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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**