Why is 1/0 = 0 wrong? If it is!

I understand why 1/0 = infinity or unknown number as division is
understood through multiplication and there is no number x that
satisfies the equation 0*X = 1.

I accept this theory but i have a different argument.

When you divide 4 units into 2 parts, you get 2 parts each. 

Or as you would tell a elementary school kid, if you divide 4 candies
into 2 children, each gets 2 candies.

If i divide 1 between 2, each gets half.

Zero is nothing. 

So if i divide 1 part by nothing or zero, the original numerator or
the whole doesnt diminish. Unlike when you divide a whole number by
another whole number or integer, the numerator diminishes into as many
parts as the denominator.

But when you divide the numerator by nothing, you are in other words
not dividing at all.

That means you dont lose anything and you dont get anything. Or simply
put, if i divide 4 candies among 0 children, no body gets anything and
i dont lose any part of the 4. Am i right?

If i do not lose or diminish any part of my numerator how can my
resultant figure that i get be anything but nothing.

When i divide 1 by nothing, i get nothing.

or i get a 0.

What is the wrong with this argument?

Because 1 divided NO TIMES remains only 1.

tutuzdad-ga

So can we also say that 1/0 = 1 !!!!!!

I just discovered something interesting that would make for a good
trivia question. When asked to calculate 1 divided by 0 my computer's
calculator says ERROR: POSITIVE INFINITY.

Ya learn something new every day. Oh well; it's not my area of
expertise which explains why I didn't try to answer it.

Let me know if this helps explain it (at the risk of making myself out
to be more of a math idiot than is already apparent)

http://mathforum.org/library/drmath/view/55976.html
(explanation geared toward third graders)

tutuzda-ga

the reason when you divide 4 candies among 0 children you do nothing
is because it's impossible to do anything.  it's impossible to divide
4 candies among 0 children just like it's impossible to divide 1 by 0.
 just like your example where you divide 4 things into 2 equal parts. 
well you can't ever divide 4 things into 0 equal parts no matter how
many times you divide it.  you'll allways get something.

"When i divide 1 by nothing, i get nothing."

Math meanings tend to be more precise than is expressed by informal
use of English words.

When we say a/b = c in arithmetic, our meaning is equivalent to a = b*c.

Since 0 times 0 is not 1, I think it should be plain that in
arithmetic one does not compute 1/0 = 0.  Indeed, there is no
arithmetic number c such that 0*c = 1, because 0*c = 0 for all such c.

Often 1/0 is left undefined, so you could describe it as "nothing" in
an informal sense of "not having been defined".  But this is certainly
a different meaning of "nothing" than arithmetic "zero".

It is also possible to define a result, but necessarily this result
must be "outside" the realm of ordinary number arithmetic.  One
possibility is to define the result as INFINITY, as tutuzdad-ga
reports his calculator does, but one has to be careful if one desires
to make a consistent distinction between PLUS INFINITY and NEGATIVE
INFINITY.

regards, mathtalk-ga

Tutuzdad, back of the class for you! [I enjoyed very much your posting
to the "are we pregnant" kid.]

Sagarch, you have a good train of thought and inquisitive mind. You
need to follow your experiment of dividing a little farther by trying
a few more numbers in the denominator.

Division by zero is undefined. I will leave it to someone who wants to
be paid for this answer to provide the proof (available on the 'net)
but I can tell you one way to look at it. Division by zero is not zero
and it is not infinity, though my explanation will make it look like
it is infinity. It really isn't and I am not going to explain why, I
just want you to see a way to look at it.

If you take one and divide it by, say, 1000, you get a rather small
number. Divide one by 75 and you get a pretty small number. Divding
one by 2 gives a half, still small.

Now if you divide one by a tenth, your answer is ten. Dividing one by
0.842 is 1.1876484. Dividing one by 0.146823 (which is a rather small
number) gives the answer 6.8109219. Finally, dividing one by
0.000000000100206234 (a terribly small number) will give an answer
that is in the billions.

So dividing any number by increasing smaller numbers yields
increasingly larger answers. As the number we are dividing by
approaches zero, the answer approaches infinity.

Now remember, division by zero does NOT equal infinity. If you say it
does, your math teacher will fail you. The simple reason for this is
division is the inverse of multiplication; if you divide any number by
zero, and then multiply by zero, you should be back to the number you
started with. But multiplying infinity by zero produces only zero, not
any other number.

Now somebody post the proof and get paid.

Another way to understand it, using words, is that when you
divide, say, 8 by 4, you are separating eight items four times,
into bundles of two.

