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Discoveries About Complex Conjugates |
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Students as
Discoverers!
Offer students the
opportunity to make some mathematical discoveries.
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The Scenario |
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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.
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