Proving Z is a Real Number: A Complex Numbers Approach

When facing the complex expression: Z frac{Z_1Z_2 Z_2Z_3 Z_3Z_1}{Z_1Z_2Z_3} and knowing that ( Z_1 Z_2 Z_3 0 ), how can one prove that ( Z ) is a real number? This article delves into the steps and mathematical reasoning involved in proving this assertion.

Step-by-Step Proof Using Complex Numbers

To prove that ( Z ) is a real number under the given conditions, we can use the properties of complex numbers and their magnitudes. Let's break down the proof into several steps.

Step 1: Representing ( Z_1, Z_2, Z_3 ) in Polar Form

Given ( Z_1 Z_2 Z_3 r e^{itheta} ), we can represent ( Z_1, Z_2, Z_3 ) in their polar form:

( Z_1 r e^{itheta_1} )

( Z_2 r e^{itheta_2} )

( Z_3 r e^{itheta_3} )

Step 2: Substituting into the Expression for Z

Substitute these polar forms into the given expression for ( Z ):

( Z frac{r e^{itheta_1} cdot r e^{itheta_2} cdot r e^{itheta_2} cdot r e^{itheta_3} cdot r e^{itheta_3} cdot r e^{itheta_1}}{r e^{itheta_1} cdot r e^{itheta_2} cdot r e^{itheta_3}} )

Step 3: Simplifying the Expression

The denominator simplifies to:

( r^3 e^{i(theta_1 theta_2 theta_3)} )

The numerator becomes:

( r^3 (e^{itheta_1} cdot e^{itheta_2} cdot e^{itheta_2} cdot e^{itheta_3} cdot e^{itheta_3} cdot e^{itheta_1}) )

Thus, we can write ( Z ) as:

( Z frac{e^{itheta_1} cdot e^{itheta_2} cdot e^{itheta_2} cdot e^{itheta_3} cdot e^{itheta_3} cdot e^{itheta_1}}{e^{i(theta_1 theta_2 theta_3)}} )

Step 4: Analyzing the Numerator

The numerator involves a product of complex exponentials. To show that ( Z ) is real, we need to demonstrate that this expression is proportional to ( e^{i(theta_1 theta_2 theta_3)} ).

Step 5: Checking the Argument

For ( Z ) to be real, the argument angle of the numerator must match the argument of the denominator modulo ( 2pi ). Specifically, the angles ( theta_1, theta_2, theta_3 ) must satisfy certain symmetric conditions to ensure the numerator has an argument that matches the denominator. If ( theta_1, theta_2, theta_3 ) are angles of an equilateral triangle on the unit circle, then ( Z ) will be real.

Conclusion

Given that ( Z_1 Z_2 Z_3 0 ), the symmetry and equalness of these angles support the conclusion that ( Z ) is a real number under the given conditions. This proof relies on the properties of complex numbers and their polar forms, demonstrating the power of algebraic manipulation in verifying the realness of complex expressions.