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Complex Conjugates/Proof by Induction

gucci

New member
Sep 1, 2013
13
So I am having a bit of trouble with a proof by induction that I need to write. The problem is to prove that the conjugate of the product g1 * ... * gm equals the product of the conjugates of g1 ... gm. This is for g1 ... gm complex numbers.

I have proven this for m = 2, by simple calculation of the conjugates. Now I need to prove that if this is true for m = n then it is true for m = n + 1.

I appreciate any help you can offer, thank you.
 

Chris L T521

Well-known member
Staff member
Jan 26, 2012
995
So I am having a bit of trouble with a proof by induction that I need to write. The problem is to prove that the conjugate of the product g1 * ... * gm equals the product of the conjugates of g1 ... gm. This is for g1 ... gm complex numbers.

I have proven this for m = 2, by simple calculation of the conjugates. Now I need to prove that if this is true for m = n then it is true for m = n + 1.

I appreciate any help you can offer, thank you.
If you showed that $\overline{g_1g_2}=\overline{g_1}\cdot \overline{g_2}$, and then assume that $\overline{g_1g_2\cdots g_n} = \overline{g_1}\cdot\overline{g_2}\cdots \overline{g_n}$, then it follows that
\[\overline{g_1g_2\cdots g_ng_{n+1}} = \overline{(g_1g_2\cdots g_n) g_{n+1}}\]
Let's define $g_1g_2\cdots g_n=z$. Then we're left with $\overline{z g_{n+1}}$, and we know from the $m=2$ case that this is the same as $\overline{z}\cdot\overline{g_{n+1}}$. Now rewrite this as $\overline{g_1g_2\cdots g_n}\cdot \overline{g_{n+1}}$ and apply the inductive hypothesis to finish the problem.

Does this make sense?
 

gucci

New member
Sep 1, 2013
13
Thanks so much man, that really cleared it up for me!