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Dividing by Zero

eday2010
As far as I am concerned, dividing by zero should not return an error in a calculator. The answer should be zero.

Why? Because if you are dividing 4 by zero, you aren't actually dividing. You're not doing anything. So you get nothing. And zero is nothing. So the answer should come up zero because you haven't done antyhing.

On the other hand, you can also say that 4 divided by zero should be 4, because you didn't do the division, so you still have 4 left.

But -Error- is not an acceptable answer to me.
Bikerman
Sorry but you are wrong. Dividing by zero is not the same as doing nothing. Division is an operator that is valid for the set of numbers - both real and imaginary. In field mathematics (operating on field values) then the operation is undefined, but in other types of maths it is well defined and valid.
eday2010
If you divide something by zero, you aren't dividing it. So you still have the same number you started with.
Bikerman
 eday2010 wrote: If you divide something by zero, you aren't dividing it. So you still have the same number you started with.

No, you don't. I'm afraid you are wrong. Dividing by 1 is the same as not dividing it at all - as should be quite obvious if you think about it.
ocalhoun
I'd say more likely the result should be infinity.

Go back to the elementary-school division where you ask, how many times does 2 go into 10? (5)

So, how many 0's could you fit into 10? An infinite number, because no matter how many you add, they still take up none of the original 10.
Bikerman
The problem with that is that the value never leaves zero no matter how many times you iterate. If the answer was infinity then you should see the answer start to converge on infinity with sufficient repeats...
The best answer (for the normal number system) is 'undefined'
ocalhoun
 Bikerman wrote: The problem with that is that the value never leaves zero no matter how many times you iterate. If the answer was infinity then you should see the answer start to converge on infinity with sufficient repeats...

Well, there are other ways to arrive at infinity than diminishing returns...
Just sayin'
 Quote: The best answer (for the normal number system) is 'undefined'

Yes, I'm aware that this opinion about it runs quite counter to accepted mathematical thought.
(There are precious few instances where 'infinity' is practically different than 'undefined' anyway.)
Bikerman
There are lots.
Try Cantor's definitions (it does get a bit heavy though).
You know what, Chuck Norris can divide by zero.

SonLight
Lim [x -> 0+] (1/x) = +inf
lim [x-> 0-] (1/x) = -inf
value [x=0] (1/x) is not defined

Simple pre-calculus logic, but it's good enough to suit me.

Bikerman, I agree with you it would be nice to have Tex available.
ocalhoun
 Bikerman wrote: There are lots. Try Cantor's definitions (it does get a bit heavy though).

So, let me redefine that as x/0=
eday2010
 ocalhoun wrote: I'd say more likely the result should be infinity. Go back to the elementary-school division where you ask, how many times does 2 go into 10? (5) So, how many 0's could you fit into 10? An infinite number, because no matter how many you add, they still take up none of the original 10.

I'd go for that as an answer since it does make sense.
ocalhoun
eday2010 wrote:
 ocalhoun wrote: I'd say more likely the result should be infinity. Go back to the elementary-school division where you ask, how many times does 2 go into 10? (5) So, how many 0's could you fit into 10? An infinite number, because no matter how many you add, they still take up none of the original 10.

I'd go for that as an answer since it does make sense.

^.^

Making sense: 1, Science: 0
Now, on to round 2!

Really though, for any serious endeavor, it's probably better to go with the advice of people who make a career out of mathematics, rather than a rank amateur like myself.
infinisa
4 / 0 has no answer.

This is because division is derived from multiplication:

When I say x / y = z, that actually means that x = z * y,
e.g. 6 / 2 = 3 means 6 = 3 * 2

So 4 / 0 = z means 4 = z * 0
However, whatever (real) value of z you choose, z * 0 = 0, never 4.

So, as I said, 4 / 0 has no answer.

As several people have mentioned, as x -> 0+ (i.e. from above), 4 / x -> infinity,
and as x -> 0- (i.e. from below), 4 / x -> minus infinity. These observations are useful in many contexts, but it doesn't mean that 4 / 0 is infinity (or minus infinity): 4 / 0 still has no answer!

Hope this is clear
gverutes
This is a classic argument/discussion. I agree with the division by zero guy that says it's undefined.
baboosaa
trying to define division by zero is like trying to destroy whole base of modern physics and engineering because whole calculus was developed on the base that- how to escape from the form 0/0 and how to give meaning to the form 0/0. 0/0 has not only troubled us but it had also troubled newton and liebnitz and they also couldnt find out what is 0/0 all they did was changed the form 0/0 to something they can put meaning into.
D'Artagnan
 eday2010 wrote: As far as I am concerned, dividing by zero should not return an error in a calculator. The answer should be zero. Why? Because if you are dividing 4 by zero, you aren't actually dividing. You're not doing anything. So you get nothing. And zero is nothing. So the answer should come up zero because you haven't done antyhing. On the other hand, you can also say that 4 divided by zero should be 4, because you didn't do the division, so you still have 4 left. But -Error- is not an acceptable answer to me.

