for making the system perfect (disallow a meaningless operation).
Your example of logic on thursdays is not the same, as that would have a notable impact on the daily use of logic. Disallowing division by zero however makes sense in the general case.
Have you ever graphed 1/x? Scroll down a bit on http://www.mathsrevision.n
Is 1/0 positively infinite or negatively infinite?
How do you calculate 1/0 + 1/0? Or 1/0 - 1/0? What is 1/0 + 3? Does (1/0 + 3) - 1/0 = 3? Does (1/0)^2 = 1/0? Does -1/0 = 1/0?
The axioms of the real numbers that we are used to dealing with just can't meaningfully accommodate the idea that the expression (1/0) has a defined numerical value. Accepting that (1/0) is a well defined value leads to all kinds of absurdities that contradict the basic axioms and theorems of algebra.
now 'we' are starting to talk about x in equations and letting x tend towards zero...
that's diferent from x=0
some history....
the work of the severnth century Indian Mathematician Brahmagupta treated '0' as a number not just a place holder and set out rules dealing with it. he was advanced for his time. The Hindu-Arabic numbering system which include zero in this way was promulgated in the west by Fibonacci in his 'Liber Abaci' (the book of counting) first published in 1202. Brought up in North Africa and schooled in the Hindu-Arabian arthimetic, he recognised the power of using the extra sign 0 combined with the Hindu symbols 1,2,3,4,5,6,7,8 and 9.
Interesting thread. Perhaps the question should be "division by zero ... why?" Mathematics has developed as a means of quantitively measuring some event or thing, initially for real world occurrences and subsequently for theoretical occurrences. What real or theoretical event would dividing by zero measure? Why isn't there a base 1 numbering system or, worse yet, a base zero? Simply because they have no use for anything we know about to date. Maybe some day we will need to define x/0 as a measure of something.
www.bbc.co.uk/berkshire/co
X/0 = Not a Number (and certainly not infinity).
Zeor is a real number and absolutely behaves like a real number.
Recall that a set R together with operations + and * is called the field of real numbers if certain axioms are fulfilled (i.e. if it is a complete ordered field). Among others R must contain an additive neutral element, 0. Hence 0 behaves perfectly like a real number because it is a real number
Division is not part of the definition of "real number". From the axioms of the field of real numbers, we can prove a theorem, namely that an equation a*x = b where a is a non-zero real number and b is an arbitrary real number always has exactly one solution in R.
You think that because of the restrictin "a is non-zero", the number 0 is not a real number?
Well, there is another theorem: The x*x = c where c is any non-negative number has at least one solution in R and exactly one solution is non-negative. Does the necessary restriction "non-negative" imply that negative numbers are not real numbers?
We use the first theorem to define b/a IF a is non-zero.
We use the second theorem to define sqrt(c) IF c is non-negative.
You may say: But we extend the real numbers to the complex numbers in order to make sqrt(c) defined for arbitrary input. Why don't we do the same extension trick to make division defined everywhere?
As I said before, you are free to make any such extension you want. However, it is impossible to make a *useful* extension. You will always loose something in the course of extension. For example, when going from the reals to the complex numbers, we still have a complete field - but it is no longer ordered. This is a pity but bearable. Whatever trick one tries to define 1/0, the properties lost are generally considered unbearable.
Without a definition of *useful* this this is just a meanlngless statement and impossible to debate.
an example of an extension was referenced in my first comment
http://en.wikipedia.org/wi
If that is unbearable to you and hagman, no one is forcing you to use it, but it would be misleading to claim that everyone will always find it unbearable.
calm down - no one said anything about being unbearable - these are comments on statements not how people feel. this just not agreeing and forces the debate to come up with evidence - I agree the word 'useful' is somewhat 'wooly' and the link you provided is interesting but doesn't really provide the definiton of useful. I made me smile the that link had restrictions on division by zero (a/0 where a is non-zero)
Excellent... so there are systems that allow division by 0, but they have limits and might not be as useful as the standard system.
ozo,
I read the Wikipedia article and the the external link there (http://mathworld.wolfram.
I also read up on Dr Anderson's Nullity idea. http://www.bbc.co.uk/berks
one property it gives up is the ability to add any element of a field to any other element of a field to get a result that is also an element of the field.
The nullity idea seems like a mixture of an affinely extended real line and a projectively extended real line, with unclear continuity properties.
It is shown being used to define 0^0, but that seems pointless when 0^0 already has a good definition.
