Thread: Why can't you physically divide by zero?

Some say 1/0 is ∞. Others say you can't divide by zero in the first place.

Outside of abstract algebra, let's look at a physical analogy of elementary arithmetic.

You have 12 cookies and you must evenly distribute them to n number of people.

If there are 3 people, 12/3=4, each person will get 4 cookies.

If there's only one person, 12/1=12, that person will get 12 cookies.

Wait, what if there are zero people? :O

12/0=∞? So the empty air gets an infinite number of cookies?! (what a lucky non-existent entity)

I think I have a possible solution. What are your thoughts?

well you've clearly demonstrated that it makes no sense to partition any quantity into zero parts, hence why others (not the people who say it is ) simply say that 1/0 does not have a defined value. On the other hand we can clearly see that some functions grow without bound when approaching a particular limit, for example or . At x = 0 however this function, has no defined value, mostly because you can't divide by zero.

First I don't believe one can divide by zero in most areas of math. Considering we can't put its value anywhere on the real number line. But remember we have more abstract values like the imaginary unit. We did know we couldn't use it in the RNS, but by assuming the existence of -1's square root, we made i. And it has been avidly applied to mathematics, making it one of the famous 5 constants.

BTW, the reason I said I don't believe 1/0 can be defined was because it's true ... and to imply that the following argument is not supporting “1/0 = infinity”, but to critically analyze this “cookies” analogy.

Okay, the argument from physical partitions of division, with cookies

12/a = b

We have the twelve cookies, “ a ” amount of persons to distribute to, and “ b ” amount of cookies per each person of “a”

This argument holds together logically for the most part, except when “a” < 1.

If there is half a person, for example a = 0.5, then that half of a person will receive 24 cookies, because 12/0.5 = 24. Already there the argument loses hold.

It's not logical that you had 12 cookies to begin with and you ended up distributing 24 cookies to half of a human body.

The cookies argument, it seems to me, is an argument using reductio ad absurdum, by making division by zero appear to be a physical impossibility or absurdity. But the same argument can be used against division of any number positive real number less than one. -

So my argument against the analogy is that it makes x/0 an absurdity while also making division by halves thirds (etc.) also an absurdity, even though they're accepted as possible to every mathematician. So since you obviously can divide by halves and thirds you should also be able to divide by “infinite-ths” aka zero.

I think 1/0 is undefined, not because of this argument (which I just argued against) but for other reasons. I'm trying to point out that this is not a valid argument.

Going back on division by zero, if one were to divide by anything less than one, it would be the same as multiplying by its reciprocal. So physically, if you were multiplying a certain number of apples by “ n ”, you could also say your dividing that by 1/n. So if x/0 is infinity, then 12/0 cookies is the same thing as multiplying those 12 cookies by infinity, yielding infinity ( ignoring the idea of giving pieces of a person some sweet treats )

This is considering ∞ is a numerical value. I guess it's only defined in the extended real number system, probably not even. Though there are situations in which it is accepted as a number, such as projective geometry, and with the Riemann Sphere.

You cannot divide by zero because there is no meaningful result to that operation. No element of any given set of numbers - real, imaginary or otherwise - would fulfil this equation. Infinity is not a defined number, it is merely a symbol meaning "exceeding all values". You cannot use it to perform calculations like these.

Originally Posted by Markus Hanke

You cannot divide by zero because there is no meaningful result to that operation. No element of any given set of numbers - real, imaginary or otherwise - would fulfil this equation. Infinity is not a defined number, it is merely a symbol meaning "exceeding all values". You cannot use it to perform calculations like these.

Let's say:


Therefore, by using the multiplicative property in inverse operations (not reduction/simplification, but the broader method) we get:



So that means x multiplied by 0 (and inversely) equates to 1. There is no real number that can equal x.

Now, we can again draw another statement:



Again, no defined value can equal x. Looking at this graph of the rational function: f(x)=1/x

We see how the function approaches both + and - infinity as x approaches 0. So if then ?

That can't be true, because logarithmically, in 10^n, if n=+∞, then it would equate +∞. If n=-∞, then it would equal 0.

Since obviously

We can also draw other statements from this, but this is enough.

So this is my solution: Without changing how the RNS works, I will define new values in my own system:

Infinitely small quantities are represented in numbers such as . I would call these numbers the equivalent of our infinitesimals, which seem to me scarcely understood or applied.

There are the finite numbers, like our usual 1, -1, pi, e, etc.

Then there are the infinites, in notation like or But there is no such symbol ∞.

So again, there is no " 0 " zero or " ∞ " infinity, only values numerically written out. (the infinitesimals are the equivalent of "0" and the infinites "∞")

So here, it would seem possible to divide by "0", which would be

Here, it's easier to imagine how because in this system it's . Appearing as though an infinitely large amount of an infinitely small amount yields a finite amount, so it's reasonable in that sense.

... instead of an infinite amount of nothing, which should yield nothing still.

Now there's one very important aspect to remember:numbers in this system are relative. So being finite, infinitesimal, or infinite, is just a comparative quality of numbers (just like positivity and negativity).

In our real number system, we agree that . We say they're equal, for one reason, because there's really nothing in between 1 and 0.(9). In this system, the difference is , but because of number relativity, it's as though there's nothing between the two values making them seem to be the same number.

Look at and . These two numbers seem to both be infinite, but the difference between them is 1. One is a finite number, not "nothing", so they're not equal? Just because there's a finite difference does not change their infinite quality.

That's why They're all representations of the same relative quantity, seen as a single point on the number line.

What about and ? They're equal. But a positive number can never ever equal its opposite negative, right? With relativity it can, in the case of infinitesimals. Infinitesimals are all infinitely small, relative to the finite numbers, so they behave like the same thing in operations.

The idea is that on a number line, where we see our usual integers, the infinitesimals would all be located on the same point, "0". That's relativity.

Do you see how this relates to indeterminate forms? which equates to That's why we would see 1 = 2 in some situations involving division by zero.