How to prove $\frac 10 \notin \mathbb R $

Question

I'm trying to prove $\frac 10 \notin \Bbb R $ by contradiction. That means by supposing $\frac 10 \in \mathbb R.$

Is there a way to do this?

Thanks.

Answer 1

To prove that $\frac{1}{0}$ does not exist, it is perhaps best to start by reviewing the definition of division.

By definition, $\frac{a}{b}$ is the unique solution to the equation $a=bx$. So $\frac{1}{0}$, if it exists, is equal to the unique solution to the equation $1=0x$.

However, that equation has no solution: using the field axioms, which the real numbers satisfy, it is easy to show that $0x=0$ for any real number $x$, and $0 \ne 1$.

Therefore, $\frac{1}{0}$ does not exist.

Answer 2

If by $\Bbb R$ you mean the usual real line then the division by zero is not defined because it is not defined either for the field of rational numbers $\Bbb Q$, what is contained in $\Bbb R$.

The rational numbers are generally constructed formally through two operations defined over the set $\Bbb Z\times\Bbb Z\setminus\{0\}$:

$$\text{Addition}\quad(a,b)+(c,d)=(a+c,b+d)$$

$$\text{Multiplication}\quad(a,b)\cdot(c,d)=(ad+bc,bd)$$

for any tuples $(a,b),(c,d)\in\Bbb Z\times\Bbb Z\setminus\{0\}$, where we generally write $(a,b)\equiv\frac{a}b$, and the operations $a+b$ and $ab$ are defined on the integers. Then observe that there is no rational number of the form $(a,0)\equiv\frac{a}0$, so it doesn't have a meaning associated to it.

However, in other constructions as the one-point compactification of the real line, namely $\widehat{\Bbb R}$, we can define $\frac{a}0:=\infty$ for any $a\in\widehat{\Bbb R}\setminus\{0\}$. But in the standard real line, as I had shown above, the symbols $\frac{1}0$ doesn't have a meaning attacked to it.

Thus this is not a question that can be "proved", it is just that it is not defined at all.