>>with unclear continuity properties. t is shown being used to define 0^0
Since it requires to be defined.
hagman: It may be that the real nunmbers form a Field, but before forming a field the ordinals must be defined and before the ordinals the cardinals. And zero is the cardinality of the so-called empty set BY DEFINITION. It therefore comes before 1 in the ordinals BY DEFINITION.
The entire of modern mathematics requires the elementary set theory first laid down by Cantor and it also requires the existance of transfinite numbers. In spite of the fact that the function 1/x can be made indefinitely large by judicious choice of x, it cannot be made transfinite (since the partition of a transfinite set is always transfinite). Therefore 1/x cannot be a transfinite number but must remain undefined.
The Real Projective Line, from ozo's link, has the property that the infinity operation + is undefined, whereas in the continuum it is not, since aleph null may be added to itself to give itself (this is based on one-to-one set correspondance, which in the RPL is not the case).
@BigRat:
You need set theory (I will loosely refer to ZF in the following) before the reals only as a matter of foundation - IF you "believe" that at least one set exists and for any two sets x,y the sets usually denoted {x,y}, Ux, Power(x) exist, that for each suitable formula Phi a set usually denoted {t in x | Phi(t)} exists, that a set INF exists that contains both {t in x | not t=t } for an arbitrary set x and for each t in INF also U{t,{t,t}} in INF ...... THEN you can construct a lot, esp. you have the notion of ordinal (a transitive set of transitive sets that are totally ordered by "in") and the notion of cardinal (an ordinal that is not bijectable to a smaller ordinal) but of course yuo also have the natural numbers N as a certain subset of INF, can define addition and multiplication on N, can proceed from there to Z, then Q, then R to have the field of real numbers. Fine.
BUT you can also discuss real numbers without founding them on set theory, namely by simply considering a complete ordered field. Moreover this is independant from your foundation (ZFC or NBG or ...) as well as deails of your construction (Dedekind sections or Cauchy sequences? But also e.g. when stepping fom N to Z by using ordered pairs: Kuratowsky or Wiener?); one can also ignore the fact that 0 in Z, Q or R is never the ordinal 0 / empty set.
Note that in the set theory approach, even addition and multiplication of natural numbers are somewhat arbitrary, hence one could hardly expect more arbitrary operations like subtraction or division to be "perfect".
>>BUT you can also discuss real numbers without founding them on set theory
That is true, basically one can make anything one likes as axiomatic. I have no idea of the ramifications of that approach, would however claim that traditional mathematics is set based which makes the operation undefined for the reasons I have given.
> surely x^0 = x/x so
That is one way to define it.
Which may be suitiable for some peoples purposes, but not others.
There are several different meanings of the symbol ^
Some of them even refer to functions that behave very similarly except at (0,0)
which can cause confusion when people don't realize they are talking about the different functions
I usually prefer to talk about the one that defines 0^0 as 1,
but I acknowledge that there may be times when a different ^ that handles 0^0 differently may be preferred.
(other uses of ^ can be so completely different that it quickly becomes apparent when different things are meant)
> 0^0 = 0/0 = 1
(Some would say
0^0 = 0/0 = undefined
I usually preder
0^0 = 1 != 0/0
> and also 0^x=0 for x>1 and has derivative 0 at x>1, so how the sudden jump to 1 at x=0, where does the energy needed to do that come from?
I don't know what "energy" might mean when referring to a function, nor why it needs to come from anywhere.
There is a discontinuity at (0,0), but many functions have discontinuities.
There are times when you want to restrict yourself to continuous regions of a function, and if so, you may want to avoid (0,0)
(But even if you do, the derivative of x^y can grow unboundedly, so if that requires some kind of "energy" that is limited, you may want to avoid an even larger region)
> If anything 0^0 =0 and 0^x has derivative 0 at 0, which would be ok
For most applications,
0^x is a relatively unimportant function, whereas x^0 turns up in many theorems.
But if 0^x happens to be important for what you are doing,
you may use the definition of ^ that is most convenient to you
I would only suggest that for the sake of clarity, when you are talking to others who may not
be dealing with that particular application, you may want to explain the peculiarities of the ^ as you are using it.
It is interesting to note that the debate recalls that of the end of the nineteenth centuary, where people like (I hope I've got this right) J.J.Sylvestre argued that "what is true up to the limit is true at the limit". Hence things like 1/x as x tends to infinity is zero, since it gets smaller and smaller but never negative, so in the limit it must be zero.
