ZERO.

> What is the meaning of having ZERO as the denominator?
It depends on what domain you are talking about
in a division ring, it is not allowed, in the projectively extended reals, a/0 is represented by a single point (when a!=0)

> What does  ( 1 / 0 ) one divided by zero / Numerator divided by Denominator ( Numerator / Denominator ) mean?
In a division ring, it is meaningless
in the projectively extended reals you can interpret it like a vertical line with an infinite slope
> What is the value of Anything divided by ( Anything / Zero ) Zero?
In a division ring, it is meaningless,
in a projectively extended reals. Anything / Zero = 1/0 (for Anything != 0 )
> Can we have Zero as denominator?
It depends what structure your division operator is working in
> Can we split / cut / separate / divide / etc; the Numerator with Zero?
it depends on the structure you are operating on, and what you mean by split / cut / separate / divide / etc

> It is to do with how the Real number field is defined:
the Real number field is a division ring, so division by zero is not defined

The real number field is a field first and foremost, where division by zero is also not defined.

division by zero is not defined (for the real number field) because nobody can come up with a useful definition which is consistent with the other postulates. There are many "proofs" of such oddities like 2 =1 which depend on trying to divide by zero in a way which is partially  hidden. When it is realized that division by zero is not allowed, one can often spot mistakes in a proof or problem on which you (or somebody else) are working.

Oh, not this again. :)

>>Please add your own similar questions...
What is the meaning of division?

For example:

Q: What is 22/11
A: 2

Q: Starting from ZERO, how many times can I add 11 while ensuring I am not greater than 22.
A: 2

Q: Starting from ZERO, how many times can I add ZERO while ensuring I am not greater than 22.
A: As many times as I want.

Paul

>>Q: Starting from ZERO, how many times can I add ZERO while ensuring I am not greater than 22.
A: As many times as I want.

Hence the answer infinity.

Anthony

To take that statement even further, I would be more inclined to reword it to be:

Q:  Starting from ZERO, how many times can I add 11 UNTIL I am greater than or equal to 22?

In that case, the answer is 2.  But if you apply that wording to ZERO, then you get:

Q:  Starting from ZERO, how many times can I add ZERO UNTIL I am greater than or equal to 22?

This answer is not defined.  It cannot be done.  

And if you are adding a negative number, then you would have to switch the signs around (i.e., until I am less than or equal to -22...).

It's a great illustration, though.

So it appears my High School math teacher was mistaken.

Anthony

To be fair, this is beyond high school math. In high school math, you arent explicitly told the assumptions made swhen you operate on a set of numbers, and teachers will only explain what they have to, sometimes even lying in order to simplify their explanation

>>>So it appears my High School math teacher was mistaken.<<<

I wouldn't say that.  As I said, it's a good illustration.  It's not a traditional definition of division, but it gets the point across.  If you get into infinite series or complex numbers, then the illustration becomes more shaky.

But I do like the example for the fact that you can show how multiplication is really just addition taken up a notch.  

I took more umbrage with the conclusion that the illustration proved that 1/0 = infinity.  As I said, that conclusion also means that 1/(-1) is also infinity.  It just needed a few more parameters.  

Division by ZERO has always been tricky.  And as zhuba said, when dealing with something this complex, it is easier for teachers to use simpler examples.  

He was actually one of my better teachers. At the time I appreciated the explanation as understanding was much more important to me than just a bunch of numbers.

Glad he attempted to make is understandable, as he made many other things he taught feel more comprehensible than they might have been with a lesser teacher.

Anthony

AFAIK, there is a difference between 'arithmetic' and 'mathematics' that needs to be understood before deciding if "this is beyond high school math".

I get a feeling that there is a lot of confusion around the teaching of 'arithmetic' when it gets called 'mathematics'. I don't think it's simple to understand the differences. (Nor am I particularly qualified.)