As a developer (a bad one for not understanding this) division by 0 is a concern to me, you cannot let a program return an uncatched divizion by 0 exception , still i never understood why you can multiply by 0 and cannot divide by 0.

as for the division by 0 problem, i'm too lazy to study right now but if you really want to know , the ansawer is a bit tricky, here are some links i had in my bookmarks marked as future reading...

http://www.math.utah.edu/~pa/math/0by0.html
http://leedsmathgeeks.com/2009/why-cant-we-divide-by-zero/
Bikerman
It is very easy to see.
2*0 = 0
5*0 = 0

So if we take a division:
5/0=?
then this means what do I have to multiply 0 by to get 5? Obviously there is nothing you can multiply zero by that will give you 5.

Alternatively
0/0=?
so you might say what number do I have to multiply 0 by to get 0? and you might answer 0. But that is wrong, because it also works for 1,2,3...infinity. So there is no number which you can multiply by 0 to give 0 - there are an infinity of such numbers.
Therefore division by zero is undefined.
eday2010
That is still flawed to me because when you do 5x0, the answer should still be 5. Why? Because when you do 5x0, you aren't multiplying the 5 by anything. You are basically not doing a mathematical equation because you are not multplying by anything. And when you take 5 and do not multiply it, you still have 5.
Bikerman
 eday2010 wrote: That is still flawed to me because when you do 5x0, the answer should still be 5. Why? Because when you do 5x0, you aren't multiplying the 5 by anything. You are basically not doing a mathematical equation because you are not multplying by anything. And when you take 5 and do not multiply it, you still have 5.
You are multiplying by zero which is not 'nothing' except in a loose everyday meaning. Mathematics is more rigorous and zero is well defined.
ocalhoun
 eday2010 wrote: That is still flawed to me because when you do 5x0, the answer should still be 5.

I've got 5 zero-dollar bills that I'd like to sell to you!
And 5 zero-acre plots of land to sell, too!
LittleBlackKitten
I agree with Chris on this one.

Let's put it this way.

To divide is to create groups of, to coin a term from my grade 3 teacher.

If you take 5, and make 0 groups out of it, how many groups are left? None - 0. This is why anything divided by 0 is still just that - 0. It is the same with ALL numbers - you are making zero groups out of any number, so how many groups have you made?

It works with all division. Take 8, and make 4 groups from it. You can get 4 groups of TWO, which is the answer. (8*4=2).

So, Chris is right.
Afaceinthematrix
 LittleBlackKitten wrote: I agree with Chris on this one. Let's put it this way. To divide is to create groups of, to coin a term from my grade 3 teacher. If you take 5, and make 0 groups out of it, how many groups are left? None - 0. This is why anything divided by 0 is still just that - 0. It is the same with ALL numbers - you are making zero groups out of any number, so how many groups have you made? It works with all division. Take 8, and make 4 groups from it. You can get 4 groups of TWO, which is the answer. (8*4=2). So, Chris is right.

You misspoke because you agreed with Bikerman but then said the exact opposite of his argument (which is correct).

"If you take 5, and make 0 groups of it, how many groups are left? None - 0." That is wrong. How can you make 0 groups of 5 if you have 5? The least you can make is 1 (assuming you're only working with integers). So 5/0 is NOT 0. Nor is it infinity or minus infinity. It is undefined.

The reason why you cannot divide by zero is that R (and C for that matter) is a field. Division by zero would break one of the field axioms. You can divide by zero in certain Algebraic rings depending on how the ring is defined.

You can look at division by zero with a limit which is often useful in many branches of mathematics but it doesn't mean that you should define x/0 = infinity (x not equal to 0). You can also look at the limit of 0/0 which will be different depending on the function you're working with.
Flakky
Bikerman wrote:
 eday2010 wrote: If you divide something by zero, you aren't dividing it. So you still have the same number you started with.

No, you don't. I'm afraid you are wrong. Dividing by 1 is the same as not dividing it at all - as should be quite obvious if you think about it.

Dividing by 1 changes nothing, but cause it's nothing should not return 0
eday2010
And dividing by zero means you aren't even dividing.
SonLight
Let me try to describe what division means in an intuitive way.

