# Division by zero. Go ahead and laugh.

Discussion in 'Undergraduate Math' started by Lefty, Oct 7, 2004.

1. ### LeftyGuest

No, I do not "divide by zero". But it did occur to me to mention it here as
an illustration of mathematical culture.

I have said before that mathematicians "do not like" paradoxes and
inconsistencies. It's all quite natural that they wouldn't, and I don't
blame them. But there are many who have voiced their disagreement with this
opinion. Please let me qualify my belief.

Division by zero creates an inconsistency in arithmetic, and any result can
be derived if such a thing is allowed. Now then, instead of always
stipulating for example, 1/x, x not 0, we simply state that 1/0 is
"undefined", and avoid it completely. We avoid it like the plague. The only
place I could find these things being treated seriously is in singularity
theory. But in general, people do not even acknowledge that this is a
singularity. They just state flatly that 1/0 "does not exist". And other
indeterminate forms are treated pretty much the same.

The fact that arithmetic falls apart when you let 1/0 is fascinating and
amazing. There is no reason to relegate this operation to the trash can. It
would be better to acknowledge the inconsistency, the fact that we don't
really understand why it exists, and to simply work around it.

The problem of division by zero is that it creates many paradoxes.
Mathematicians hate this, and therefore they have defined division by zero
to be "non-existent". Why didn't you guys just throw away zero altogether ?
I'll tell you why. Because you want to keep zero around because it's a handy
thing to have, but those damned paradoxes have got to go, so we'll just
"define them away".

It's the biggest cover up since Watergate.

There are no paradoxes or inconsistencies in mathematics, unless you remove
all the ad hoc stipulations and band aids which hold it all together.

Confess.

Lefty, Oct 7, 2004

2. ### Zodness AGuest

Wrong. 1/0 = infinity

Computers have problems with it, and they "hate" it.

What does 0/0 = ? infinity or 1 or some number n ?

Zodness A, Oct 7, 2004
in2infinity likes this.

3. ### LeftyGuest

Wrong. 1/0 = infinity

.....and I thougth that _I_ was picking a fight .....(!) heh heh -

Lefty, Oct 7, 2004
4. ### bhGuest

I know this is not a reasonable thing to say, but I will say it
anyway... Why is x/0 considered 'undefined' instead of 'infinity'...
or worse, it is considered to mean 0... ie 1/0=0, 2/0=0,
400y^2/0=0.... If I divide 4 by 1, I take 1 out and that leaves 3,
then I take it out again, that leaves 2, then again 1, and finally a
4th time I get zero, and that means that 1 goes into 4 exactly 4
times... By the same method, if I divide 4 by 0, I take 0 out once and
have 4, twice 4, three times 4, to infinity... It is counter-intuitive
to work toward a lower and lower divisor and get a higher and higher
answer, only to finally reach 0 and it becomes 'undefined' or 0... how
annoying... 4/1=4, 4/.1=40, 4/.01=400, and on and on... Oh well, I
guess this rambling probably belonged in a philosophy forum instead...

bh, Oct 7, 2004
5. ### Brock NashGuest

I know the people who argue about it believe me
1/0 is in fact defined! It is infinite. Think about it logically and
you'll understand.
Here's two reasons to support why Real#/0=Infinite
1: If you graph the equation y=2/x, when x = 0 y falls at infinite!
2: If you have 2 donuts and are trying to split them up into groups of
0, you can make infinite groups of 0. Here's your group of zero
donuts, here's yours....

It seems so obvious, I hate when people say it is undefined!!!!

Brock Nash, Oct 7, 2004
6. ### Brian M. ScottGuest

in <in

[...]
This is simply incorrect. There is no inconsistency; see
below.
No. What you apparently don't quite understand is that
division is a *defined* operation, defined in terms of
multiplication. The statement that a/b = c means _by_
_definition_ that bc = a. If the equation bx = a has no
solution in the domain of the multiplication operation, then
there simply is no c for which bc = a, and a/b is a fortiori
just happens that when the domain is the rational, real, or
complex numbers, 0 is the only choice of b for which this
happens. If you define division similarly in the ring of
integers mod 10, say, you'll find that 5/2 is similarly
undefined, because there is no solution to the congrence
2x = 5 (mod 10).