Tom

Arithmetic is the branch of mathematics dealing with integers
(which are not restricted to just the Rational Integers that one may be familiar with from Elementary Arithmetic)

I always believed arithmetic was the addition, subtraction, multiplication, and division segment of mathematics. Mathematics was the all inclusive teaching of everything else like Algebra, Geometry and so on.

Anthony

This is a discussion of mathematics and science, not of religion, so what you "believe" shouldn't come into it.

zhuba,

Relax.

*grin*

WaterStreet,
Your sometimes friendly EE Core Zone Advisor here

many branches of mathematics share concepts with other branches, and many mathematicians work in more than one branch or talk to mathematicians working in other branches, and the the boundary between branches aren't always rigorously defined.  And school curricula don't always follow those divisions anyway.
(you wouldn't expect a literature class to spend one semester focusing just on synecdoche and a separate semester focusing just on metonymy)
So what is taught in a particular class does not necessarily correspond exactly to a particular branch of mathematics.
Integral domains define an addition operator and a multiplication operator, but not necessarily a division operator.
Peano integers don't even define subtraction.
Division rings will have a division operator.

But structures that define division by 0 are not fields or division rings, or integral domains.

also, words like "arithmetic" can have a technical meaning that is different from its common meaning.
and it may have more than one technical meaning and more than one common meaning that is understood (or   not) in different contexts.

WaterStreet

If I was any more laid back I would be horizontal

:-))

What is the answer?

1. Infinity

2. Undefined / Not defined

3. Impossible. Division by zero is not possible. Zero should not be used as denominator.

>AnthonyRusso  http:#a23530701

I do not doubt that your High School teacher told you that n/0 gave infinity as a result. He has to explain a complex subject in terms comprehensible to children who knows even less math than he does. A bit of what he says is bound to be 'lies for children' or even just plain wrong.

Here most of what he said was right. the curve really approaches infinity as the divisor decreases towards 0. That is true. but it very carefully do NOT tell you what happens when the divisor actually reach zero.

There is a second complementary curve we should look at here, the one we get when the divisor is negative and goes towards 0 from there.

n/-1 = -n
n/-0.5 = -2n
n/-0.25 = -4n
As the divisor approach 0 FROM BELOW the result approaches negative infinity

So if we just project the curves we get the result that n/0 have both the value of infinity and the value of -infinity. That is quite a margin of error there; enough to tell us that that we cannot just project the curve like that.

And that is why n/0 is a special case, it has no value and it has whatever value we want at the same time. It is undefined.

regards JakobA

Let us consider x, y and z as any three numbers. If we take an equation such that x / y = z then x = y * z. 4 / 0 is said to be infinity. If 4 / 0 is equal to infinity, 4 / 0 = infinity then 4 = 0 * infinity, that is 4 = 0 but 4 is not equal to 0 therefore 4 / 0 is not equal to infinity. What is infinity? Infinity means not finite. If infinity means not finite then infinity can not be a number. Therefore infinity is not a number. When we divide a finite number with another finite number we get a finite answer but not infinity. Therefore any finite number divided by any finite number is equal to some finite number. For example 4 divided by 4 is equal to 1. The numerator is 4 which is a finite number. The denominator is 4 which is again a finite number. Now 4 / 4 is equal to 1 and once again 1 is a finite number. Therefore the result of dividing a finite number with another finite number is also a finite number. 4 / 0 is sometimes said to be undefined or not defined. If it is so then let us define it. We already know that 4 / 0 should give us a finite number. 0 / 4 means we are dividing the numerator 0 with the denominator 4. Zero means nothing. Dividing 0 with 4 means dividing nothing into 4 parts, it means we are not dividing anything. Therefore 0 in the numerator means we have nothing to divide. What does 4 in the denominator mean? 4 in the denominator means we are dividing the numerator into 4 parts that is we are converting the numerator into 4 parts. Now that we know what denominator means we will discuss 4 / 0. 0 in the denominator means dividing the numerator that is 4 into zero parts. Mind you, 0 in the denominator does not mean we are not dividing. Zero in the denominator actually means we are converting the numerator into zero parts. If we can divide the numerator into zero parts then each part is equal to zero. But sum of zeroes does not give us 4. Therefore 4 can not be divided by 0. In fact numerator can not be divided by 0. Therefore 0 can not exist as denominator. It is true that the numerator can not be divided into zero parts and it is also true that sum of zeroes does not give anything but zero. Therefore
1. Zero can not exist as denominator.