On the other hand, when you divide any number of items 0 times,
you have not divided at all. Thus it can be called an "invalid
function", since no function has been performed.

i can provide a simple proof (answer) for you question "Why isnt 1/0 = 0 ?".

by definition, multiplication and division are inverse operations of
each other. take for example:

4/2 = 2

So, by using the inverse operation of multiplication 

2*2 = 4

so in general, if we have a/b = c, then b *? = a.

In your proposed equation, where 1/0 = 0, then 0 * 0 = 1, and thus the
universe as we know it would collapse. And i would take by your
reasoning, you would consider 4/0 = 0. so 0 * 0 = 4 also. Therefore:

1 = 4

Or any other number you wish. So as you can see, thru this reasoning,
we abandon all logic.

This takes care of your question, but you may want to consider the following:

think beyond integers. 4 / 2 = 2, but 4 / (0.5) = 8. which does not
necessarily deminish. Actually if you divide by numbers closer and
closer to 0 you will get higher and higher answers. this would lead
some people to believe the answer is more towards the infinite than 0.

Here's the simplest explanation that I know as to why 1/0 is undefined...

One of the laws of math says that zero divided by anything is zero. 
Take 0 objects and distribute them evenly to X people, and each person
gets 0.

In mathematical terms, 0/X = 0.  So far, so good.

Another law says that anything divided by itself equals 1.  Take Y
objects and distribute them evenly to Y people.  Each person gets 1.

In math terms, Y/Y = 1.

So, let X = Y = ZERO.

Our first law tells us that 0/X = 0.  Therefore 0/0 = 0.
Our second law tells us that Y/Y =1.  Therefore 0/0 = 1.

So, which law do we apply?  We can't apply both and say that well,
sometimes it's 1 and sometimes it's 0 (you can only get away with that
in Quantum Mechanics and/or with respect to the definition of light),
and we can't arbitrarily choose one over the other.

So, years ago, mathematicians all agreed that anything divided by zero
is undefined.  While most calculators show it as positive infinity,
this is technically incorrect.  The correct way to say it is the LIMIT
of 1/x approaches positive infinity as x approaches 0.  By itself,
however, the expression 1/0 is undefined and has no answer.

Another way to look at this is using limits. 

Notice that: 
1/1 = 1 
1/(0.5) = 2 
1/(0.25)=4 
1/(.0001)= 10000
1/(0.000,000,000,000,1) = 1,000,000,000,000. 

So notice that for 1/x as x gets smaller (without reaching zero) 1/x
gets larger and larger......so when 1/x, x=0;  is undefined and not
equal to 0.

Hope this helps a little

A previous comment stated:
"[M]athematicians all agreed that anything divided by zero
is undefined.  While most calculators show it as positive infinity,
this is technically incorrect.  The correct way to say it is the LIMIT
of 1/x approaches positive infinity as x approaches 0."
Not exactly.  That is only half the story.  Note that if one divides 1
by -0.000001, one gets -1000000.  Making the divisor progressively
closer to 0 will result in quotients approaching *negative*, not
positive, infinity.  One would say instead that as x approaches 0 from
above, the expression 1/x approaches positive infinity; and as x
approaches 0 from below, 1/x approaches negative infinity.  Since 1/x
has a left-sided limit and a right-sided limit that differ, we simply
say that the limit of 1/x as x approaches 0 does not exist.

0 / 0 = 0
and
any number / 0 = 0

Everyone is trying to satisfy calculus and fractions by flawing
division by zero to suit their LIMITED FUNCTIONS.

Division by zero is a statement that NO DIVISION IS HAPPENING
You all seem to get that point but seem to miss the point that it does
not have to happen to be valid statement of no division.

It's the same as multiplication by zero.  When there it none you state
there is zero multiplication.  You then state you have zero product of
multiplication.

With division you can look at the answer of zero as a product of
division.  So by not having any division you have zero division as
your product.
Ex:  Cake / 0 = 0 Divided Cake  (0 portions of divided cake)
The cake exists but there are no divided portions.

If you divide a cake for every kid who comes to your party and nobody
comes to your party then you wont divide that cake for any kid.

You have a cake / For Zero People = 0 Portions Of Divided Cake

Sad like this topic

Check out http://members.lycos.co.uk/zerobyzero/
for more info:

Division by zero has infinite possibilities if you give it a try.

And these reversion answers they feed you poor souls is horribly flawed to:

Because!!! Multiplication does not reverse also when multiplication of
zero is involved!!
a = 1 * 0
a = 0

The reverse of
a = 1 * 0
becomes
a / 0 = 1
and we have a flaw!!

Should we state multiplication by zero is an error because we can't reverse it?

Who wasn't to wake up in a world where they ban multiplication by zero.

The simple truth is that is can be used and you are all dealing with
LIMITED FUNCTIONS and trying to make them unlimited by removing the
limited end of your function from the entire spectrum of mathematics.

