Problem 9 - Complex numbers from A to...Z

Question

The pair \(\left(z_{1}, z_{2}\right)\) of nonzero complex numbers has the following property: there is a real number \(a \in[-2,2]\) such that \(z_{1}^{2}-a z_{1} z_{2}+z_{2}^{2}=0 .\) Prove that all pairs \(\left(z_{1}^{n}, z_{2}^{n}\right), n=2,3, \ldots\), have the same property.

Step-by-Step Solution

Answer

Question: Prove that if a pair of nonzero complex numbers (z_1, z_2) has the property that there exists a real number a in [-2, 2] such that z_1^2 - az_1z_2 + z_2^2 = 0, then all pairs (z_1^n, z_2^n), where n = 2, 3, ... also have the same property. Solution: We showed that for any given n = 2, 3, ..., the generalized equation (z_1^n)^2 - a'(z_1^n)(z_2^n) + (z_2^n)^2 = 0 holds for a' = a, where a is a real number in the interval [-2, 2]. This proves that the pairs (z_1^n, z_2^n) also have the same property as the original pair (z_1, z_2).

Step 1: Write down the given equation

Our starting point is the equation \(z_{1}^{2}-a z_{1} z_{2}+z_{2}^{2}=0.\)

Step 2: Generalize the equation for powers of \(z_1\) and \(z_2\)

To prove that \((z_1^n, z_2^n)\) have the same property, let's generalize the equation by replacing \(z_1\) with \(z_1^n\) and \(z_2\) with \(z_2^n\). Our goal is to show that this new equation also holds for some \(a' \in [-2, 2]\). The generalized equation is: \((z_1^n)^2 - a'(z_1^n)(z_2^n) + (z_2^n)^2 = 0\)

Step 3: Manipulate the generalized equation

We can rewrite the equation from Step 2 as follows, factoring out \(z_1^{2n}\) and \(z_2^{2n}\): \(z_1^{2n} - a'z_1^nz_2^n + z_2^{2n} = 0\)

Step 4: Observe the connection with the original equation

Notice that the equation from Step 3 can be rewritten in terms of the original equation as: \((z_1^n)^2 - a'(z_1^n)(z_2^n) + (z_2^n)^2 = (z_1^{2n} - a'(z_1^n)(z_2^n) + z_2^{2n}) = (z_1^{2})^n - a'(z_1z_2)^n + (z_2^{2})^n\) Now, let's consider the given equation: \(z_1^2 - az_1z_2 + z_2^2 = 0\) Raising both sides of this equation to the power of \(n\), we get: \((z_1^2 - az_1z_2 + z_2^2)^n = 0^n\) \((z_1^2)^n - (az_1z_2)^n + (z_2^2)^n = 0\)

Step 5: Conclude the proof

Comparing the result from Step 4 with the equation obtained in Step 5, we see that the expressions are identical for \(a' = a\). As \(a\) is a real number in the interval \([-2, 2]\), this implies that \(a' \in [-2, 2]\). Thus, we've shown that the pairs \((z_1^n, z_2^n)\) where \(n = 2, 3, \ldots\) also have the same property, as desired.

Key Concepts

Complex Number Equations

Complex number equations are the foundation of working with complex numbers in mathematics. A complex number is often written in the form \( a + bi \) where \( a \) is the real part and \( b \) is the imaginary part, and \( i \) represents the square root of -1. When dealing with complex number equations, we apply algebraic rules similar to those for real numbers, but with special attention paid to the property of \( i \), which is \( i^2 = -1 \).

When solving complex equations, it is crucial to separate the equation into its real and imaginary components. This method allows for the solution of not only linear equations but also more complex polynomials and systems of equations. It is important to familiarize oneself with operations like addition, subtraction, multiplication, and division of complex numbers to navigate these equations effectively.

For example, in the exercise provided, we are tasked with proving a specific property of complex numbers raised to integer powers. The ability to manipulate and simplify complex equations is on full display as the unknown, \( a \), is a real number that governs the relationship between \( z_1 \) and \( z_2 \) when squared and multiplied together.

Real Numbers in Complex Equations

Incorporating real numbers into complex equations often involves using a real constant to relate two complex numbers, as seen in the exercise's given equation \( z_{1}^{2}-a z_{1} z_{2}+z_{2}^{2}=0 \), where \( a \) is a real number. This is an integral aspect of working with complex numbers because it links the behavior of complex numbers to our more intuitive understanding of real numbers.

Real numbers when squared, summed, or multiplied always yield real results. This is useful when seeking to prove properties of complex numbers because it allows us to use real coefficients to express constraints and relationships between them. For example, \( a \) in the equation acts as a scalar that adjusts the interaction between \( z_1 \) and \( z_2 \) to fit the required property. It's a bridge across which we can translate properties of real numbers into the realm of complex numbers and use the stability of real numbers to anchor the often less intuitive complex number operations.

Powers of Complex Numbers

Understanding the powers of complex numbers can seem daunting at first, but it's an essential skill in advanced mathematics. The powers of complex numbers follow the same exponentiation rules as real numbers but with additional twists due to the nature of the imaginary unit \( i \).

Powers of \( i \) cycle through a repeated pattern: \( i^1 = i, i^2 = -1, i^3 = -i, \text{and} i^4 = 1 \) and then repeat. For higher powers of complex numbers, like \( z^n \) where \( z \) is complex, one typically uses the polar form or Euler's formula to simplify the computation. This is particularly true when \( n \) is a large integer.

In the context of the given exercise, we raise both the individual terms and the entire equation to the power of \( n \), which allows us to use the properties of exponents and demonstrate the desired property. This manipulation not only highlights the cyclic nature of complex power but also emphasizes the importance of keeping the structure intact when raising complex numbers to higher powers to preserve their relationships.