The two numbers that are the same as their reciprocal are 1 and -1.
The reciprocal of a/b is b/a.
The reciprocal of 1 is 1.
1 is nothing but 1/1. And if you flip the numerator and denominator, you get 1/1 which is 1.
The reciprocal of -1 is -1.
-1 is nothing but -1/1. And if you flip the numerator and denominator, you get 1/-1 which is -1.
The only two numbers that have their own reciprocals are 1 and -1.