Zero divided by Zero equals 0 & 1 & ∞

[Philosophy Explorer]

Wolfram says the same thing I just said:

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

PhilX

[Hobbes' Choice]

To answer the question, it depends on the context. Now it can be argued that 0/0 = 2 too because 0 x 2 = 0 and any other number you can think of. Since 0/0 = any number on the real number and there are an infinite number of them, then it's meaningless to talk about division by 0.

Nope.

Apparently you don't understand that 0 multiplied by any number is the same as 0/0. Why don't you check this out with a math teacher to verify?

PhilX

Anyone with any sense knows that dividing by nothing makes no sense. That's why in computer languages there is a specific error message designed to alert the programmer when one of their variables reaches zero otherwise it invokes the "Division by Zero Error" message.
Zero is not a quantity. The whole thread is without merit.

[Philosophy Explorer]

Anyone with any sense knows that dividing by nothing makes no sense. That's why in computer languages there is a specific error message designed to alert the programmer when one of their variables reaches zero otherwise it invokes the "Division by Zero Error" message.
Zero is not a quantity. The whole thread is without merit.

That's my point. And zero not being a quantity is besides the point as zero is regarded as a number by mathematicians since you can add and multiply by it to get an answer (part of math axioms). Division is the opposite operation of multiplication which means that 0/0 takes in the entire real number which is why it's meaningless (Wolfram says the same thing).

PhilX

[wtf]

Apparently you don't understand that 0 multiplied by any number is the same as 0/0. Why don't you check this out with a math teacher to verify?

That's as wrong as wrong can be. If 5*0 = 0/0 then dividing both sides by 0 gives 5 = 1/0. And multiplying both sides by 0 gives 5 = 0.

[Moyo]

LETS SEE what WE HAVE here :^)

Recall that an equivalent realtion partitions a set into equivalence classes.

We start of with the cartesian product X = Z X Z* , where Z is the integer numbers and Z* is the integers without "0".

We then have an equivalence relation (a,d)R(b,c) iff ad=bc (think a/b = c/d)

(a/d) = E(a,b) (i.e the equivalence class), E(a,b) = {(c,d) | ad=bc}

e.g E(1/2) = {(1,2);(2/4);(3,6);(4,8)....}

Now here's why we cannot divide by zero

If we try have (1,0) R (a,b) i.e. we treat zero fairly and let it generate other equivalent quotients like here we have say (a,b).

In the Equivalence claasses named above there is a distinguished element where the greatest common divisor is 1. Thats the rule of quotient classes so in the relation (1,0) R (a,b) then a cannot be 0 and b cannot be 1.

But if

1/a = b/0

we can solve this by

0x1 = ab

therefore

a= 0/b = 0

therefore

a=0

since we could not have

a= 0

in our quotient class but having this fact we are led to conclude that

a is in fact = 0 ,

Then 1/0 cannot be expressed with factors, even its own.Or equivalently (stay with me here) it can only be expressed with irreducable factors.

This means that E(1,0) is not related to any number since E(a,b)={(a,b);(2a,2b);(3a,3b)...} but for E(1,0) we only have {(1,0)} period, meaning E(1,0) does not exist as it is not related to anything .either (2,0) or even itself and hence cannot be an Equivalence relation even in the most general way.


Understand that this stems from the way we constructed the rationals...their very construction (making them equal to themselves ) forces us to include a blind spot...if you change the way of construction then you have "imagined" a new set of numbers that is not equivalent to itself and not the "rationals".

Just saying.

[Hobbes' Choice]

Apparently you don't understand that 0 multiplied by any number is the same as 0/0. Why don't you check this out with a math teacher to verify?

That's as wrong as wrong can be. If 5*0 = 0/0 then dividing both sides by 0 gives 5 = 1/0. And multiplying both sides by 0 gives 5 = 0.

This statement is wrong. 0/0 = 1. therefore not 0, not 5x0.

[Philosophy Explorer]

Apparently you don't understand that 0 multiplied by any number is the same as 0/0. Why don't you check this out with a math teacher to verify?

That's as wrong as wrong can be. If 5*0 = 0/0 then dividing both sides by 0 gives 5 = 1/0. And multiplying both sides by 0 gives 5 = 0.

This statement is wrong. 0/0 = 1. therefore not 0, not 5x0.

Again 0/0 equals any number along the real number line (and +/- infinity) because division is the opposite operation of multiplication.

PhilX

[Hobbes' Choice]

That's as wrong as wrong can be. If 5*0 = 0/0 then dividing both sides by 0 gives 5 = 1/0. And multiplying both sides by 0 gives 5 = 0.

This statement is wrong. 0/0 = 1. therefore not 0, not 5x0.

Again 0/0 equals any number along the real number line (and +/- infinity) because division is the opposite operation of multiplication.

PhilX

10 divided by 5 is 2. Or how many fives in ten. There are two fives in ten.
How many zeros in zero ? One!

