Why is $1 =1 $? why is it so why cant be $1 =$ something else?

Question

It may sound stupid but why is $1=1$ or $n=n$ if thats the case does $1/0 = 1/0.$

Answer 1

Because we define equals as an equivalence relation which needs to fulfill three properties

1. reflexive meaning $x\sim x$
2. symmetric $x\sim y \implies y\sim x$
3. transitive $x\sim y $ and $y\sim z$ imply $x\sim z$ where $\sim$ means it fulfills the relation.

Answer 2

One of the defining properties for equality is that for any $x$ we have $x=x$. Therefore $1=1$ and $n=n$ by definition.

Now we would also have $1/0=1/0$ if $1/0$ existed. However, there's no such number as $1/0$, and for something that does not exist, obviously no equality can be defined.

Note that for equality we additionally demand that there's no other $y$ besides $x$ so that $y=x$. However this is in some sort a "soft" demand because we can always enforce it from an equivalence relation by simply considering equivalence classes. Also, to make that demand, you'll have to have an independent notion of what it means for two things to be "the same".

One example of the "soft" sameness is fractions: Formally, we define fractions as pairs of numbers with certain calculation rules (and a special notation of the pair to signify that we mean the fraction, i.e. those special rules for calculation). Now $\frac12$ and $\frac24$ are clearly different pairs, yet we consider them the same number. Technically, this means that our fractions are actually the equivalence classes of all $(a,b)$ under the equivalence relation $(a,b)\equiv(c,d)\Leftrightarrow ad=bc$.

Answer 3

Sticking to the title of your post, I think it is okay for you to define $1$ to be something else. But in that case, you have to deal with consequences of your definition.

Answer 4

I think a non math answer would be that if $2 = 1$, then what is $1 + 1$? It isn't $2$ anymore (or is it? ), but we have $4 = 2$.

We are going to arrive at some inconsistencies.

EDIT: Actually to be more precise, we have to redefine '$+$' don't we? Should we go with your definition.

Answer 5

Take the number 0 then for axioms we have for all n exist -n then we know n-n=0 then n=n

Answer 6

The symbol $1$ is reserved for neutral element of multiplication. The answer to the topic comes from uniqueness of multiplicative neutral element. Assume that $1$ and $n$ are neutral elements of multiplication. Then \begin{eqnarray} n = 1\cdot n = n\cdot 1 = 1 \ . \end{eqnarray} The first equation comes from the properties of $1$ and the third from the properties of $n$. That's why $1$ cannot be anything else.

The questions "Why is $1=1$?" and "Why it is so?" and "Why can't be $1=$ something else?" are separate and the first holds trivially. The second question refers to the first. Hence it is also trivial. The third question was answered above.

Also $n=n$ holds trivially. The equation $1/0=1/0$ is not defined according to standard definitions.