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Thursday, March 11, 2010

Division By Zero

This post is actually a response to someone on a different blog. But, as I've tried to get this point across to others before, I've decided to post it on my blog (now that I have one) so as to have it written up once. The point I indend to get across to Gentle Reader is this --

Division by zero yields the infinite number.

Now, admittedly, due to the word "yields," that phrasing is less than perfect. But I can't think of a better way to succinctly phrase it. If one prefers, one can think of the issue this way: The answer to the question, "What is 'X' divided by zero?" is "the infinite number" (technically, it's the first of the trans-finite numbers, that is, it's the trans-finite number known as "aleph-null").


Following is the recent exchange on Edward Feser's blog which prompts this curent post:
Martin: Any formal system that is interesting enough to formulate its own consistency can prove its own consistency iff it is inconsistent.

[Note: 'iff' is a technical term meaning "if-and-only-if" see here]

Ilíon: Is it not the case that an inconsistent formal axiomatic system can prove *any* statement which can be formulated within the system?

Martin: Yes, if you allow inconsistency then you can prove anything. This is why division by zero is not allowed in arithmetic and there undefined operations. It order to be useful it must be incomplete.

Ilíon: Actually, the reason division by zero is -- by axiom -- not allowed in arithmetic is that division by zero yields "the infinite number." It's not that division by zero makes arithmetic inconsistent (*), it's that arithmetic is not robust enough to deal with "the infinite number."

(*) If division by zero made arithmetic inconsistent, then it is arithmetic itself which is inconsistent.


Martin: Division by zero does not produce an infinite number! It is undefined because it allows for an inconsistent arithmetic and using it you can prove any two numbers equal each other. This point is discussed at length in the book "Zero: The Biography of a Dangerous Idea" by Charles Seife. This is also why the IEEE floating-point standard has the constant NaN (not a number) and integer division by zero throws an exception on most computers.
Now, as for calculators and computers, the reason they are designed to throw an exception when a division by zero is attempted is two part:
  • the operation cannot be completed arithmetically in finite time;
  • and, even if the operation could be completed arithmetically, the answer cannot be "held" in the computer's or calculator's register -- by the same token, one can encounter the same issue when trying to use these devices to multiply large numbers
Well, that and the fact that arithmetic isn't robust enough to deal with "the infinite number," even could the operation be completed or the result be represented in the physical device.

====
Now, to the proof that The answer to the question, "What is 'X' divided by zero?" is "the infinite number".

First, Gentle Reader must understand that when I say "Now, to the proof ..." I am not using the term 'proof' in the (limited) mathematical sense.

Recall, Martin and I were talking about the concepts of 'consistency' and 'completeness' as applied to "formal axiomatic systems," such as arithmetic. Arithmetic is 'incomplete' (and thus, all of mathematics is incomplete) -- what this means is that there exist true arithmetic statements which cannot be derived from the axioms and rules of arithmetic; or, to put it another way, there exist true arithmetic statements which cannot be proven via arithmetic.

However, these true-yet-unprovable arithmetic (and higher math) statements can be grasped and understood to be true by a human mind -- after which one may treat them as axioms. So, I hope to prove my claim by showing Gentle Reader *how* to go about grasping the truth of the claim. It's up to Gentle Reader whether he will pick it up.

Arithemetic is counting, that's all it is; each of the four arithmetic operations (addition, subraction, multiplication, division) is counting. The first two are simple counting: counting upward (addition) and counting downward (subtraction). The other two are more complex counting, they are the counting of counting: multiplication is the counting of additions; division is the counting of subtractions.

Now, long division, is simply a shortcut technique for doing the series of counting of subtractions to solve a specific division problem. Once we have been taught long division, we habitually use it to solve a division problem -- after all, the long division technique is quicker and far easier than doing division in its basic form. And, I suspect, we tend to forget what division really is.