[...]

Brian

Brian M. Scott, Oct 7, 2004
7. ### Brian VanPeltGuest

mathematician Euler. Evidently, Euler allowed 0/0 to equal anything
he wanted. In fact, I read that he once said that 0/0 = 2 because
"the first is twice the second". I thought that was quite amusing,
and I am wondering how much of his prolific writings used 0/0.

Brian

Brian VanPelt, Oct 7, 2004
in2infinity likes this.
8. ### Keith A. LewisGuest

A couple of analogies come to mind.

A singularity is a like hole in a function. Asking what the value of the
function is in the hole is like asking what color a hole in a piece of paper
is. There's no color there; it's undefined.

1/0 is the answer to the question "How many trips to the well will it take
to get 1 gallon if my bucket holds 0 gallons?" You can go there as many
times as you wish but you won't get any closer to 1 gallon. In that sense,

--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.

Keith A. Lewis, Oct 7, 2004
9. ### LeftyGuest

The inconsistency which I refer to is division by zero, itself. It has been
defined to not exist, out of laziness.

No kidding. In fact, maybe it _is_ multiplication - eh ? And we'd all be in
trouble if it were'nt defined. But the fact that your definition must
contort itself to avoid a catastrophic breakdown of arithmetic, this is very
suspicious.

You really have to bend the definition quite a bit to say that 1/x is ok
everywhere in R, except at 0. We're waving a magic wand and simply claiming
that it dosent exist, and so just please pretend that you didn't see it
because it dosen't exist.

Hmmmmmmmm.....

Lefty, Oct 7, 2004
10. ### LeftyGuest

dated Thu, 07 Oct 2004 01:18:44 GMT:

Very true. Or you could say that you were a billionaire for zero minutes.
There are lots of wierd things like that.

Lefty, Oct 7, 2004
11. ### David W. CantrellGuest

Division can be -- and of course very often is -- defined in that way.
But in a system (such as that of the extended complex numbers, for example)
in which division of nonzero quantities by zero _is_ defined, division is
not defined in that way.

Division of nonzero quantities by zero being defined in such a system does
not lead to "paradoxes". But of course, if one were mistakenly to assume
that 1/0 and 0 were multiplicative inverses, then one would indeed be in
trouble quickly.

David Cantrell

David W. Cantrell, Oct 7, 2004
12. ### Brian M. ScottGuest

This is a false statement, as I explained in the previous
post. Evidently you did not understand the explanation.
Did you even bother to read it?
No, it is not.
No, you do not. That is an immediate and obvious
consequence of the definition. It's a very straightforward
definition; the fact that you understand it is rather
telling.
No. Division has no existence independent of its definition
as an inverse to multiplication. The symbol a/b is given
meaning only by its definition. Its definition says that
the statement 'a/b = c' is true if and only if a = bc. To
put it even more simply, 'a/b = c' is by definition
synonymous with 'a = bc', an expression whose meaning is
already known. Consequently, if the equation a = bx has no
solution, then there is no value of c for which the
statement 'a/b = c' is true.

In the ring of real numbers the equation a = bx has no
solution if b is zero and a is not, and as a result there is
no real number c that makes the statement '2/0 = c' true,
*because* there is no real number c that makes the statement
'2 = 0*c' true.

In other rings, like Z_10, the integers modulo 10, there are
other pairs of a and b for which a = bx has no solutions. I
already gave the example of the congruence 5 = 2x (mod 10),
which has no solutions, pointing out that for this reason
the expression 5/2 is undefined in Z_10.

But I'm wasting my time, since you obviously have no desire
to be confused by the facts.

