Zero divided by Zero equals 0 & 1 & ∞

[Philosophy Explorer]

Jaded, if the two operations are not opposite, this is what can happen.

E.g. 1 x 2 = 2 is the opposite of 2/2 = 1 when you divide both sides by 2, right? Now 1 x 3 = 3 and 3/3 = 1 are also opposite. More generally, 1 x a = a is the opposite of
a/a = 1, right (because you are dividing instead of multiplying or multiplying implies dividing).

Now look at these examples. 0 x 2 = 0 implies 0/0 = 2; also 0 x 5 = 0 implies 0/0 = 5, right? More generally,
0 x b = 0 implies 0/0 = b, no matter what b is. If division weren't the opposite of multiplication, then this wouldn't hold. This is why 0/0 can equal any point along the real number line (along with +/- infinity).

Is it becoming clearer to you?

PhilX

[Scott Mayers]

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.

Simplistically, to divide by nothing is to not divide by anything. And I'd imagine that even right now you are just as equally dividing by zero in this way even without thinking of what you might be dividing! That is, you are equally not dividing an infinity of numbers at this very moment as you would be if you were too try.

Another way?:
Division is the process of subtracting some absolute value of a number until you reach zero. But zero is already zero and therefore completed in rule by default.

Using the same example of 15/5. We divide this by 5 until we get to zero. If division by zero counts, then we also have to do this for even this example. Here we'd have 15/5 = 3 with 0 remainder. Then, if you think it appropriate to still divide by 5, we either have

0/5 = 0 remainder 0

OR

0 - 5 = -5 remainder -5,
-5 -5 = -10 remainder -10,
etc, etc,
to - infinity remainder -infinity.

Either way, we remove the meaning of division for other numbers as meaningless if you accept 0/0.

[Scott Mayers]

Out of respect of your title and your OP, you are thus 'correct' that zero divided by zero = 0, 1, and ∞. But this includes all numbers infinitely in this way if allowed.

[Jaded Sage]

to divide by nothing is to not divide by anything.

But zero is already zero and therefore completed in rule by default.

First statement: I disagree, and I'm quite certain that is incorrect. That is where math and english stop being the same.

Second statement: there might be something in that, as there might have been in the idea that there are both 3 and 4 zeros in 0.

[Jaded Sage]

Out of respect of your title and your OP, you are thus 'correct' that zero divided by zero = 0, 1, and ∞. But this includes all numbers infinitely in this way if allowed.

Gracias, but I still don't get how it equals other numbers. I didn't get the remainder thing.

[Scott Mayers]

to divide by nothing is to not divide by anything.

But zero is already zero and therefore completed in rule by default.

First statement: I disagree, and I'm quite certain that is incorrect. That is where math and english stop being the same.

Second statement: there might be something in that, as there might have been in the idea that there are both 3 and 4 zeros in 0.

To do the ACT of dividing, you need something to divide. For instance, you can say, "Let's divide some pizza into pieces between us". Yet, what does it mean to 'divide' what you don't have? So if I have a non-existing pizza (pretend or invisible pizza?), how many times can I divide it? Once? Twice? ...Infinity? In some cases, 0/0 can have meaning, though. Where used, it means "infinity". This is because you CAN take some non-existent thing and also not-divide it (meaning to divide it 0 times) infinitely. But it reduces to the same idea. Infinity means "unable to make finite or finish" The idea of "definition" means, "to make finite". Zero can be 'defined' only in part as "that which has some origin without natural or absolute value but real in meaning." But it is not defined by any process that is infinitely complete. It doesn't mean it has no use. It just means that where it can be processed such that it has no unique solution, it cannot be captured as one idea but many. 'x/0' means every number no matter what x is simultaneously.

Another way to help you understand this is to take any number in the numerator (top of the fraction) and divide it by a small number that is close to zero.

Now you know that 1/(very large number) approaches zero, right? Take 1/2 = 0. 5. Take 1/4 = 0.25. take 1/100 = 0.01. Take 1/1000000 = 0.000001. Well, then you should recognize that if this 'very large number' is infinity, we have

1/∞ = 0

This is actually interpreted as the limit of 1/x as x approaches infinity.

Since we want to find what any number is divided by zero, let y be the number in the numerator and we have

y/(very smallest number that approaches zero) = y/0 = y/(1/∞)
=
y(∞/1) [to divide a fraction is to multiply by its inverse: y/(1/∞) means y divided by 1/∞ which means y multiplied by (∞/1)
=
∞/1 = ∞

This is the 'approach' or limit of dividing any number by a number that gets eternally smaller (close to zero). The same can be done by approaching zero from the negative numbers. Only this time, the result is -∞ (negative infinity). This proves that AT zero, all numbers between -∞ and ∞ is the answer to y/0. In other words, it is ALL the numbers collectively at once! This is the graph [Wikipedia, "Division by zero" uploaded by Ktims"]:

[attachment=0]1280px-Hyperbola_one_over_x.svg.png[/attachment]

[Jaded Sage]

I'm sorry, man. I just can't follow that. I still think it is mistaken to say that dividing by zero is the same as not dividing. It seems like you are really close to explaining what I mean tho.