The problem is immediately obvious that if 1/oo is zero, then 1/0 is infinity. If one says no, then it follows that infinity may be used in one place but not another and the question arises as to why (as Dan did at the beginning). It was (historically) the set theoretical considerations of Cantor and others who put all of this on a firm foundation, that results in the inability to define 1/0. This had the side effect of claiming the existance of transfinite numbers.
There just doesn't seem to be a way of avoiding either infinities or infinitesimals, although for the latter there seems to be a limit in the Planck length (at least a physical limit).
Both chilternPC and Infinity08 referred to my interest in the idea of "what if we replace the zero with a very small number (e.g. 0.00001)" but the formulas in the Taylor series and the discussion of Limits in calculus tend to zip right over my head. For this I apologize profusely. Nevertheless, soldiering on... I'm still interested in this facet of the issue.
What if, every time we see x/0 we replace the zero with "h-bar" or 1.6 × 10^-35 or some other infinitesimal. Instead of getting "undefined", we get an answer ("really, Really BIG"). How will that screw us up when doing normal calculations?
> Instead of getting "undefined", we get an answer ("really, Really BIG"). How will that screw us up when doing normal calculations?
For at least some things, we'll simply get wrong answers. Answers will indicate something as being possible when it isn't.
Tom
p.s. Of course, there's maybe a _chance_ of the opposite being true too. For something; not sure what.
>> How will that screw us up when doing normal calculations?
Depends what the calculations are for. But when a division by zero is involved, there's a good chance that it was intentional, and that you can't get away with replacing zero with a very small number, since that would change the meaning of the calculation.
when doing normal calculations on a computer, sometimes what looks like 0 is really floating point underflow
and dividing by floating point underflow can be a floating point overflow.
But that can screw up in that you don't know whether it was a really, Really BIG positive number, or a really, Really BIG negative number, and trying to do more arithmetic with it only creates more uncertainty
Dividing by 0 also lets you do things like
a=b
a^2 = ab
a^2-b^2 =ab-b^2
(a-b)(a+b) = b(a-b)
a+b = b
b + b = b
2b = b
2 = 1
>2 = 1
but that's right isn't it?
If you find a value for x in 2x=x it could be 0.
What you actually did there is divide zero by zero and decided the answer was 1, I thought we were trying to assume that a division by zero gave 'a really big number', and multiplying this by zero would give 0.
I'm blaming your division of zero for that result, not the division by zero.
How about : There's no such thing as zero.
I've argued that zero is the cardinal of the set which matches no other set and have argued that zero amongst the ordinals is a matter of definition. I'll contend that except for the cardinal concept, there's no such thing as zero.
How? Well quantum mechanically there is no such thing as "zero" energy - the lowest state is always energetic. Assuming that there is something like the Planck Length, a length which is the smallest possible length, and assuming the relativistic invariance of it, then there is no such thing as zero length, nor zero area nor zero volume and it follows on that there is so such thing as zero time.
All the zeros which we know are cardinal zeros, and are therfore conceptual. That is why it is not possible to divide by zero, because zero does not exist.
BigRat, my head is spinning. Would I be incorrect if my translation of what you said was something like:
It is possible to have an amount of zero, "There are zero apples in the basket." Conceptual because I need to imagine an apple to see that there are none there.
It is not possible do any calculations using zero. To do any math(s) you need to have actual values and zero has no value so it cannot be used.
> What I'm proposing is that mathematics should reflect nature...
My (imperfect, incomplete, etc.) understanding is that arriving at a point in calculation where division by zero is indicated, essentially it also indicates a physical impossibility. Perhaps in that sense your proposal is just what happens.
Tom
> mathematics should reflect nature
nature is way too complicated.to reflect in mathematics.
In terms of human experience, there's not such thing a a google.
But that means that the result of adding two numbers is not necessarily a number that exists,
so the rules of addition would have to be different for different numbers.
Mathematics simplifies that to very basic rules.
so mathematics doesn't become invalid when we discover quantum mechanics or figure out what happens at the Plank length,
Mathematics can be difficult enough just dealing with the implications of simple rules,
Trying to reflect nature is the job of science, which should use the appropriate mathematics in appropriate ways for a given situation.
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by: chilternPCPosted on 2008-12-23 at 17:20:46ID: 23238078
"Extracting the square root of -1 is obviously impossible and nonsensical" - actually it is sensible and is explained when using vector (geometry) maths.(see http://en.wikipedia.org/wi