MHB Is z + z¯ and z × z¯ Real for Any Complex Number z?

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For any complex number z, both z + z¯ and z × z¯ are confirmed to be real. The expression z + z¯ simplifies to 2a, where a is the real part of z. The product z × z¯ results in a^2 + b^2, which is also a real number since both a and b are real numbers. The solution provided is accurate and demonstrates that these operations yield real results for any complex number. Thus, the claim is validated.
Yordana
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I apologize in advance for my English.
I want to know if my solution is correct. :)

To verify that for every complex number z, the numbers z + z¯ and z × z¯ are real.

My solution:
z = a + bi
z¯ = a - bi
z + z¯ = a + bi + a - bi = 2a ∈ R
z × z¯ = (a + bi) × (a - bi) = a^2 + b^2 ∈ R
 
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Yep. All correct.
 

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