want to Know why you can't divide with 0?

[iPwn]

Now, it would seem as x becomes a really, really small number, y becomes a very, very large number. It is only natural to plug 0 and ∞ in x and y, respectively, as 0 carries a certain attribute of smallness, and ∞ carries a certain attribute of largeness.

As above
So below

If indeed:
1/0 = ∞

Then:
1/0 = ∞/1

We should have:
1/1 = 0 × ∞

"Post hoc ergo propter hoc"

The third step is false.

1 / 1 != 0 x ∞

1 / 1 = 1
0 x ∞ = 0

The equations do not equal, therefore neither do 1 and 0

 

[dale]

Well yes, precisely. The third step, "1/1 = 0 × ∞", is exactly the same as the second step, "1/0 = ∞/1" (and of course, the second step the same as the first step, "1/0 = ∞"), from a mathematical point of view because a/x = y/b can be written as ab = xy.

This is because we are basically multiplying them by both of the denominators, i.e.:
1/0 = ∞/1
1/0 × (0 × 1) = ∞/1 × (0 × 1)

And so simplified by cancelling out the red bits:
1/0 × (0 × 1) = ∞/1 × (0 × 1)
1 × 1 = ∞ × 0
1 = 0 (which, yes, I agree, is not true)

So if the third step is not true, the second step (which is just the same as the third step) is not true, and so 1/0 = ∞ must not be true.

To put this into perspective, although "0 carries a certain attribute of smallness, and ∞ carries a certain attribute of largeness", 0 probably has other attributes that hinders it from fitting into the above acrobatics nicely. If we were to make 0 fit, we would have to alter the very basic concepts of mathematics, e.g., multiplication, division, etc. Way too easier to say 0 is undefined.

 

[iPwn]

The third step, "1/1 = 0 × ∞", is exactly the same as the second step, "1/0 = ∞/1" to t" (and of course, the second step the same as the first step, "1/0 = ∞"), from a mathematical point of view because a/x = y/b can be written as ab = xy.

So if the third step is not true, the second step (which is just the same as the third step) is not true, and so 1/0 = ∞ must not be true.

First step is a condition and the topic of debate:
IF 1 / 0 = ∞

Second step is causation, but ambiguous in nature:
1 / 0 = ∞ / 1

Third step is hypothetical causation and false:
1 / 1 = 0 * ∞

Here, you are trying to say that because these two equations are not equal, then the entire line of thought that brought you here cannot be plausible, making the original question impossible.

IMO, that's like hitting a dead end in a maze and stating the maze is unsolvable.

 

[dale]

First step is a condition and the topic of debate:
IF 1 / 0 = ∞

Second step is causation, but ambiguous in nature:
1 / 0 = ∞ / 1

How is ∞ = ∞ / 1 ambiguous?

Third step is hypothetical causation and false:
1 / 1 = 0 * ∞

Yes, it is false.

It is false because 1/0 = ∞ is false.

 

[iPwn]

How is ∞ = ∞ / 1 ambiguous?

I see that as ambiguous because it invites an entirely new debate of infinity. Can you divide infinity?

Yes, it is false.

It is false because 1/0 = ∞ is false.

You're saying that 1/0 = ∞ is false because some other equation is false.

If 1/0 = ∞ were true, it doesn't automatically mean that 1=0. The 1=0 argument is a different question that you're using to disprove the first.

If given directions with three steps.

1. Turn left
2. Turn left again
3. Turn left again

If step 3 leads you into a ditch, that doesn't mean that steps 1 and 2 led you into a ditch.

 

[dale]

Okay, so you have doubts with ∞ = ∞ / 1 (which is getting philosophical, i.e., is it possible to have the whole of infinity?); surely you see this, too, is problematic?

Consider again:
1 / 0 = ∞

Multiply by 0 for both sides:
1 / 0 × 0 = ∞ × 0

Now what do you make of this? How do you multiply 0 by an infinite amount of times?

 

[iPwn]

Okay, so you have doubts with ∞ = ∞ / 1 (which is getting philosophical, i.e., is it possible to have the whole of infinity?); surely you see this, too, is problematic?

I don't see it as problematic nor do I have doubts, just something I don't completely comprehend. Just because I cannot comprehend infinity, doesn't mean it doesn't exist.