If you want to define how elementary arithmetic operations work, it is possible to start with the ideas of "one" and "one more". Addition and multiplication can be defined constructively as repeated use of these. Since our issue is division, I will say no more about the details.

Subtraction and division must be defined indirectly.

What number must be added to 3 to yield 5? The answer is of course 2, so we define 5 - 3 = 2.
Not all such questions can be answered in positive integers. If negative integers and zero are introduced in an intuitive way, then all such questions involving integers have an integer as an answer.

What number must be multiplied by 2 to yield 6? The answer is 3, so we define 6 / 2 = 3.
We find many such questions do not have an integer answer. Since the process of cutting an item into pieces is intuitive (with some kind of items at least), we are motivated to define fractions in a way that intuitively fits our number system. There are a few questions that still have no answer, however. One of them is,

What number must be multiplied by 0 to yield 5?

If we had an answer, we would define it as equal to 5 / 0 or five divided by zero.
Since there is no answer to the question, the operation of dividing by zero is meaningless and undefined.

We can examine why it is undefined, perhaps by dividing by smaller and smaller numbers and observing that the answers to 5 / x get very large as x gets very small. We might see that for negative values of x close to zero, 5 / x is very small (negative but large in magnitude).

It is an inherent limitation of our number system that no meaning can be given to 5 / 0 that is consistent with other rules and values of the system. Trying to say it's "infinity" or "plus or minus infinity" is a way of expressing why it fails, but is not a definition and will lead to logical errors if you try to use it like a number.
polly-gone
If dividing by zeros was possible, calculus would be impossible. The whole process of solving a derivative the long way is just taking 0 out of the denominator. If we could divide by 0, the derivative of everything would either be itself or 0 depending on your exact logic.
Dennise
All mostly true.

Dividing by zero is undefined and zero or any other number is certainly defined.

Therefor, the operation cannot result in anything other than "error'.

Actually the result of dividing by zero is infinity .... but this too is undefined.
Flakky
I think it's much more logical that dividing by 0 should lead to infinity (or negative infinity).

1/1=1
1/0.5=2
1/0.25=4
1/0.125=8

As you see the closer it gets to zero the result gets bigger. So in my understanding this makes more sense:

1/0=infinity
5/0=infinity
-1/0=-infinity
0/0=0

Also in my logic:
1/infinity=0
0/infinity=0
-1/infinity=0
Afaceinthematrix
 Flakky wrote: As you see the closer it gets to zero the result gets bigger. So in my understanding this makes more sense: 1/0=infinity 5/0=infinity -1/0=-infinity 0/0=0 Also in my logic: 1/infinity=0 0/infinity=0 -1/infinity=0

This just isn't true. I'm going to first point out the first logical error that you made:

0/x = 0 when x does not equal 0. I think we can both agree on that.

You said that x/0 = (+ or -)infinity when x does not equal 0.

But 0/0=0? How do you know which is more powerful - the denominator or the numerator (for this reason, it's in an indeterminate form)? In fact, you cannot even look at dividing anything (even 0) by 0 without using a limit. Here's a simple counter-example to your logic.

Look at (sin(x))/x at x = 0. Putting in 0 for x yields sin(0)/0 and sin(0) = 0 so it yields 0/0=0, right? WRONG. The value of sin(x)/x as x approaches 0 is actually 1. So what you said there was clearly wrong.

Furthermore, 1/0 is NOT infinity. It is undefined. You can look at the limit and it will approach infinity, however, it's undefined. To truly explain why, you'll need some real analysis and that's just too rigorous for your level (Hell, it's even rigorous for me).
inuyasha
Of course it returns an error. You said divided by zero isn't really dividing. So you're really dividing or not? That's a logical error. You are going to divide, but you actually doesn't.
D'Artagnan
it's pretty seasoned question , i don't question the math, what i question is WHY DO THEY HAVE TO BE SO A*AL RETENTIVE ABOUT THIS.

why cant they just let division by 0 result in 0 at least on high level programming languages, ive seen this like \$number/\$divisor stoping whole companies just because the freaking programmer forgot to treat a division by zero exception. if the guy ain't gona test the result he ain't gonna test the freaking variable... nobody will be writing nuclear test software on clipper!
Bikerman
 D'Artagnan wrote: it's pretty seasoned question , i don't question the math, what i question is WHY DO THEY HAVE TO BE SO A*AL RETENTIVE ABOUT THIS. why cant they just let division by 0 result in 0 at least on high level programming languages, ive seen this like \$number/\$divisor stoping whole companies just because the freaking programmer forgot to treat a division by zero exception. if the guy ain't gona test the result he ain't gonna test the freaking variable... nobody will be writing nuclear test software on clipper!

