Just as with pointer arithmetic (because that's what this actually is) you don't multiply by the size of its elements. This:

  array[2]

is the same as this:

  *(array + 2)

The compiler knows what the type of data the pointer refers to and can produce the byte offset itself. Also:

  "...it's always pointing at the first element of the the array"

Eh... an array can degrade into a pointer when needed, but what does the following produce?

  char arr[10];
  ??? x = &arr;

Is "x" a pointer to pointer to char? From your assessment it would seem so, but in reality the type of "x" is

  char (*)[10]

i.e., pointer to array of char 10. An array is an array, and arrays can degrade into pointer types.