# zero divided by zero

Discussion in 'Undergraduate Math' started by Turco, Jan 27, 2006.

1. ### TurcoGuest

Is there anything special about 0/0, I mean is that any different from
1/0? I know they are both indeterminations, but I would like to know
if 0/0 is different from k/0 in any way... thx

Turco, Jan 27, 2006

2. ### majaminGuest

There *is* a system of arithmetic in which mathematicians do allow such
things as (0*infinity) and (infinity + infinity), but there it is
mearly a custom of convenience and not of much fundamental importance.
Firstly, 0/0 does not have a meaning if one considers it as the
operation 0*(0)^(-1). That is, zero multiplied by the inverse of zero.
Mathematicians cringe when they see something like this because
*usually* real numbers (*ahem*) have unique inverses. That is, say,
0.25 is the unique inverse of 4. Hence, we cringe, because zero times
*any* number is zero (weird, right? ;-) )Consider this:

(1) a = b
(2) a - b = 0
(3) (a-b)/(a-b) = 0
(4) 1 = 0

Obviously, we cannot allow division by zero (line (3) ) for the sake of
consistency (and sanity?).Now, as for 1/0 and k/0 as you mentioned, the
same problem creeps up: what number do you multiply 0 by to get a
non-zero number k?

If you are interested numbers the idea of the 'limit' and '0/0' and
'infinity/infinity' are important in earlier stages of analysis (read
'calculus'), especially if you know what a derivative is, etc. Refer to
L'Hospital's rule here:

http://mathworld.wolfram.com/LHospitalsRule.html

Good luck,

M.M.

majamin, Jan 28, 2006

3. ### TurcoGuest

thank you.

Turco, Jan 28, 2006
4. ### Brian M. ScottGuest

On 27 Jan 2006 13:45:05 -0800, Turco <>
wrote in
In the context of arithmetic of real numbers, both are
undefined. In general, if x and y are real numbers, we
define x/y to be the unique real number z such that x = yz,
if such a real number exists; if not, x/y is undefined. (It
isn't actually indeterminate; that's a slightly different
concept that doesn't really apply here.) 1/0 is therefore
undefined because there is no real number z such that 1 =
0*z; 0/0, on the other hand, is undefined because every real
number z satisfies the equation 0 = 0*z, and therefore there
is no *unique* real number that does so.

There is another context in which there *is* a very
important difference between two things that are often
symbolized by '0/0' and '1/0', though this is very sloppy
notation, and the context doesn't actually involve any
attempt to divide by 0. Suppose that you're considering the
quotient of two functions, f(x)/g(x), and you're interested
in what happens to it as x gets very close to some number c.
You may find, for instance, that f(x) approaches 1, while
g(x) approaches 2; in this case the quotient f(x)/g(x)
approaches 1/2, and we know this as soon as we know that
f(x) approaches 1 and g(x) approaches 2.

Or you may find that f(x) approaches 1, while g(x)
approaches 0. This is a very different kettle of fish: if
you divide numbers that are very close to 1 by numbers that
are very close to 0, you get quotients that are very large
in absolute value (i.e., either very large in the positive
direction, or very 'large' in the negative direction).

For instance, consider f(x) = x, g(x) = x - 1, and c = 1.
Then f(1.01)/g(1.01) = 1.01/0.01 = 101, f(1.001)/g(1.001) =
1.001/0.001 = 1001, f(1.0001)/g(1.0001) = 1.0001/0.0001 =
10001, etc., while f(0.99)/g(0.99) = 0.99/(-0.01) = -99,
f(0.999)/g(0.999) = 0.999/(-0.001) = -999,
f(0.9999)/g(0.9999) = 0.9999/(-0.0001) = -9999 etc. Clearly
f(x)/g(x) isn't settling down near any particular number as
x gets closer and closer to 1: for x just a tiny bit bigger
than 1, f(x)/g(x) is a huge positive number, while for x
just a tiny bit less than 1, f(x)/g(x) is a huge negative
number.

More generally, if f(x) --> 1 and g(x) --> 0 as x --> c, the
quotient f(x)/g(x) is going to 'blow up', and in particular
it can't possibly have a limiting value. We know the result
-- there is no limit -- the moment we know that f(x)
approaches 1 and g(x) approaches 0.

But if f(x) and g(x) *both* approach 0 as x approaches c,
it's a different story. Depending on exactly what the
functions f and g are, the quotient f(x)/g(x) might have a
limit, or it might not; and if it has one, that limit could
be any real number. For example, suppose that g(x) = x^2
and c = 0. If f(x) = x, then f(x)/g(x) = x/x^2 = 1/x as
long as x isn't 0; as x approaches 0, 1/x 'blows up' like
the previous example, so, as before, f(x)/g(x) has no limit
as x approaches 0. Now suppose that f(x) = kx^2 for some
real number k. Then f(x)/g(x) = kx^2/x^2 = k as long as x
isn't 0; in this case f(x)/g(x) is constantly k as x
approaches 0, and the limit of f(x)/g(x) is therefore k,
which could be any number at all. That is, knowing that
f(x) and g(x) both approach 0 gives us *no* information
about the limit of f(x)/g(x); in order to find it (or show
that it doesn't exist), we must get our hands dirty working
with the specific functions f and g.

The limit of a quotient with a 0 limit on top and a 0 limit
on the bottom is therefore _indeterminate_: its value cannot
be inferred from the limits of the numerator and
denominator, but must be worked out individually in each
case. The limit of a quotient with a 1 limit on top and a 0
limit on the bottom, however, is determined just by those
two limits: it doesn't exist. Note, though, that although
people often talk about these as '0/0 limits' and '1/0
limits', those are just verbal shorthand: there's really no
0/0 or 1/0 involved here at all.

Brian

Brian M. Scott, Jan 28, 2006
5. ### OwenGuest

n/0 is an arithmetic concern and not a concern of 'limits'.

How we define x/y determines what x/0 means.

If we define x/y as (the z: x=y*z), then x/0 does not exist for all x,
including x=0.
(because (the z: x=y*z) is not unique)

That is to say, 1/0 and 0/0 are defined but, they do not exist.

If we say: ~(y=0) -> x/y = (the z: x=y*z), then
x/0 is not defined for any x, including x=0.

That is to say, 1/0 and 0/0 are not defined.

Either way, 1/0 and 0/0 are not sensible terms!

Owen, Feb 10, 2006
6. ### in2infinity

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Yes the amount of number space that is traversed. 0/0 = 0 no number space traversed. whereas 1/0=0 means a unit of one is traverse in order to arrive at the result.

in2infinity, Apr 21, 2022

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