> If we take an equation such that x / y = z then x = y * z.
> 4 / 0 is said to be infinity
This is confusing two statements that are meant to apply in completely different contexts.
There are strictly defined conditions under which each applies, and you cannot just blindly combine the two.
To do so would be is about as fallacious as saying.
"Nothing is better than eternal happiness"
"A sandwich is better than nothing"
therefore, A sandwich is better than eternal happiness.

> If infinity means not finite then infinity can not be a number.
Only of you define "number" to mean finite.
Which is fine, as long as you are consistent about your self imposed restriction, and don't try to mix any statements derived within those restriction with statements that may apply in other areas.

> 1. Zero can not exist as denominator.
Yes, you have said this several times.
We are already familiar with many fields in which zero is not allowed as denominator, and would have no objection if you wish to keep your statements within those fields.

way12go: >>  "4 / 0 is said to be infinity."

NO !!
  4/0 is NOT infinity
  4/0 is undefined, it is an illegal operation that have no defined value.

That's right. 4 / 0 never equals to infinity.
Addition of infinite zeroes never equals to one. The answer is not infinite. The answer is " NEVER ".
What tends to what? What ever it may ( something tends to something else ) tend to, division never equals to zero or infinity.
1 / 0 never tends to infinity and
1/ infinity never tends to zero. Infinity is not a number, we can not use it as we use any finite number.
Therefore the answer can not be " UNDEFINED ", it is " Zero can not exist as denominator ".
We need an answer and " UNDEFINED " is not an appropriate answer.
Therefore the answer is " ZERO CAN NOT EXIST AS DENOMINATOR ".

Please visit

http://www.experts-exchange.com/Other/Math_Science/Q_23768109.html

http://www.experts-exchange.com/Other/Puzzles_Riddles/Q_24002461.html

http://sagargorijala.blogspot.com/

I too have a fundamental theory of existence:

             Good is better than bad becuse its nicer.

Yeah, I'm too have one: 1 + 1 =2

Chucks, LOL,...

Thanks a lot for your " Fundamental Theory Of Existence" (s).

Infinity means not finite.
We know that
Numerator / Denominator = Result.
Infinity is not a number.
Therefore Numerator can not be Infinity,
Denominator too can not be infinity and
Result too can not be infinity.
Therefore we have a conclusion and it is
Any finite number ( Numerator ) / Any finite number ( Denominator ) = Some ( Result ) finite number.

Numbers are a virtual concept.
Zero is a number but when it comes to existence
" ZERO " means " NOTHING ".
There can be infinite numbers ( Virtual Existence ) but there can not be ( Real Existence ) infinite Apples.
Zero can not exist as denominator.
1 / 0 is not equal to infinity.
1 / 0 is not " UNDEFINED ".
1 / 0 means we are dividing the numerator with nothing and it is not possible.
Therefore Zero can not exist as denominator.

> Numerator / Denominator = Result.
> Infinity is not a number.
> Therefore Numerator can not be Infinity,
That only follows if you assume that a number must be finite and that Result must be a number

Which again is ok, if that's what you want to talk about,
But if you are talking about fields that disallow division by 0, then it is already a well know fact that, in a field,
Zero is not used as a denominator

Just as everyone already knows the "fundamental theorem" that,
1 + 1 = 2
(at least in the Peano Integers, if not in Galois Fields)

I will raise this forum to 500 points and then award points. Thanks a lot for all your time and participation. Please keep participating. Once again thanks for your most valuable time.