I see (∞ / 1) as an infinite array. Again, no doubts, just too human.

Consider again:
1 / 0 = ∞

Multiply by 0 for both sides:
1 / 0 × 0 = ∞ × 0

Now what do you make of this? How do you multiply 0 by an infinite amount of times?

You don't. It will be zero regardless of where on the number line you are, from 0 -> ∞. If the answer is always going to be zero, why would I concern myself with how to write it out?

Just because we can't write it down doesn't mean it is not possible. It's amazing that scientists can quantum pair photons and yet the concept of infinity is like the end of the flat earth. Many things are possible, we just have our reservations and rules. I remember reading an argument somewhere about dividing by zero. The author stated that according to math, dividing by zero was undefined and therefore impossible. He further went on to make the comment, "Not just because some rule someone made up." Well, math is indeed a bunch of rules that man made up. Rules that we use to define the world we live in. The only problem with being so zealous about those is that we cannot possibly understand the world we live in, in totality, and it's not impossible that something comes along and shatters the foundation of what we "know."

 

[dale]

You don't.

The difficulty in multiplying 0 by ∞ is not in the number of times that you would have to labour your hand and other writing resources to expand the equation, but in the concept of infinity itself.

It will be zero regardless of where on the number line you are, from 0 -> ∞.

Why? (I think this is a valid question, especially now that you have mentioned rules because most mathematicians would agree -- which makes it more of a rule, that it will not be zero.)

 

[iPwn]

The difficulty in multiplying 0 by ∞ is not in the number of times that you would have to labour your hand and other writing resources to expand the equation, but in the concept of infinity itself.

Fair enough, and I guess my point was poorly made. I simply mean to point out that anything multiplied by zero is zero. So even infinity, zero times, is nothing. If I were given infinite power a total of zero times, I would not have infinite power. Therefore, my point was simply that debating the details of how ∞ * 0 would work out is IMO moot.

Why?

I don't entirely understand your question here. My point was that whatever number you could possibly decide on to define ∞ would still equate to 0 when multiplied by 0.

(I think this is a valid question, especially now that you have mentioned rules because most mathematicians would agree -- which makes it more of a rule, that it will not be zero.)

My thoughts on a bunch of sir's in a smoke filled room agreeing on something making it law are funny but irrelevant.

Please explain, as I am curious, how (∞ * 0 != 0).

 

[dale]

Haha, very picturesque, the smoke-filled room.

I just want to point out that I find this discussion very enticing. I think it's great that these questions are raised, and by no means am I just saying: look, this is how the text book has it, so neh (because then it really depends on where the text book is from; e.g., I know the French defines the square root of x as x approaches 0 as 0; I also know that some schools define 1 / 0 as ∞ to help students with solving limits -- but this is more of a shorthand for convenience than -- and I digress). It really is getting (and probably has to be) philosophical, which is great fun.

Moving on, I'll first try to describe one consensus (?) of the smoke-filled room. But only because I hope that it will help in moving the later discussions.

Infinity is a concept. So trying to multiply 0 by infinity is a lot like making this multiplication literally with a programming language (of course, a lot like, but not exactly like). You could usually safely declare 0 as int, but what would you declare infinity as? The very nature of its inability to be accounted for prevents it from being declared as int.

Going a bit deeper. Why do we have trouble accounting for infinity? What sort of infinity is it? This makes a difference. One tangible example would be the calculation of the probability of near-Earth object hitting Earth causing the end of the world as we know it. We know that it will be a number of extreme smallness (i.e., the probability of one near-Earth object hitting Earth) multiplied by a number of extreme largeness (i.e., the number of near-Earth objects for years to come). And the result? -- we really do not know, until we know it for sure. Another tangible example would be the measurement of something extremely light with an electronic scale. Each of the extremely light item would read as having the mass of 0 on the scale. And there really isn't any way to know if it really has 0 mass or it is just very light. But as we duplicate this extremely light item, which is infinitely small from the electronic scale's point of view, i.e., it is unable to comprehend or process how small it is, to the infinite amount of items, then there might be a comprehensible result -- or not -- we do not know, until we know it.

I think this vaguely insinuates why in calculus the answer would be indeterminable, as it really depends on how this infinity (and why it is infinite) is approached.