[Philosophy Explorer]

10 divided by 5 is 2. Or how many fives in ten. There are two fives in ten.
How many zeros in zero ? One!

How about 0 + 0 + 0 = 0? Isn't that three zeroes in zero?(beginning to remind me of Abbott and Costello)

PhilX

[Hobbes' Choice]

10 divided by 5 is 2. Or how many fives in ten. There are two fives in ten.
How many zeros in zero ? One!

How about 0 + 0 + 0 = 0? Isn't that three zeroes in zero?(beginning to remind me of Abbott and Costello)

PhilX

No it's not three zeros in zero. It's adding nothing to nothing three times.

[Philosophy Explorer]

10 divided by 5 is 2. Or how many fives in ten. There are two fives in ten.
How many zeros in zero ? One!

How about 0 + 0 + 0 = 0? Isn't that three zeroes in zero?(beginning to remind me of Abbott and Costello)

PhilX

No it's not three zeros in zero. It's adding nothing to nothing three times.

What do you mean by zero in zero?

PhilX

[Jaded Sage]

I forgot all about this. WTF came closest to getting it. I think maybe we are onto something with the 0 + 0 + 0 being three zeros in zero, which could be two or four just as easily. Maybe we're onto something.

But the fact is: we are challenging mathematical rules. It is assumed that every number is unequal to every other number. It is also assumed that two quantities that are equal to the same quantity are equal to each other. These two assumptions cannot be held simultaneously, given what we've shown.

Also, no. It can't be argued that 0/0=2, unless you've done some work with infinity that you haven't shown us, or something with ratios.

[Philosophy Explorer]

Jaded Sage said:

"Also, no. It can't be argued that 0/0=2, unless you've done some work with infinity that you haven't shown us, or something with ratios."

Are you denying that multiplication and division are opposite operations?

PhilX

[Scott Mayers]

To answer the question, it depends on the context. Now it can be argued that 0/0 = 2 too because 0 x 2 = 0 and any other number you can think of. Since 0/0 = any number on the real number and there are an infinite number of them, then it's meaningless to talk about division by 0.

Now if you talk about approaching 0 as they do in calculus, that's a different matter which may be defined and is acceptable to mathematicians. Also note that for a quadratic equation, there may be two answers for the variable which mathematicians find acceptable.

So unless you have further qualifying information, the answer is division by 0 is meaningless.

PhilX

This confusion by most, even by mathematicians throughout time, still haunts many. I still see that we have a conflict with this in regards many apparent paradoxes which I struggle to show the same problem [that Monty Hall thing I don't want to go back into].

Another way that might help others to think of this is to ask if given a perfectly empty or absent-of-absolutely-anything container [ideal of course], you might think of this as containing all that NOT TRUE. Thus, if absolutely nothing is inside such a container it also includes all that it NOT TRUE infinitely. If some X does not exist or is untrue, then it lies in the 'container' that is perfectly empty or non-existent. For instance, I may say that "I have nothing in my pocket" as implying that I should also have no thing in mind that we are positively thinking, like money, which is a real idea. It thus can be true that I also do not have a car in my pocket. So I have 0 Car(s) in my pocket.

The numerator of a fraction "posits" what is true, just as a Car is an idea that exists.

Someone mentioned above in error that division is NOT subtraction. This is false. It IS the measure of repeated subtraction.

15/5 means "posit 10" to begin with, then subtract 5 and count each ACT of subtraction.

15 - 5 = 10
10 - 5 = 5
5 - 5 = 0

Count the fives until you reach no Remainder. This is three.

A fraction that is less then one is the Remainder form or "Ratio" of comparison that is minimized. This is why we call them "rational". So a fraction like "1/2" means only that given a posited non-zero thing [a non-nothing], here a '1', that we are thinking of a unit of something that has 2 parts of a whole.

Now take "x/0".

This means x - 0 = x. So this would imply that x/0 = x + no remainder of zero. Using this equation, multiply both sides by zero in kind to what someone else above though that means to cancel the zero on the left as

(0)x/(0) = (0)x
(0/0)x = 0x
x = 0.

So if given any 'x', it would always force 'x' to also equate to zero. But this contradicts any number given to 'x' except zero itself. So, by convention, 0/0 is assigned to mean infinity. In fact the very symbol of "infinity" is likely derived from this as "00" to remind us (or original users) that is meant both that 0 x (any infinite number of '0's) = 0 AND that 0/0 = infinity.

[Jaded Sage]

Phil X: I think I might be, in this instance exclusively. But you'll have to forgive me, because it's literally been over a decade since I've done math in school. Would you please explain to me, as you would a child, how that pertains to this case exactly, again, as you would a slow-learning child.

Scott: While I'm very impressed by your prowess and devotion, the point I mean to make is that convention seems to be at odds with itself in this instance, or at least that this instance seems to defy convention, or that convention hasn't properly addressed it yet. It looks like it might be beginning to tho.