Question

which operation will not change the value of any nonzero number ?​

Answer 1

Multiplying by one will not change the value of any non - zero number.

You can multiply any number (including zero) by one without changing it. Similarly, you can raise any number (once again, including zero) to power one and it won't change.

Answer 2

Answer:

Multiplying by one will not change the value of any non - zero number. You can multiply any number (including zero) by one without changing it.

Step-by-step explanation:

Multiplication by 2 will change the value of any non-zero number. In fact multiplication by any (real/complex/anything appropriate) value will change the value of any nonzero number with the exception of 1. This is because we can prove (with the right set of axioms/assumptions) that 1 is the identity under multiplication (again it would probably best to specify a field here), that this identity element is unique, and zero is the only entity that can be multiplied a non-identity element and return itself.

Let us look at the special cases of multiplying by 2 and 0:

Given an input of x, the first multiplication returns 2x, and this is only equal to x iff it is a solution to 2x=x. However, this has only 1 solution (provable using the Fundamental Theorem of Algebra) and it is zero, which we aren’t concerned with. Hence, for all other values we get a different value then what we started with. Multiplying by zero is even easier, as we always get zero as our return (zero product property as you might find in some introductory algebra texts) and our input is guaranteed to be nonzero by assumption. It is not difficult to show that this holds for an arbitrary non-one real, and it immediately follows to show that this gives us a set of countably infinite number of operations that do what is being asked. From here, it actually not difficult to show that the number of such operations is certainly uncountable using other common types of operators, but I will leave that to you.