Consider the multiplication problem: "5 * 4 = x" and let's forget that we have memorized the multiplication table. How can we solve the problem? By multiple additions, of course, same as when you used to do arithmetic on your fingers. The problem "5 * 4 = x" is equivalent to "5 + 5 + 5 + 5 = x" (or, if one *really* wants to get down to basics, it is equivalent to "(1+1+1+1+1) + (1+1+1+1+1) + (1+1+1+1+1) + (1+1+1+1+1) = x" )

Similarly, a division problem can be solved by reiterative subtractions ... because that's what division really is. The division problem "20 / 5 = x" can be solved by counting how many itertions of subtracting 5 from the result of the previous iteration are required to get the result to a number less than 5. Thusly:
  • 1] 20 - 5 = 15
  • 2] 15 - 5 = 10
  • 3] 10 - 5 = 5
  • 4] 5 - 5 = 0
As it takes 4 iterations to reduce the dividend to a number less than the divisor (and, in this example, there is no remainder), the quotent/answer to the problem "20 / 5 = x" is 4.

So, let us apply what we have learned -- or, realized that we already know, but have overlooked -- and divide some number by zero. It doesn't matter which number we choose, the correct answer is always "the infinite number." Consider the division problem "20 / 0 = x." We can attempt to arithmetically solve it thusly:
  • 1] 20 - 0 = 20
  • 2] 20 - 0 = 20
  • 3] 20 - 0 = 20
  • ...
  • 99] 20 - 0 = 20
  • ...
  • 999] 20 - 0 = 20
  • ...
  • 5359] 20 - 0 = 20
  • ...
  • 999...999] 20 - 0 = 20
  • ...
As Gentle Reader can clearly see:
1) division by zero does not make the arithmetic we are doing inconsistent;
2) as the result of the subtraction at each iteration continues to be equal to the dividend, the problem can never be completed via arithmetic -- there is always at least one more iteration to be performed
3) thus, the answer to the division problem "20 / 0 = x" is "x = infinity"
3a) however, this answer can never be arrived at via arithmetic (for arithmetic is 'incomplete' )
3b) nevertheless, a human being can see/understand that the result of division by zero is always "the infinite number."

As I said at the first, I can show Gentle Reader *how* to grasp the truth that the correct answer to any division by zero is "the infinite number." But it's up to Gentle Reader to let go the incorrect assertion about division by zero that he was taught so many years ago, and go ahead and grasp the actual truth of the matter.

========
The reasons I have categorized this post under "free will" and "pious myths" and "reason" are as follows:
  • Firstly, I'm trying to keep the categories to a manageable minimum, so for this post I didn't create new categories such as "mathematics" or "logic"
  • As they are so connected, I'm tending already to use the category "reason" to also include "logic"
  • I've categorized this under "pious myths" because it's a myth, a mis-teaching about arithmetic, that I'm intending to help the reader see through
  • I've categorized this under "free will" because the reader must *choose* to see through the mis-teaching we were all taught in our schools

21 comments:

Martin said...

Google division by zero undefined.

There's a lot written on this topic and division by zero is not allowed because it produces results as follows:

3 * 0 = 5 * 0

3 * 0/0 = 5 * 0/0

0/0 = 1 as any number divided by itself is 1 if you allow division by zero.

3 = 5

I fully realize that computers have finite precision, but why is the result called NaN (not a number) instead of some form of infinity?

Martin said...

There's also the issue that outside of arithmetic the concept of dividing by zero is nonphysical. For example:

How many apples does each person get if you distribute 10 apples among 0 people?

There are a finite number of apples so zero people do not magically get infinite apples. In mathematical jargon, a set of 10 items cannot be partitioned into 0 subsets. So the concept is meaningless, or undefined. So there's no deeper concept for the mind to grasp.

Ilíon said...

You know, I wrote up something which already answers your objections. Did even you read it?

And, some of your objections simply reflect faulty logic. For instance:
Martin: "Google division by zero undefined.

There's a lot written on this topic and division by zero is not allowed ...

I fully realize that computers have finite precision, but why is the result called NaN (not a number) instead of some form of infinity?
"

These objections are relevant, how? I presented an argument showing that the commonly-taught position is incorrect ... and you responded, in effect, "Everyone else disagrees with you, so you're wrong."


Martin: " ... division by zero is not allowed because it produces results as follows:

3 * 0 = 5 * 0

3 * 0/0 = 5 * 0/0

0/0 = 1 as any number divided by itself is 1 if you allow division by zero.

3 = 5
"

I'm aware of that little trick ... the "result" follows from making some arithmetic and logic errors.