0 / 0 = 0

Put quite simply. How many times can you take 0 out of 1 (or any other
number)? answer: infinite. if you take zero out of 1, you have 1. do
it again, you still have one. you can do it forever.

It's not about taking 0 out of 1.  The logic behind the operation is a
long one and it's best you see the site.  I hate repeating myself.

In short division can be looked at through many angles, views or
opinions.  The construct this system various in certain logical
conditions.  It does not change take any thing away from the system
but adds to it by enabling functions that were previously not allowed.

If you look at the system in terms of making portions then division by
zero would be a statement that NO PORTIONS ARE MADE.  This system
works by taking the result and placing it on a separate scale.

So you can put the entire volume on the scale
When One Portion Is Made
Object / 1 = 1 Object   ( Entire Volume of Object On The Scale )
When Two Portions Is Made
Object / 2 = .5 Object   ( Half Volume of Object On The Scale )
When Four Portions Is Made
Object / 4 = .25 Object   ( Quarter Volume of Object On The Scale )

When No Portions are Made
Object / 0 = 0 Object   ( No Volume of Object On The Scale )

This means there was nothing of the object taken and put on the scale.
 It does not mean the object vanished.  This system measures in a
sense change in the state of object.  While the object is not divided
physically when using 1 and 0, it is in two different states.

Division by one can have various conditions of state like being moved.
Object / 1 = 1 Object Moved
Object / 0 = 0 Object Moved

An application of this formula in science is measuring voltage.

Voltage / Resistance of Path = Current
So
12 v / 1 ohm = 12 amp

Now this 12 amp is in motion.  There is transfer of electricity. 
Notice that we have 12 amp as the result and not 12 volts.  There was
a transformation during the use of division by one.

So how do you state that three is no current?

12 v / 0 ohm = 0 amp
This way you can see that the result of 0 amp means there is no current.
And we still have 12 volts.
And by allowing division by zero we can agree the reason we have no
amps is because we have no path of current.  The reason we can state
this is because the less wire the less ohms we have.  And finally when
the wire is missing, like when a light bulb burns out.  We no longer
have any wire for the path of current.  Since it needs the path to
conduct over.  We can say the lack of necessary path of wire is
missing.

Well that's one of many practical applications that can use division by zero.

You have used an example - the one of candies and children - which
doesn't really work for the mathematical implications dealing with the
division operation. Do you know why? Because you couldn't consider the
case in which the number expressing the quantity of children is less
than 1, as long as in natural world - which is also the world of our
natural language, the one we use for communicating and making examples
taken fron everyday real life - it is impossibile thinking about "half
a chil". But in Maths it really is, because "child" is a logical
concept. And it can range from minus infinite till plus infinite (the
range of your example is: natural numbers equal or higher than 1).

So let's take you example. And let's make it also more easy. Consider
you have a cake and 10 children which are supposed to equally eat the
cake. 1 cake divided 10 children gives us the result of one tenth of
cake for each of them. Fair enough... let's go further. Now children
become 5. Each of them will have one fifth of the whole cake.
Only one child? The whole cake for himself! Lucky guy...

Let's consider now the case in which we have "half - i.e. 1/2 -
child". Absurd?!? Absolutely no! My hypotesis is right this: how many
children are we supposed to consider for dividing the cake in such a
way that every child will have and equal part of it? How many
children? 1/2 children.

In this case how much of cake each 1/2 child will obtain? Exactly 2
cakes! It seems impossible that the cake has been multiplied as fishes
with Jesus... But calculation is correct: 2 cakes times 1/2 children
is equal to the original 1 cake.

So let's "approach" zero by considering the new "1/100 child" example.
We have the same cake which has to be equally divided for... how many
children? 1/100 children! In this case every "1/100 children" will
have 100 cakes. Same as before: 100 cakes times 1/100 children is
equal to the original 1 cake.

As much as we reach the zero number (1/1000; 1/10000; etc...) strange
things happen: the original cake still remains of its own dimensions,
but portions of cake given to "children" become bigger than the whole
cake, and really BIG as long as "children" becomes higher in number.

So: what do you think is gonna happen when those "children" become
infinite, i.e. when we consider the limit of (1/n)children as n
increases to infinite? A lot of "children" (i.e. a lot of
(1/n)children), whose "dimension" is reaching zero. In that case: each
of them - thus beeing SO little - will eat a quite infinite cake,
whose dimension is exactly the original one times the inverse of the
dimension of a generical "little child".

So... answer by yourself: why isn't 1/0 = 0. Because otherwise no one
of that "little child" would eat the cake! And they are - as
demonstrated -REALLY hungry! And you are NOT so cruel till the point
of not giving food to those little creatures... Aren't you?