I have given hard copies of my theory to people whom it may concern. I hope I will get somewhere.

way12go,

"Therefore Numerator can not be Infinity,"
"Therefore Zero can not exist as denominator."

Each statement is untrue

Of course, there can be expressions in the numerator that equate to infinity. For example, Tangent (90 degrees).
and
Of course, there can be expressions in the denominator that equate to zero. For example (the expression X-3 for X = 3)

What you mean to say is that the results of operations with those are undefined.

I meant
Zero can not exist as denominator
and that means
anything that equals to zero can not exist as denominator.
In the first place " Tan 90 can not exist ".
Tan 90 is 1 / 0 and 1 / 0 can not exist.
Also Cot 0 can not exist. Anything that leads to " Zero as the denominator " can not exist.

Wrong

perhaps not wrong per se, so much as true in a limited sense.

and in that limited sense, it is trivially true,
(as is perhaps befitting a fundamental axiom)

If two angles are 90 degrees then it is not a triangle. If any angle is equal to zero then it is not a triangle. If it is not a triangle can we still call it trigonometry?

Yes

If an angle of triangle is reduced to zero then it is no longer a triangle.  This is what is studied in trigonometry.

many trignometric identities continue to hold when triangles become degenerate.
if those identities are expressed in terms of division operations, then care man be required in applying them
in order to avoid division by zero.

This topic seems to have generated a lot of response. It took me a long timeto figure this out too. Here's my reasoning:

Simple algebraic manipulation states the if:

   c = ab

(c = a multiplied by b) then it sould be possible to solve this for either a and for b as in:

   c / a = b

or

   c / b = a

Having said that, let's suppose we take the last one and substitue b with zero as in:

   c / 0 = a

Now, to solve this for c, we have to move the '0' to the other side of the '=' sign giving us:

   c = a x 0

'a' multiplied by zero is just 0. Do you see the problem?

One final point. Speaking theoretically now, the only time you would be able to divide by 0 is in the following case:

   c = a / b

where a = 0 (and ONLY 0), b = 0 (and ONLY 0) and c = 0 (and ONLY 0) because this would translate to:

   0 / 0 = 0

and this would have no function. But that's theoretical!

Please visit the below website.

http://www.grc.nasa.gov/WWW/K-12/airplane/sincos.html

The sum of all the angles of a triangle is 180 degrees. No two angles can be 90 degrees. Also no single angle can be zero. Sin 0 can not exist. Cos 90 can not exist. Tan 90 can not exist. If Sin 0 exists it means two sides are overlapping (without any angle between them) and it is not possible. We live in a three dimensional World where everything is three dimensional and three dimensions only. We neglect the third dimension for our convenience. Sometimes we neglect two dimensions and continue studying as if everything is occurring as a single dimension.

>    0 / 0 = 0
even in those contexts in which a / 0 is defined for a != 0,  0 / 0 generally remains undefined.

> Sin 0 can not exist
That seems rather absurd.
Does sin(-1) exist?
Does sin(400) exist?
If you want to declare that you never want to talk about sin(0), that's fine,
no one can force you to do so,
But you have no right to cause those people who do deal with sin(0) and have no problem with it to ll disappear.

And in fact, the very website you cite lists tables for sin and cos, including sin(0) and cos(90)

I have a theory.  I can fly like Superman.  Want to see me prove it?

A bullet is fired at me from 4 feet away.  As it travels it gets to 2 feet away, then 1 foot then 1/2 foot, then 1/4 ft. then 1/8, then 1/16, 1/32, 1/64, 1/128, 1/256, 1/512, 1/1024, 1/2048, 1/4096, and so on to an infiniite degree.  Therefore, since you can always calculate a smaller distance between me and the bullet, it never reaches the point where it's 0 feet from me (hence touching). And therefore the bullet will never hit me, and therefore I'm bulletproof.  

and if we apply the commutative property of addition:
me=bulletproof
bulletproof=Superman
Superman=Fly
Fly=Me

Now if you'll excuse me, I have to fly to the grocery store and pick up some apples.  Lois is making Apple Pie just like we used to have on the Kent farm when I was growing up, as a special thank you for all the times I saved her over the years.