For instance, the first step ("3 * 0 = 5 * 0") *really* equals ("0 = 0") -- and we are done.

Also, "0/0 = 1 as any number divided by itself is 1 if you allow division by zero" is not actually true. In fact, "0/0 = 0" -- as I've actually shown in the OP, were you to have paid attention to what I wrote. "0/0 = 0" for the very same reason that "0/1 = 0" -- the dividend is *already* zero, and thus there are no possible iterations of subtracting the divisor from the dividend to get a quotient.

Ilíon said...

Martin: "There are a finite number of apples so zero people do not magically get infinite apples. In mathematical jargon, a set of 10 items cannot be partitioned into 0 subsets. So the concept is meaningless, or undefined. So there's no deeper concept for the mind to grasp."

Now you're just being silly.

Does dividing 10 apples by a non-zero number change the number of apples which exist? No; so why would you pretend that dividing 10 apples by a zero changes the number of apples which exist?

Must mathematical truths oe concepts be instantiable in "the real world" before they are true? If so, them most of mathematics is false.

Did I even say or imply that we can bring "the infinite number" into "the real world?" No: if fact, I explicitly said that arithmetic is not robust enough to deal with "the infinite number."

Martin said...

I read your post but it was fairly long and I may have missed something. But I don't think you understood my objections and that I didn't make them clear enough.

Computers have values they can place into their registers which are symbols to represent numbers that can't be represented directly. The floating point standard defines symbols for positive infinity, negative infinity, and NaN. NaN is not a number and is not the same as either form of infinity. So everyone else is saying that division by zero yields NaN. You claim this isn't relevant, but it is very relevant as the foundation problem in mathematics is unsolved. So we don't know if math is objectively real, purely a human invention, or a mix of the two.

Now you admit that arithmetic is not robust enough to handle division by zero, but then claim a discussion of the physicality of the concept is out of bounds too. If you are outside of arithmetic and outside of physical reality, then what is the foundation you are grounding your answer in? You're speaking as if mathematical truths are grounded in something that people can agree on, but that's not true until the foundation problem is solved. That's why I was making an appeal to the two grounds of this problem that have some common meaning.

Now I disagree with your subtraction proof that infinity is the value when you divide by zero. My counter proof is that division is actually the reciprocal operation of multiplication. So to understand division by zero we should multiply not subtract. So

r = a / b

a = r * b

Set b = 0 so this is equivalent to the division by zero problem and r is the number that satisfies division by zero.

a = r * 0

Now any number can be placed in r and make this equation work. So I counter claim that division by zero yields a finite number in a finite time. This is obviously the first number that satisfies the above equation. I also claim that at the present time mathematics has no foundation and only those bits of it we can verify against physical reality are objectively real. At this point you'll need to solve the foundation problem to state what you are grounding your view that there's a higher truth which says division by zero yields infinity. Also you are using arithmetic in your subtraction proof and then stating counter claims based in arithmetic are out of bounds. This means that you're invalidating your subtraction proof too!

Crude said...

"You're speaking as if mathematical truths are grounded in something that people can agree on, but that's not true until the foundation problem is solved."

"I also claim that at the present time mathematics has no foundation and only those bits of it we can verify against physical reality are objectively real."

Some comments.

* What does "something people can agree on" matter to the question of whether the foundation problem is solved? Are you saying that when the problem is solved, we'll know because no one will disagree? But why should anyone believe that?

* Not everyone agrees about the existence and nature of "physical reality" either. What about idealists? What about simulation theorists? What about neutral monists, or panpsychists? If universal agreement is the standard, the list of problems that are unsolved are going to quickly grow, possibly more than you realize yet.

Martin said...

Crude, mathematical truths are claimed to be either self-evident, postulated, or proven. Ilíon is correct that there are truths that we can't prove from our axioms that are still true. However, to be useful such truths would need to be of the self evident nature. You might consider making them postulates and be very carfeul when you use them in proofs. Now most people consider division by zero to be a meaningless concept, both within arithmetic and in the physical world, so it doesn't fall into the catagory of truths we can't prove yet are true anyway.

I suppose I have the naive hope that if the foundation problem is solved it will be self-evident that the solution is true. We could then use that as the external thing to verify unprovable truths with. Now you could be correct that we never solve the problem or fail to realize when we do. But that would leave us in our current state.