Letting division by zero equal zero would be a catastrophe. The whole basis of mathematics would be destroyed and you would not be able to have any faith in the results of any calculation. The reason for being 'anal retentive' is that maths is either consistent (and self-consistent) or it isn't. If it isn't then it is important to understand WHERE it isn't and WHAT THAT TELLS US about maths as a useful tool for modelling reality. If you suddenly say 10/0 = 0 then you have invented a whole new type of mathematics which is internally conflicted and quckly leads to non-resolvable paradoxes.
chatrack
Hi,
dividing by zero, means dividing by a near zero value, that is a value very close to zero.
so it should give an infinite magnitude a result.

I think it is just like thinking that Two parallel lines meet at infinity.. that is maths and science like
metalfreek
Only logical thing that we can say when a number is divided by zero is undefined or in case of calculus indeterminant.
Afaceinthematrix
 metalfreek wrote: Only logical thing that we can say when a number is divided by zero is undefined or in case of calculus indeterminant.

Indeterminate would be where the numerator and denominator are both 0. That's a big difference because the numerator and denominator are both affecting the limit and you don't know which one is affecting it more strongly than the other and the limit might end up being 4 or something other than 0 or infinity...
sermonis
Maybe it sounds weird but it's better to understand then you say you devide it into 1 or you devide it into 2.

Like when you have pizza you devide pizza cake into 1 piece so you have 1 pizza
if you devide pizza into 2 pieces you have 2 * 1/2 of pizza cake.
If you like to devide pizza cake into 0 pieces you can't cause you have one pizza not zero pizzas that's why you will get an error saying 'Hello we have this pizza, you wanna make it desapire?'

And like you said doing nothing is trying to devide our pizza cake to 1 piece cause it is in one piece already.

ps. Math is easier when you compare it to food ;D
timetorock
But the case becomes different if you take limits.
_AVG_
Here's an interesting thing about dividing by zero (of course, only if you're interested in complex abstract mathematics!)

Have you heard of different algebraic structures in mathematics called "Hypercomplex Numbers"? These generalize even the Complex Numbers and use a Cayley-Dickinson construction. There are Quaternions (famous as one of William Rowan Hamilton's discoveries), Octonions and Sedenions, etc. ....

As we become more and more Hypercomplex, normal algebraic rules break down. For instance, the multiplication of Quaternions is non-commutative; the multiplication of Octonions is non-commutative and non-associative. As far as Octonions are concerned, a consequence of this fact is that zero divisors do exist!

To explain this better, imagine that you have xy = 0 and neither x nor y is zero (like in Matrix Multiplication!) Then multiply (1/y) with xy and you get x! So, you've multiplied something by zero and not got zero!

I guess mathematics gets a bit absurd when it's abstract (and probably I shouldn't have posted this in the Basics forum, but anyway, hope you find this interesting)
mazito
Bikerman wrote:
 D'Artagnan wrote: it's pretty seasoned question , i don't question the math, what i question is WHY DO THEY HAVE TO BE SO A*AL RETENTIVE ABOUT THIS. why cant they just let division by 0 result in 0 at least on high level programming languages, ive seen this like \$number/\$divisor stoping whole companies just because the freaking programmer forgot to treat a division by zero exception. if the guy ain't gona test the result he ain't gonna test the freaking variable... nobody will be writing nuclear test software on clipper!

Letting division by zero equal zero would be a catastrophe. The whole basis of mathematics would be destroyed and you would not be able to have any faith in the results of any calculation. The reason for being 'anal retentive' is that maths is either consistent (and self-consistent) or it isn't. If it isn't then it is important to understand WHERE it isn't and WHAT THAT TELLS US about maths as a useful tool for modelling reality. If you suddenly say 10/0 = 0 then you have invented a whole new type of mathematics which is internally conflicted and quckly leads to non-resolvable paradoxes.

i see some videos made it in Spain, about pre calculus, and the narrator uses the sacred rules of Math (speaking about real numbers), and i loved that

1.- is a sin divide by 0 (in math such try led us to no real number)

2.- if you tray the par root of any x<0 there is no real number too

3.- there is not natural logarthim for x<0

at first i need a precision, the numbers are ideas, a number reprecents an amount, but a number can write it in many ways, I often ask my students that if they know the number five and everyone tells me yes, but the five can be represented by, 5, V, +++++, 25/5, 125/25, iiiii, etcetera, then the numbers are not the things, people, weights, coins.

