Why is $ (\frac{1}{0} )$ not defined?

Question

Why is $ (\frac{a}{0} ), a \in R $ not defined?

if :$a=0$ what?

Answer 1

First you need to know one thing:

What does "$\frac{a}{m}$" mean?

The answer is: "$a/m$ is the one and only unique solution to the equation $mr = a$." ($r$ is the unknown that I have considered).

Given that, let's consider the case under consideration:

For $m=0$, the equation clearly states that it has no solutions: for something to be equal to $\frac{a}{0}$, with $a\neq 0$, we would need $0\cdot r = a$. But $0\cdot r = 0$ for any $r$, so there are no solutions to the equation and since there are no unique solutions to the equation, there is no such thing as $\frac{a}{0}$. So $\frac{a}{0}$ does not represent any mathematical number.

Rigorously speaking, we simply do not define "division by $0$" for real analysis.

Answer 2

By $R$, I assume that you mean the real numbers $\mathbb R$.

Let's think about what division means.

We know that $14 \div 7 = 2$ because you add $7$ together twice to get $14$, i.e. $7 \times 2 = 14$.

We know that $100 \div 5 = 20$ because you add $5$ together twenty times to get $100$, i.e. $20 \times 5 = 100$

What about $1 \div 0$? How many times do you add zero together to get one? Solve $0 \times x = 1$.

You can't do it. It doesn't make sense.

Answer 3

Another interesting variation is the following: In commutative algebra, if you are allowed fractions (via the construction known as localisation), then you are always allowed to expand a fraction by any number that is a valid denominator. Within that framework, if you allow $0$ as a valid denominator, you automatically say that any fraction may be expanded by $0$. If we take an arbitrary fraction $\frac ab$, and use that, we get $$ \frac{a}{b} = \frac{0\cdot a}{0\cdot b} = \frac{0}{0} $$ which means that every single number is equal to $\frac00$. That's kind of boring, don't you think?