In mathematics, division by zero is undefined.  However, why can't a
number divided by zero be defined as equal to infinity?

Hi halejrb,


   Good day. This is a very interesting question which has troubled
all of us at one stage or the other. Some of us have learnt that that
x/0 is undefined, some of learnt that x/0 is infinity (x not equal to
0, in which case it is undefined). I have learnt the latter.

   However, I'm not going to answer based on what I've learnt. I've
browsed a few internet sites, and I've come across some interesting
explanations. One particular site says that,

"There are, however, contexts in which division by zero can be
considered as defined. For example, division by zero z/0 for (z
belongs to C*, z not = 0) in the extended complex plane C-Star is
defined to be a quantity known as complex infinity."
( URL :: http://mathworld.wolfram.com/DivisionbyZero.html )

Related URLs

Division by Zero and Continuity
http://www.math.utah.edu/~alfeld/math/precise.html


University of Toronto Mathematics Network - Answers and Explanations
Is there really such a thing as "infinity"?
http://www.math.toronto.edu/mathnet/plain/answers/infinity.html


Why You Can't Divide Nine By Zero
http://www.math.toronto.edu/mathnet/plain/questionCorner/nineoverzero.html


Division by Zero
http://mathforum.org/library/drmath/view/58497.html


I hope this answers your question. Have a good day :-)

Cheers,
aditya2k

Wow, that's a great answer.  The U. of Toronto site gives the typical
mediocre high school answer that I got when I was that age.  But I
always suspected that certain higher level math applications allowed
for division by zero and you've satisfied my curiosity.

Hi again halejrb !

"But maybe you're thinking of saying that 1/0 = infinity.  Well
then, what's "infinity"?  How does it work in all the other
equations?  

Does infinity - infinity = 0?  
Does 1 + infinity = infinity?

If so, the associative rule doesn't work, since (a+b)+c = a+(b+c)
will not always work:

1 + (infinity - infinity) = 1 + 0 = 1, but
(1 + infinity) - infinity = infinity - infinity = 0.

You can try to make up a good set of rules, but it always leads
to nonsense, so to avoid all the trouble we just say that it
doesn't make sense to divide by zero.

What happens if you add apples to oranges?  It just doesn't make
sense, so the easiest thing is just to say that it doesn't make
sense, or, as a mathematician would say, "it is undefined".

Maybe that's the best way to look at it.  When, in mathematics,
you see a statement like "operation XYZ is undefined", you should
translate it in your head to "operation XYZ doesn't make sense"."
http://mathforum.org/library/drmath/view/53336.html

Hope it helps !

THX1138

The last comment explains this quite well. It can be defined as
infinity however that doesn't make sense. However, even that is not
true because similar to the creation of negative and complex numbers,
there is another more universal form of arithmetic called non-standard
arithmentic which uses non-standard numbers (i.e. multiples of
infinity of infintesimals).

dear halejrb;
  i have read your question & also an answer provided & comment.
what i feel is that equality(=) is something that gives an association
of exactness, whereas infinity is a relative thing.definately one
infinity can be greater than other.if we go by the defination of
division , then division by zero is the number of zeroes in which the
given number can be broken into. logically a larger number can be
broken into larger number of zeroes(relatively). so if we define any
number divided by zero as "=" infinity then there shall be
ambiguity.since infinity is not exactly a value but a range of values
( as we saw that one infinity can be greater than other) hence there
cannot be the equality association while dealing with infinity.

The problem is one of consistency.

If A/B = C, this implies that A = B * C.

Suppose A = 1 and B = 0

If we set C = infinity, then the first equation is fine, but the
second becomes false since 1 <> 0 * infinity.

Therefore division by zero is undefined.

The past comment is correct however: 
When graphing x/0 (x<>0) you would graph an asympote to a line on the
point with a value of something over 0.

0/0 =0 not undefined.

While this isn't a direct answer I think it will convince you ...