Flawed Logic is flawed logic, no matter how hard you try to disguise it.

skcandko,

Are you suggesting that it's time for the asker to bite the bullet?

regards.

ps.  You ought to try posting in the other P&R Zone where sometimes we can use that kind of humor.

P&R is where I found this...

And are you suggesting that I'm not who I say I am?  Clearly I proved mathematically that I'm  the Man of Steel!

okay

I always thought there ought to be a separate zone on EE for nutters. I think skcandco has just confirmed this.

Well at least I was able to prove SOMETHING.  
In recognition of that I think the new Insanity Zone ought to be named after me.

Water:  Yeah I think the OP needs to bite the bullet on this one and stop trying to extend this question back into his other two posts. To sum up:

What is the meaning of having ZERO as the denominator?
It means that you are trying to preform a function that cannot be performed.  Much like standing in the path of a bullet and relying on math to keep the bullet from hitting you.

What does  ( 1 / 0 ) one divided by zero / Numerator divided by Denominator ( Numerator / Denominator ) mean?
It means take 1 and try to divide it into Zero parts.

What is the value of Anything divided by ( Anything / Zero ) Zero?
Well, since the function cannot be performed, the value of a thing is the same as when you started, since you have not preformed any action on it.  OR a little more abstractly the value of trying to divide something into zero part is a fruitless endeavor, and has no value.

Also I think you need to change your wording.  Anything should really be worded more correctly as A THING.  ANYTHING is better defined as 1 (or a specific amount up until and including) of whatever you want. You've also made the assumption in other posts that ANYTHING is the same as EVERYTHING.   Everything is Unlimited, and is all encompassing.  

Can we have Zero as denominator?
Yes, but as an undefined function we don't know what to do when this occurs.

Can we split / cut / separate / divide / etc; the Numerator with Zero?
You can try, but it's a pointless endeaor.
From your own post:
Therefore the answer is " ZERO CAN NOT EXIST AS DENOMINATOR ".

In a right angle triangle one angle is 90 degrees and the sum of the other two angles is 90 degrees. The sum of all angles in a triangle is 180 degrees. Therefore in a right angle triangle ( One angle is already 90 degrees ) the other angle(s) can not be 0 and or 90. For a straight line Sine 0 is 0. We can observe that is the Sine graph. Therefore Tan 90 can not exist for a right angle triangle. Tan 45 ( Tan theta = opp/adj and for Tan 45 opp=adj - Tan 45 is clearly equal to one ) is 1 and

tan(A+B)=(tanA+tanB) / (1tanAtanB)

Tan 90 = (Tan 45 + Tan 45 ) / (1-Tan 45.Tan 45 )
Tan 90 = ( 1+1) / ( 1-1.1)
Tan 90 = 2 / 0
Zero can not exist as denominator
Tan 90 can not exist.
Sin zero exists but not for a right angle triangle.
If a right angle triangle exists then one of its angle can not be zero!?

http://www.mathalino.com/reviewer/derivation-of-formulas/the-double-angle-formulas

some of us are still failing to see the inevitability of the way that conclusion is lead to.

====================================================================================
Xxavier:  Clearly Zero can not exist as denominator ie division by 0 is meaningless this inevitably leads to the conclusion that anything that exists can not be destroyed into nothingness only change of form is possible.

   Extending this idea with a little further leads inevitably to all motion being relative and there being only 3 dimensions.

Just a theory I am working on, feedback appreciated.
====================================================================================

I too think so.

More information is available from the fallowing websites...

Fundamental theory of existence.
http://www.experts-exchange.com/Other/Math_Science/Q_23768109.html

0.Fundamental Theory Of Existence.
http://www.experts-exchange.com/Other/Puzzles_Riddles/Q_24002461.html

It's like being drawn into an argument with a five year old.

