Discoveries About Complex Conjugates
 Topic Index | Algebra2/Trig Index | Regents Exam Prep Center


Students as Discoverers!
Offer students the opportunity to make some mathematical discoveries.


The Scenario
Ask your students to prove (or disprove) the following scenario:

The square of any complex number and the square of its conjugate are also conjugates.

The scenario is true.

Possible Proof: 
Let a + bi be the complex number.
Let a - bi be the conjugate of the complex number.

(a + bi)2 = (a + bi)(a + bi) = a2 +abi +abi + b2 i2 = a2 +2abi - b2 = (a2 - b2) + 2abi

(a - bi)2 = (a - bi)(a - bi) = a2 -abi -abi + b2 i2 = a2 -2abi - b2 = (a2 - b2) - 2abi

(a2 - b2) + 2abi  and  (a2 - b2) - 2abi  are conjugates.