X * Y is defined to be the area of a rectangle with sides X and Y.
Division is defined to be the inverse of multiplication. In other
words if A = X * Y then A / X = Y. Consider rectangles R1 which is 5
by 0 and R2 which is 7 by 0. By multiplication we see that R1 has area
5 * 0 = 0 and R2 has area 7 * 0 = 0. If division were defined for zero
then 0 / 0 would have to be both 5 and 7.

More generally in any system that is not trivial, undefined has to be
an outcome in at least one situation. Otherwise you would be able to
resolve equations where sufficient information is lacking (which would
mean that the system is trivial). The stuff above is a good example;
having just the area and that one side was length zero, you lack
sufficient information to know the length of the other side. If you
could solve 0 / 0 then you would be able to calculate values that you
don't have sufficient information to know.

I've always been taught that infinity is not a number, but a limit
concept.  If infinity were a number then 1 = 2: Because 1*Infinty =
2*Infinity but if infinity were a number then you could divide by it
and the above equation would be 1 = 2, which is wrong.  So I guess you
would have to say that dividing by zero is undefined, although in
certain engineering applicaitons you can take it to be infinity to
help the numbers come out correctly.

-TheMabbit

I have to disagree with the spirit of some comments. There is a
sensible and consistent way of doing maths, different from the
standard one, where some division by some "zeros" does make sense, and
equals some "infinity". According to

Renling Jin, professor of mathematical logic, College of Charleston

Nonstandard analysis allows us to use "infinitely large" numbers and
non-zero numbers which are "infinitesimally close to" zero, in an
enlarged universe called nonstandard universe. Taking this advantage,
one can derive a result in the nonstandard universe and then push down
the result to the standard world to get an interesting theorem.

http://math.cofc.edu/areas.html

search nonstandard analysis

http://members.tripod.com/PhilipApps/nonstandard.html

http://mathforum.org/dr.math/faq/analysis_hyperreals.html

Additionally, division by zero reaches new bounds of inexactitude when
we divide zero by zero.  There are three conflicting definitions here:

X/X = 1
X/0 = undefined (or infinity, depending on your textbook)
0/X = 0
X/Y = X * (1/Y)  [Such that 0/0 = 0 * infinity/undefined, and 0 *
infinity is a question unto itself]

So, as a short answer: working with the idea of infinity as a concrete
number like 3 or 42 will always lead to heartbreak, and dividing by
zero is synonymous to multiplying by infinity.

Cheers!

Here's my 2 cents' worth, and bear in mind I happen to be a
professional mathematician.

For thousands of years, mathematicians debated this and related
questions. In the beginning, only the existence of the natural numbers
(1,2,3,..) was acknowledged, since they were used for counting and
bore an obvious correspondence to "real" objects i.e. 3 apples, 4
oranges etc. The concept of zero did not exist, nor that of infinity,
the negative numbers, or the real numbers. "Does it make sense to
subtract 7 from 3?" someone asked one day in a prehistoric "google
answers" forum. Obviously not, since if you have 3 apples you cannot
take away 7! Someone answered. However, it was eventually realized
that the _concept_ of subtracting 7 from 3 can be useful sometimes,
for example if I have 3 apples and want to buy from you something in
exchange for 7 apples, we can both agree that I should owe you 4
apples and give them to you at a later time, so in a sense for the
duration of the debt I will have "-4" apples.
And so the negative numbers, and zero (3-3=0) were introduced. The
fractions, or rational numbers, were soon to follow as the solutions
to the equation x*a=b, where a and b are integers. Then it was
realized by the Pythagoreans that even they were not enough, since
equations such as x^x = 2 (which describes the diagonal of a square
whose side is 1) cannot be solved in rational numbers, so the real
numbers were introduced. Then, a millenium and a half later, it was
eventually realized that even the enigmatic and seemingly meaningless
equation x*x=-1 can be made to "make sense" if one introduces the
concept of imaginary and complex numbers.
As for the concept of infinity, it must have been introduced in the
last few centuries as mathematics progressed to more and more advanced
notions. It does not have one single meaning but again it is a useful
concept in various contexts. It is "the number that is greater than
1,2,3,4,..." (again, the precise meaning of this differs with the
context, in higher mathematics there are many different notions of
infinity such as cardinal numbers, ordinal numbers, orders of
magnitude of growth of a function, the pole of the Riemann sphere and
many more)