[...]

Oh, by the way: if you want at least to pretend to be
serious, learn to mark your snips.

Followups set.

Brian M. Scott, Oct 7, 2004
13. ### The World Wide WadeGuest

Brian, he's a troll.

The World Wide Wade, Oct 7, 2004
14. ### Richard HenryGuest

Have you met James Harris, by any chance?

Richard Henry, Oct 7, 2004
15. ### David W. CantrellGuest

One must be careful in dealing with analogies. Consider the following:

1/2 is the answer to the question "How many trips to the well will it
take to get 1 gallon if my bucket holds 2 gallons?"

Of course, that's absurd. It will take 1 trip, and yet we certainly don't
want 1/2 = 1 !

What I said above was largely in jest. But seriously, if one chooses a
better scenario (similar in spirit to what you had in mind), it's easy to
see why 1/0 is indeed infinite. Below my signature is a copy of a response
of mine from an old thread.

David Cantrell

-------------------------------

[Copy]
It is true that the notion of division is normally
linked with the notion of multiplicative inversion. That works beautifully
if you restrict yourself to, say, the real number system and avoid
division by zero. However, if we wish to be able to speak reasonably of
dividing a nonzero value by zero, then we cannot restrict ourselves to
the notion of multiplicative inversion (since zero cannot have a
multiplicative inverse).

Consider this very simple idea. Take a nondegenerate interval A on the
real line. Say, A = [0, 36].
How many intervals of length 18 are contained in A? 2

N.B. Here and below, I am thinking of _distinct_ closed intervals,
pairs of which would have at most one point in common. So, for the
above example, the two intervals are [0, 18] and [18, 36].

How many intervals of length 12 are contained in A? 3
How many intervals of length 1 are contained in A? 36
How many intervals of length 1/2 are contained in A? 72
How many intervals of length 1/100 are contained in A? 3600
How many intervals of length 1/1000000 are contained in A? 36000000
How many intervals of length 0 (which are, of course then, degenerate
intervals) are contained in A? This is essentially the same as asking
how many points are in A. The answer, clearly, is that there are
infinitely many, oo.

Just as the second example above corresponds with 36/12 = 3,
the last example corresponds with 36/0 = oo. Very simple, and not
based on the notion of multiplicative inversion. (But I wouldn't call it
"mathematical magic" either.

Note that, despite what is often said, taking x/0 = oo for nonzero x
does not lead to contradictions -- that is, _if_ you know what you're
doing. Of course, for those who don't know what they're doing, all sorts
[End copy]

David W. Cantrell, Oct 7, 2004
16. ### Lewis MammelGuest

I think you've got a problem here.

Lewis Mammel, Oct 7, 2004
17. ### Jon HaugsandGuest

*
What color has number 10? What is the least sad number? There are a
lot of operations that the mathematicans "avoid like the plague".

Jon Haugsand, Oct 7, 2004
18. ### Norm DresnerGuest

A properly programmed modern computer that uses the right representation
system for numbers can produce the results
1.0 / 0.0 = "infinity"
1.0 / "infinity" = 0.0 IIRC
and it will also give NAN (not-a-number) in most appropriate situations
too.

It all depends on how it's programmed. The hardware could care less.

Norm

Norm Dresner, Oct 7, 2004
19. ### Richard WrigleyGuest

--
Richard.

"I have yet to see any problem, however complicated, which when looked at in
the right way, did not become still more complicated"
Poul Anderson

In this case the calculation should be carried out in the ring of Intigers,
trips can only be whole numbers, and the rules of division should be framed
accordingly, i.e. that odd / even = (odd + 1) /even.

Richard Wrigley, Oct 7, 2004
20. ### JustinGuest

: It's the biggest cover up since Watergate.
....
: Confess.

You're either stupid or trolling. I would venture the latter, but you
are free to suggest it may be the former.

Best,
Justin

Justin, Oct 7, 2004