Is it correct that $\frac{1}{0}=\frac{1}{-0}$ and if it is, why is $\frac{1}{0} \neq 0$?

Question

This is a genuine question, I am not trying to convince anyone. But I'm sure hundreds of people already considered this, so if you can point out where I'm wrong, it would be much appreciated.

If we forget about limits and infinitesimals, then $0 = a - a$, where $a \in R$

Then $0=-0$

$\frac{1}{0}=\frac{1}{-0}$

$\frac{1}{0}=-\frac{1}{0}$

$\frac{1}{0}+\frac{1}{0}=0$

$2 \frac{1}{0}=0$

$\frac{1}{0}=0$

It also means that $\frac{0}{0}=0$

Then the usual reasoning that $\frac{0}{0}$ can't be determined because for example $2*0=0$ and also $3*0=0$ doesn't hold. We can divide both sides by $0$, but we will get only the trivial $0=0$.

$\frac{1}{0}=0$ may seem counterintuitive, but since the limits for 1/x approach $\infty$ from one side and $-\infty$ from other side, it makes sense that the value for the exact zero will be exactly in the middle.

I can offer one counterpoint for my reasoning. The 'proof' I provided is incorrect if and only if 1/0 does not belong in the set of real numbers. Then the addition of this value may not be commutative. But I'm not sure that there is any proof that 1/0 is not in R.

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Edit:

Thank you all for your answers, and I apologize for trying to argue with some of you in the comments. I am not trying to start a discussion. I consider my question answered now, in fact, I have already answered it myself in the last paragraph.

Answer 1

What division really means is the inverse of multiplication. So 1/0 means the number you multiply by zero to get one. However, it is a property that x*0 = 0 for all x. This is the reason we say 1/0 is undefined.

This is all a consequence of the interplay of operations (addition and multiplication) on the space of real numbers. Have you studied rings?

Answer 2

Both expressions

$$\frac 10 \quad\text{ and }\quad \frac 1{-0}$$

do not make sense with the usual rules of the real numbers.

(If they did, then $0$ would have a multiplicative inverse. Namely, there would exist a real number $x$ such that $0\cdot x = 1$. But it is straight forward to show there is no such number $x$, since $0\cdot a = 0$ for all real numbers $a$.)

As such we cannot perform algebraic operations with those expressions and reach valid conclusions.

To put it more bluntly: we cannot start with a nonsense statement

$$\text{"Meaningless expression"} = \text{"Meaningless expression"}, $$

which is not even a false statement, and logically conclude a true statement.

Answer 3

The whole reason behind 1/0 not being defined is BECAUSE of this problem. 1/0 cannot be calculated, but we can do our best. We can say the LIMIT of 1/x as x approaches 0.

However, try graphing 1/0. You'll notice that at x=0, it goes both up AND down. This is called a vertical asymptote.

This is because the limit as x approachs 0 from the left and the limit as x approaches 0 from the right are not equal. As you'll learn in calculus, this means that the limit cannot exist.

As a result, we can't assign a value to 1/0. Not only does it have no logical meaning, if we try to assign it logical meaning, it still doesn't exist.

Answer 4

In first-order logic, where all function symbols denote total functions, if you want to include division as function symbol in the language of fields, then it is, in fact, very convenient to adopt the axiom that $1/0 = 0$. Results such as the decidability of the theory of real closed fields fail to hold if you include division in the language unless you adopt some convention and $1/0 = 0$ works well.