another example is through the usage of complex numbers:
A=(a+j), A* = (a-j)
A.A*=(a^2 - aj + aj- j^2)
= a^2-(-1^(1/2))^2
=a^2 - (-1)
= a^2 + 1 which is always larger than 0
Complex Numbers
The word conjugate is one of those words in mathematics which has many different meanings. There is a standard algebraic/number theoretic definition. Let be an extension field over a smaller field . Given two algebraic elements we say are conjugates iff they has the same minimal polynomial. So and ( ) are conjugates because they have the same minimal polynomials.
Though that definition above is applied to fields we can extend it a little and apply it to (which is not a field) as well. Given two algebraic number we say and are conjugates if they have the same minimal polynomial over . In your case and have as their minimal polynomial.