For any real number taking it is the same as the identity function.  
The dot product of x and y multiplies component-wise x and this of y.  When in polar form it is produced by negating the exponent.  For any real number taking it is the same as the identity function.  The addition of a number to this for that number produces a real number which is equal to double the real component of the previous number.   It is distributive over multiplication and addition and it is represented with a bar above the number, FTP, name this type of number formed by taking the negative of the imaginary part of a complex number.
The dot product of x and y multiplies component-wise x and this of y.  
{Complex Conjugate}