Notice that, in all the cases described above, the introduction of the
new concept always followed a stage where everybody thought that such
a concept can not exist, is "meaningless", "undefined" etc. The mere
mention of the possibility of existence of such a concept would
provoke very emotional, even outraged responses. (There are some
well-documented examples of this, specifically for the case of the
introduction of complex numbers, you can read about this in the
wonderful book "The Mathematical Experience" by Reuben Hersh and
Philip J. Davis, you are all big boys so I don't need to give an
amazon.com link).

Finally, sometime in the 19th century, mathematicians began to realize
that there is really not much point in debating whether something is
meaningful, makes sense etc. This was wasting a lot of time and energy
and hindering the development of new mathematics. Do you want 1/0 to
be defined as 5? Someone suggested. Go ahead! See where that gets
you... So, a new approach was introduced, the so-called axiomatic
approach, according to which one may define any new concept one
wishes, and explore the logical consequences thereof, without having
to answer to silly questions such as "does it make sense". One may
even redefine addition as 3+4=5 if one should be so inclined. The
ultimate test for the value of a mathematical theory was whether it
contained beautiful ideas, had concepts and results that were
esthetically appealing, and also it helped (but was not necessary)
that the ideas be useful.

After this very long introduction, I will come back to the original
question of division by zero: You may define it anyway you like.
However, there are some definitions, and in some given contexts, that
have been found to be useful and to possess properties that are
appealing. Infinity is the most common of them, practically anybody
can understand why intuitively it is a logical choice. However, in
some contexts there doesn't really seem to be a satisfying answer, one
that appealing properties, and this is why traditionally it is
explained to laymen (rather than give the long explanation I gave)
that division by zero is "undefined".

Hope this helps,
-d

I thoroughly agree with this previous comment. The earlier commentary
seems to miss the point.
 However there is a problem with all Axiomatic formulations of
mathemaitcs that was brought up ealier this century by that chap named
Goedel. But you probably don't want to go there.

dsico

This is tongue in cheek, but:

Sherman's theorem (named after my friend Sherman Tangga) states that
any number divided by zero is the number itself. i.e. n / 0 = n

The rationale? If you have n bananas and your divide it by zero (in
other words, you don't divide it at all ... the operator is redundant)
you will have n bananas left.

Needless to say, many people hit him on the head for that... ;-)

Arguments (against division by zero) like "you cannot divide an apple
into 0 pieces" are not very good. After all, with that interpretation
you can only divide by positive numbers: what does it mean to divide
an apple into -3 pieces or imaginary pieces? The question is whether
there are OTHER sound interpretations or whether it works
mathematically.

At http://www.matematik.su.se/~jesper/research/wheels/ it is shown
that one may extend the set of numbers so that division by zero works.
Moreover, it is shown that this can be done for any commutative ring
(and in fact, any commutative semiring).

For more info in infinity (esp. the history) you might try:

The Mystery of the Aleph by Amir D. Aczel

He is a quite good writer, and makes things very clear - it's not a
very high-level book, but for the non-mathematician, it is a good
exposure to the concepts.

Chris

Well, let's see why we can't divide by zero. Let a, b be real numbers:

a = b
aČ = ab
aČ - bČ = ab - bČ
(a - b)(a + b) = b(a - b)
a + b = b

a+b=b works when a=b=0 :-)

Hi Aditya2k:

Sure, but a and b are two arbitrary and equal numbers. If it is always
true that a = b => a + b = b, then one is necessarily presupposing
what a (and as a consequence, b) is (zero). If that were the case, x =
y, h = g... ad infinitum will always be zero. By implication then,
every unknown then has the value of zero, which is an absurdity.

But I'm sure you knew that... you were just nitpicking, weren't you?
;-)