The funny thing about the underlying nature of physical reality is that it isn't really that important other then it exist and be consistent. If we're in a giant simulation we can still agree that division by zero isn't real within that simulation.

Now it would be really cool to know the metaphysically true nature of physical reality, but that lack of knowledge doesn't stop me for making use of it.

Martin said...

Ilíon, a tanget within a tanget. Your claim that "0/0 = 0" is also not consider true. While it might be true using a subtraction definition of division. It is indeterminite using the reciprocal operation of multiplication definition.

So now your rebuttle of my little trick isn't true and I claim 3=5.

Crude said...

Martin,

"The funny thing about the underlying nature of physical reality is that it isn't really that important other then it exist and be consistent. If we're in a giant simulation we can still agree that division by zero isn't real within that simulation."

That doesn't seem to wash - so if idealism is true, or simulation theory is true, or panpsychism is true, all of them are talking about 'physical reality'? That's emptying 'physical' of practically all meaning. Then again, that's pretty common nowadays.

"I suppose I have the naive hope that if the foundation problem is solved it will be self-evident that the solution is true. We could then use that as the external thing to verify unprovable truths with. Now you could be correct that we never solve the problem or fail to realize when we do. But that would leave us in our current state."

Not really. I mean, if what I'm saying is possible, then "[...] that's not true until the foundation problem is solved" fails. That's a pretty different state of affairs than what was started with here (that we'll know when we find the solution, because everyone will again.)

Martin said...

Crude said, "That doesn't seem to wash - so if idealism is true, or simulation theory is true, or panpsychism is true, all of them are talking about 'physical reality'? That's emptying 'physical' of practically all meaning. Then again, that's pretty common nowadays."

The question is what are the properties which we can percieve. Now there are a lot of theories of reality, but they all must be consistent with what we perceive. In our current perceptions I can make a pile of ten stones. I can sub divide that pile into 2 smaller piles, or five smaller piles. But I can't sub-divide it into zero smaller piles. Now all possible theories of reality have to be consistent with that observation or we would discard them.

Using Ilíon's subtration aporoach to division I can remove zero stones from ten stones an infinite number of times and do it for the rest of my life while I'm doing other things. But that statement doesn't seem to have any meaning either.


Crude said, "Not really. I mean, if what I'm saying is possible, then "[...] that's not true until the foundation problem is solved" fails. That's a pretty different state of affairs than what was started with here (that we'll know when we find the solution, because everyone will again.)"

I don't understand you. My statement is that at the present time arithmetic only has meaning in either mathematics or physical reality (like my concrete enumeration examples). Within those domains the concept of division by zero is considered to be undefined and without meaning.

Now what other domain is there to apply the concept of division by zero and get meaning out of it? Ilíon implied he knew of another one. Now maybe there is, but what is it? To me that seems like a question only a solution to the foundation problem can provide.

Now I don't know how to solve the foundation problem. It seems like something far beyond my ability and possibly my understanding. My hope is that if someone else solves it I might understand their solution and it made sense to me.

Foxfier, formerly Sailorette said...

...Could this not be phrased much more shortly by referring back to the old way of saying "X divided by Y," "how many times does Y go into X?"

You can put more and more and more of "zero" into five, and never reach five; thus, it is infinite.

Equally, three of nothing is still nothing; nothing to the tenth power is nothing.

Gotta keep in mind, though, that math at the very least started as a way to convey observed fact, that just because something can be stated as a word problem doesn't mean that it can actually be modeled, and that just because something can be modeled for stated knowns doesn't mean it's going to be right for unstated knowns. (The old turtle-- tortoise?-- and arrow thing comes to mind.)

Ilíon said...

Foxfier , of course this can be less verbosely stated by asking, "how many times does Y go into X?" -- or, since division is really just multiple subtractions, by asking "how many times can Y be subtracted from X?" (taking proper account of the signs of both numbers, of course).

I’ve already encountered extreme resistance (to the point of insisting that I am stupid) (*) to a less verbose explanation of this arithmetic truth. So with this post, my intent is prove the point -- as nearly as one can prove an inherently insoluble arithmetic truth -- by undertaking division at its most basic level.