Bye, Enrico.

Another lovely and beautiful point to ponder,

This is again a statement in normal english which we will then
extrapolate into mathametical language..

"When your company grows from 100 to 200, you have grown 100% or you
have doubled, when you grow from 10 to 15, say in profits or revenue,
you have grown by 50% or 0.5 times. But when you take the first step
that is when you have grown from 0 to 1, how much have you grown?"

I dont think you will like it if someone tells you, you havent grown at all. 

Nor, can i say you have grown 100% because you have grown far more than that!

That i felt explains the concept of division by zero and as division
is a inverse of multiplication and inverse of inverse is the original
so it also should be the concept of multiplication by zero.

We always approach zero and fail to explain what happens when we actually hit zero.

Excuse me for interpolating the normal english into the math !

But hope you are getting my point, as x approaches zero, limit of 1/x
approaches a large undefined number.

This is true as long you are approaching zero. But what when you reach
zero. Why havent we ever reached zero?

Why is the train running into zero for all the life and never reached
zero? And why do we get dumbfounded responses when we actually go to
zero?


Answer - Since there is no point called zero that you and I can reach.

Every point can be a zero for another plane of numbers. Yet the same
point can be a different number for a different plane.

Zero is purely an imaginary point of reference that is assumed to be
the starting point for a series!

So when you divide a number on the plane by this imaginary point of
reference, or multiply by this imaginary point of reference. You end
up getting no answers !!! Simply because you are running up and and
into and imaginary point !!!


So Lt of 1/x as x approaches 0 is an undefined large number!

Value of 1/x as x finally has approached 0 doesnt exist since that
point never existed!

Isnt it beautiful that such a large number that was spiralling out of
bounds collapsed as we reached the point of refernce!

May be number line is not linear :-) Can it be spiral ? Can it be
spherical ? or a mix of both???

Something new to explore!!!

I like zerobyzero's assertion:

Cake / 0 = 0 Divided Cake  (0 portions of divided cake)
The cake exists but there are no divided portions.

But, to me, this simply means that the answer is both
0 and 1, since, while there are 0 portions, there is
still 1 cake.

I take issue, however with his assertion that:

12 v / 0 ohm = 0 amp

0 ohms means a direct short, and, until the power
source burnt up, you'd have extremely high current!

That speaks to the validity of a number approaching
infinity, rather than zero.

However that may be, I'm more in favor of the idea
that the answer is both 0 and 1, since I can at least
prove that they're equal:

Given a = b

multiply both sides by a:

a^2 = ab

subtract b^2 from both sides:

a^2-b^2 = ab-b^2

factor:

(a+b)(a-b) = b(a-b)

divide both sides by (a-b):

(a+b)(a-b)/(a-b) = b(a-b)/(a-b)

a+b = b

substitute b for a, since a = b

2b = b

divide by b:

2 = 1

subtract 1 from each side:

1 = 0

There...it's all settled.  ; )

1/0 is not infinity. It is undetermined. Why is it undetermined? If
you start dividing 1 by very small positive numbers, you will get very
large positive results. Now if you start dividing 1 by very small
negative numbers you will get very large negative numbers. Therefore
1/0 is not infinity it is undetermined...Your calculator says it's
infinity 'cause some programmer made a shortcut to operations that
have no defined result.

As has been mentionned before, 1/0 isn't really infinity, we say it is
undefined. Though your arguments about breaking numbers into parts is
pertinent, try thinking about the problem this way:

If you've got 10 units, you can separate them into five groups of 2.
Similarly, if you have 5 units, you can separate them into five groups
of 1. But then, how many groups of 0 can you separate 1 into (or any
other number for that fact)? No real number will satisfy this
requirement.

Why is 1/0 undefined? We can prove this using limits.

Let f(x)=1/x

We look at what happens as x approaches 0 by taking the limit of f(x)
as x approaches 0 from below (0-) and from above (0+)

Lim f(x) = -Infinity
x>(0-)
As you choose values of x increasingly close to 0, from below (i.e.
negative values of x increasingly close to 0) you'll notice that f(x)
becomes infinitely large and negative.

Lim f(x) = +Infinity
x>(0+)
As you choose values of x increasingly close to 0, from above(i.e.
positive values of x increasingly close to 0) you'll notice that f(x)
becomes infinitely large and positive.

In other words, f(x) can be made unboundedly large and positive or
unboundedly large and negative by choosing values of x sufficiently
close to 0. As neither the limit approaching from above or the limit
approaching from below exist, f(0) is undefined.

Interestingly, if this was not the case, several interesting proofs
would follow. Among other things, one could for example prove that
2=1.

CK