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

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SUMMARY

The discussion confirms that for any complex number z, both z + z¯ and z × z¯ are real numbers. The solution provided demonstrates that if z is expressed as a + bi, then z + z¯ simplifies to 2a, which is a real number. Additionally, the product z × z¯ simplifies to a² + b², also a real number. Therefore, the assertions made in the discussion are mathematically valid and accurate.

PREREQUISITES
  • Understanding of complex numbers and their representation (a + bi)
  • Knowledge of complex conjugates (z¯)
  • Familiarity with basic algebraic operations (addition and multiplication)
  • Concept of real numbers in the context of complex numbers
NEXT STEPS
  • Study the properties of complex conjugates in more depth
  • Explore the geometric interpretation of complex numbers on the complex plane
  • Learn about the polar form of complex numbers and its applications
  • Investigate the implications of complex numbers in advanced mathematics, such as in calculus or signal processing
USEFUL FOR

Mathematicians, students studying complex analysis, educators teaching algebra, and anyone interested in the properties of complex numbers.

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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