The complex conjugate of a complex number *a + bi* is *a - bi*, where *i* is the square root of negative one. As we learned back in first year algebra, multiplying (*x *+* y*) * (*x - y*) yields *x*^{2} - y^{2}, and since *bi* squared is *-b*^{2}, (*a + bi*) * (*a - bi*) = *a*^{2} + b^{2}, a real number. This makes the complex conjugate a useful number.