As you can see, even with this longer demonstration, some persons will still resist even understanding the proof/argument (to say nothing of admitting its truth).


(*) The first fellow I tried to whom I tried explaning this, someone with a great deal of advanced mathematical training, kept trying to deny the truth of whatI was stating by pointing to the "approximation to infinity" used in calculus (1/x, where 'x' --> 0). His misguided reasoning was I was, in effect, claiming that "positive infinity" (1/x, where 'x' --> 0) and "negative" (-1/x, where 'x' --> 0) are equal. About half a dozen tines, I pointed out that (1/x, where 'x' --> 0) is never *actually* division by zero, for 'x' only approaches zero. And he'd make the same misguided objection; and finally "objected" that I am obviously too stupid to understand math.

Foxfier, formerly Sailorette said...

Ah, should've figured you'd have a reason for using so many pixels.

Martin said...

Foxfier, exactly just because something can be stated doesn't make it objectively real.

Real Numbers for example are probably not really real as space, time, matter and energy likely can't be subdivided forever. Eventually you'll hit the Plank scale and the game is over. So the arithmetic precision of the universe is likely fixed, although we obviously can't prove that from inside the universe.

I've also been ignoring the phrase "the infinite number", but technically infinity isn't a number either, it's a concept.

Ilíon, I'm not name calling or even hostile. But strictly out of curiosity, how many people have you convinced that when you divide by zero you get something meaningful?

Also, it dawned on me after ponding your blog entry to ask a question. Are you talking philosophically or are your trying to link this back to the domain of mathematics in some way?

Ilíon said...

Well, I do almost always have a reason (and, generally, a good one) for the pixels I expend.

Foxfier, formerly Sailorette said...

...Which would be why he calls it "the infinite number" instead of "infinity." If this is simple English, rather than a mathematical term, then "infinite" is a characteristic of the number, not the number itself. Especially when you remember that "infinite" is mathematically defined as "not finite," and "finite" includes the definition "not zero" among its meanings.

are probably not really real as space, time, matter and energy likely can't be subdivided forever
Irrelevant. Unless it can be shown, you're effectively guessing. Not a good thing to do when trying to prove something.

For your apple example, it can be rephrased: "distribute ten apples into zero groups." Or, for a more personally relevant example: "spend ten dollars zero times." Makes perfect sense, then, for 10*0 to equal zero.

You seem to be trying to change the topic.

Martin said...

I'm not trying to change the topic as I intend to bow out and stop discussing the concept. This isn't math or reality so I really don't care.

Foxfier, formerly Sailorette said...

...If you were going to bow out, why did you bring up new topics?

Topic one:
Physical reality of numbers, based on unproven beliefs.

Topic two:
Conflating the concept of a number that is infinite with the claim that "infinity" is a number.

Topic three:
Attempted topic change to the number of people whose minds are changed-- irrelevant on the topic of truth. Possible mis-statement of topic.

Topic four:
Asking a question, which I'll presume you actually wanted an answer to.

Awful lot of topics for someone that wants to "bow out."
Unless you were going for that cheezy "I get to say my piece on X topic and then it's over, and you're the one that can't let it go if you reply" thing? While I understand the tactical advantage of that one, I can't say I think very highly of it....

Martin said...

Actually I am curious about the answers to the questions. But I'll get those as an e-mail.

I brought up new topics because I wasn't being smart about time management at work. So when I say bow out I mean that I'm going to stop posting because of lack of time.

I'm also losing interesting in the topic because the difference between infinity and the infinite number doesn't make much sense to me. I wasn't even clear there was some kind of distinction until you claimed there was. So it's hard to debate a point you don't understand.

I thought saying I was going to stop posting would be less rude than vanishing.

Foxfier, formerly Sailorette said...

As you said yourself, infinity isn't a number.

Martin said...

That's the part that doesn't make any sense. In order to count the number of iterations in Ilíon's iterated subtraction method of division would require the summation of an infinite series which diverges to infinity. So it's the same thing as infinity.

Infinity is not a number because it's not possible to actually reach it. No matter how long the iterations go on in you're still as far away as when you started. Unless you sum the infinite series.

It doesn't make sense to talk of summation of an infinite series outside of mathematics because you can't do it.