"I'm better than everybody."
"Why?"
"Cuz I said So."
"No way."
"Yes, way."
"No Way!!"
"Yes, Way!!"
"NO WAY!"
"YES, WAY!!"

One more thing:

If there are only three dimensions, (I'm assuming you mean the Traditional Length, Width, depth etc. when you say there are only three dimensions) then how do you explain yesterday?  You want to say there's no such thing as time, then there's no such thing as yesterday, and no such thing as tomorrow.  Time is a dimension just like anything else.

You want to change your course through space?  Apply a force that causes your momentum through space to change.

You want to change your course through time?  Apply a force that causes your momentum through time to change.

Don't ask me how to apply that force, cuz I'm not tellin'.  Geez I gotta do all the work for ya?

Actually, one theory is that what we perceive as 3 dimensional space may be equivalent to information on 2 dimensional boundary, or a 3 dimensional boundary including time.
http://en.wikipedia.org/wiki/Holographic_principle

But what connection there may be between the properties of division rings,
and the  properties of physical space, or how one can draw any conclusions about one
based on statements about the other, still eludes me.

And with that, I am taking a 1° turn along the 10th dimension... *unsub*

1/0 = blurg

Blurg is not a number in the traditional sense.  It is greater than infinity and less than negative infinity.  Graphing blurg is seemingly gray, but it isn't - it is all black from all the attempts made and all white from all the results validated - all at the same time.

The best way to understand blurg is this:
If I cut a pie into 0 pieces, the pie tin is empty and I am full.

Yes, but you could refill the pie tin with the used pie contents in a few hours/days depending on your regularity.  Therefore blurg is only temporary?

Sounds like a load of crap to me....

Are you saying this question is a load of crap?

Let's see -

This post = ZERO
Zero = The contents of the pie tin.
The amount in the pie tin = used pie after digestion time.
(but since time does not exist according to previous stipulations in this thread, the contents are both zero and used pie simultaneously)
Used pie = load of crap

and since everything is equal: this post = load of crap.

Looks like you were right.

Excellent!! I like it.

Ok, I'm putting your whole to theory to rest as false right here, right now.

Velocity = distance/time

without time you get

Velocity= distance/ZERO

Division by Zero is impossible according to your theory of everything.

What you should have tried to come up with was a theory on conservation of time, much like the laws governing conservation of matter and momentum.

Shouldn't that be:

without time you get

   Velocity = Distance / Zero

but as we can't divide by Zero, everything stands still.

This seems to be a contradiction in terms because, the needle will never reach the floor but even if it did, we would never hear it because the above formula also applies to sound waves however, if we don't stand still and listen carefully then we would never hear the needle drop anyway.

In conclusion, we get:

   Sound = Distance / Zero                        which equates to just Zero

and

   Sound = Distance / NOT(Zero)              which also equates to just Zero


therefore,

   Distance / Zero = Distance / NOT(Zero)

which proves an important point.... if we cancel out Distance on both sides of the equation we get:

   Zero = NOT(Zero)

and because Zero is NOT Zero, we can infact divide by Zero!! - So it appears the scientists had it wrong all the time.

And that's why we're no closer to discovering time-travel.

I was listening to the radio the other day and it occured to me (but probably to nobody else) that during the sports news, the announcer said there was a "Zero result"..... A match that results in zero! - As though it never existed or never took place.

That's pretty scary because my mate who had tickets for the match swears he was there however, if the match didn't really exist then where the hell was he last Saturday afternoon?

One theory is, he was abducted by aliens.

Please see my reply (23774401) and award point accordingly.  Thank you.

Thanks a lot, one and all. Please post your conclusions before I award points. Your participation is appreciated.

Please post your final conclusions and please do make better conclusions.

I have examinations and I will probably award points after I finish my examinations.

Thanks again and see you soon.

A wise calculator once told me: "Cannot Divide by zero"

Thanks a lot. I hope your time and participation is appreciated.