1. Of one root of an equation, a conjugate is another number that is a root of the same equation. Thus, if x ² + 2x – 3 = 0, the numbers 1 and –3 are conjugates. If one root of an equation is a complex number of the form a + bi, then it is a fundamental theorem of algebra that it has a complex conjugate of the form a – bi, also a root of the equation. Complex binomials are those such that (a + b) and (a – b) which differ only by one sign. Another conjugate of (a + b), though not of (a – b), is (–a + b).

 

2. Conjugate angles add up to 360°.

 

3. Conjugate lines of a conic section have the property that each contains the pole point of the other, while conjugate points of a conic have the property that each lies on the polar line of the other.

 

In general, conjugate indicates that there is a symmetrical relationship between two objects A and B; in other words, there is an operation that will turn A into B and B into A.