then move on to the division, divide something is split into pieces to see how much is up to each receiver, if you divide the 0 between several, ther is anything for echa one of the recivers, but instead if you try to divide by 0, loses its reason to exist, there is not such thing.

is so hard to write in english for me, to explain this kind of ideas, and i remains silent in some themes that i want to particpate, so please forgive me if i write some that is a nonsense
KaczuH
I've just watched something which may clarify the topic a bit

They expalin some problems which occur in previous posts and give idea how to think about zeor and infinity
asnani04
In everyday mathematics, any number divided by zero is undefined. Also, in the basic definition of a rational number, most textbooks say, " a number of the form p/q, where p and q are integers, and q is not equal to zero, is a rational number." So, it's clear.
codersfriend
In cartoons, some people put when you divide by zero, a distortion in the universe occurs
jajarvin
Zero is the first number! Try no to forget it. One is only the second number.
kelseymh
 jajarvin wrote: Zero is the first number! Try no to forget it. One is only the second number.

Sorry. Zero is the first natural number. There is no "first number", since the integers cover the full range from -infinity to +infinity, and the reals cover the full range from -aleph_0 to +aleph_0. In both cases, zero is the middle number. Try not to forget it.
codersfriend
some people joke that dividing by zero causes distortion in the universe or great bug on computers
tW_Studios
shashwatblack
you are soo wrong... i'm afraid that's the wrong-est you could ever get...
when you say something like 4/2, you're adding 2 to itself. and the number of times you do that is your answer.. in this case it's 2. you add 2 twice to reach 4.
similarly you add 2, 2.5 times to reach 5, so 5/2 = 2.5.
when you say 4/0, you're NOT doing nothing, the thing is, no matter how many times you add 0 to itself, you always get 0. Infinity is an abstract concept that says that if you add 0 to this many times, you will get to any number you want. thus 4/0 is not 4, but it's equal to infinity. of course you'll never actually get to infinity if you were to keep counting, because you can always add one. as said earlier, infinity is just an abstract concept.
as for 0/0, you can get it to be anything
say,
2 * 0 = 0 implies 2 = 0/0
3 * 0 = 0 implies 3 = 0/0

you can get it to be anything, thus is is called indeterminate state. and you cannot solve any equation in this state. your calculator shows it as error. some calculator can distinguish between these forms and display result as `inf` for infinity.. just an example..
hspb
Division is the function of our mind.
We can devide so, as we determine.
ethan_brown
Division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as a/0 where a is the dividend (numerator).
harrer
Let's understand it like this:

You've been Given a loaf of bread and a knife.

You're asked to cut it into 1 piece. But the loaf is one piece itself. So you return it without cutting.

But what if you're asked to divide it into zero.
You have to return something but zero means you have to return nothing.

This gives an ambiguity.
SonLight
In the first case, technically you cannot _cut_ it into one piece, because you would not be cutting it. You could offer to give them the full loaf uncut instead, and they might agree that you had provided them what they want.

While cutting (or not-cutting) it into zero pieces is mathematically impossible, in this case you could eat the loaf and say, "here's nothing".
Arrogant
Dividing by zero means you are doing nothing
Leaving it in one piece
But Dividing by zero doesnt mean you are doing nothing
Do you even have a slight concept of calculus?
spinout
In the numberphile video they wonder how/if a calculator is coded to deal with zero?
I presume my fairly new texas instrument calculator I have is coded to spot the zero.
But older calculators (before LCD ...) I presume not are coded to spot zero directly and present an overflow error after the subtration operation. But I am not an expert in calculator design so please fill in! ... I once had a calulator with numbers in red dots instead of LCD - yes I got grey hair
Bikerman
I remember my Grandad giving me a CASIO 501P programmable calculator in 1978. It was the dog's bollox - fully programmable 7-segment display and 128 step memory ( ie you could store 128 instructions per program)

1/0 would be partially trapped (and produced Err in the display from memory)
Afaceinthematrix
_AVG_: You can divide by zero in any ring that has a zero divisor; that's definition. The maths doesn't get absurd when you start talking about these things; it just gets tricky. The point is to abstract it because in many fields, such as physics, you need to look at something in something other than Euclidean space. I personally find algebra to be extremely boring to study as a subject but it is extremely important in other fields - such as topology - and so I see it as a tool to be used.
cabenqc
IMHO
Primary school won't touch this topic in general.
Secondary school introducing some good concepts if got a good teacher.
University can have a full course to discover that, but not so common nowadays.