[Scott Mayers]

The first of the following is true for all cases from negative infinity to positive infinity by the rule 0*n = 0. The second of each of the following shows that given any particular equation from the first should allow you to divide like terms on both sides. This demonstrates that 0/0 = all numbers. This makes it indeterminate everywhere.

0*0 = 0
0 = 0/0 [divide zero both sides]

1*0 = 0
1 = 0/0 [divide zero both sides]

2*0 =0
2 = 0/0 [divide zero both sides]

...

0*n = 0
n = 0/0, where n is any number.

See http://mathforum.org/dr.math/faq/faq.divideby0.html for an exhausted explanation with examples.

[Jaded Sage]

Yeah, I also got that by the transative property too. It equals all numbers.

[Scott Mayers]

Yeah, I also got that by the transative property too. It equals all numbers.

Note that while the process doesn't allow division by zero to mean anything, it is still useful if one wants to assign it the indeterminate infinity as can be the case in some maths one could develop. That is 0/0 can be 'piecewise' defined, as it is called to be a type of infinity. In fact, I believe that the origin of infinity is derived from placing two zeros together to represent this: '00'. I find it a useful device in my own metalogical theory in an altered way.

[Jaded Sage]

I still don't understand the process. Unless you are talking about your most recent post after this one. But just because the process doesn't allow it doesn't mean it doesn't have an answer. I'm only aware of two types of infinity: potential and actual. But that's only because I dabbled in aristotle's metaphysics instead of math. Metalogic sounds fun.

[Scott Mayers]

I still don't understand the process. Unless you are talking about your most recent post after this one. But just because the process doesn't allow it doesn't mean it doesn't have an answer. I'm only aware of two types of infinity: potential and actual. But that's only because I dabbled in aristotle's metaphysics instead of math. Metalogic sounds fun.

I used the graph up above to aid in understanding. I'm guessing that you are not familiar or comfortable with graphs. But it should help understanding if you can follow this:

The process of math acts like a machine. So think of a calculator, for instance as one type of machine that takes a set of inputs you want to manipulate in a simple process, like division (to relate to this example). The 'inputs' are the available numbers called the 'domain' (think of your home as your 'domain') where you begin. The graph then uses the x-axis (horizontal line) to represent this domain of possible inputs.

A 'function' is the simplest kind of relation(ship) between the domain and the solution you aim for, we call the 'range' (think a firing range where you intend the target away from you to end on). A 'function is one number input to one number output. For this 'output' the range is represented by the vertical line from '0'. So for each input, the graphed line above represents all possible number inputs that give you an output.
Because this is a function, only one possible solution is allowed for each input. To test this on a graph like this, you imagine a vertical line going through each and every point. If it crosses the drawn graph of your solutions MORE THAN ONCE, it indicates more than one solution and so would NOT be a function.

Division is a function type relation. [A circle would be an example of one that is NOT a function but can be made into two.]

So the graph above two or three posts up show that as the graph of the curve on the left side of '0' gets closer to '0', it approaches that vertical line through zero, the y-axis. But it goes down off the page. It should be drawn with an arrow to show this but is usually understood. So this means that as you get closer to zero from the left side, it forever gets lower towards negative infinity.

On the opposite side, as you find solutions (the curve) that get closer to zero from the right, it approaches positive infinity. But AT '0', it either never reaches ANY number, OR the solution is any and all numbers that lie on the y-axis at zero. This pictatorial representation shows this. So technically, if you demand a solution, it is each infinity of numbers from -∞ to +∞. So this may be represented as each real number from -∞ to +∞ OR simply, ALL REAL numbers at once!

This is not illegal to think about. But just as infinity itself in not defined [de- means 'of', -fine means 'fin(al)' or end; and in- means NOT ever, -finite means 'fin(ite)' means ending], math simply places it in the category of non-defined ideas because they don't represent any fixed or certain number.....because is never ends. It is an ACT rather than a STATE, a VERB rather than a NOUN.

Does this help clarify things better?

[Jaded Sage]

Yeah, the issue is that I want to call into question the rules of math (or 0/0 does). I mean, that's the heart of the question. Doesn't an asymptote represent infinity?

[Scott Mayers]

Yeah, the issue is that I want to call into question the rules of math (or 0/0 does). I mean, that's the heart of the question. Doesn't an asymptote represent infinity?

Yes, it does where it is a vertical line. An asymptote is also the x-axis from negative infinity to positive infinity on that graph too. But since the x-axis represents the real input numbers, the only one not allowed as 'defined' is the '0' there. That is, the domain is all Real Numbers except for '0' in the graph for division. But an asymptote is sometimes also a sloped line OR many curves even.

[Jaded Sage]

I suppose that is part of the issue then. Why is an exception made for zero?