0/0

$0⁄0$ is a indeterminate form commonly left undefined. However, this proof will show that $0⁄0$ can equal any real number.

Proof
Let's say that $0⁄0$ = x. Multiplying $0⁄0$ by 0 to isolate the 0 in the numerator gives 0($0⁄0$) = 0(x). This simplifies to 0 = 0. Therefore, $0⁄0$ can equal any real number.

Examples
Let's say that $0⁄0$ = 2.

0($0⁄0$) = 0(2).

0 = 0.

Now let's say that $0⁄0$ = 18.

0($0⁄0$) = 0(18).

0 = 0.

Now let's say that $0⁄0$ = 42.

0($0⁄0$) = 0(42).

0 = 0.