Problem 16 - Complex numbers from A to...Z

Question

Let \(z_{1}, z_{2}, z_{3}\) be complex numbers such that $$ z_{1}+z_{2}+z_{3}=z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}=0 $$ $$ \text { Prove that }\left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right| $$

Step-by-Step Solution

Answer

Answer: Three complex numbers z1, z2, and z3 will have equal magnitudes if the following conditions hold: 1. z1 + z2 + z3 = 0 2. z1 z2 + z2 z3 + z3 z1 = 0.

Step 1: Write down the given equations

We are given the following two equations: 1. \(z_1 + z_2 + z_3 = 0\) 2. \(z_1 z_2 + z_2 z_3 + z_3 z_1 = 0\)

Step 2: Use equation (1) to rewrite equation (2)

From the first equation, we have \(z_3 = -(z_1 + z_2)\). We will substitute this into the second equation to eliminate \(z_3\): $$z_1 z_2 - z_2(z_1 + z_2) - z_3(z_1 + z_3) = 0$$

Step 3: Simplify the equation and rearrange the terms

Now, we can distribute the terms and simplify: \begin{align*} z_1 z_2 - z_1^2 z_2 - z_2^2 z_1 - z_1 z_3^2 - z_3^3 &= 0 \\ z_1 z_2 - z_1^2 z_2 - z_2^2 z_1 - z_1(-z_1 - z_2)^2 -(-z_1 - z_2)^3 &= 0 \end{align*}

Step 4: Expand the equation and collect like terms

We will now expand the equation and combine like terms: \begin{align*} z_1 z_2 - z_1^2 z_2 - z_2^2 z_1 + z_1(z_1^2 + 2z_1 z_2 + z_2^2) - z_1^3 - 3z_1^2 z_2 - 3z_1 z_2^2 - z_2^3 &= 0 \\ z_1^3 + z_1^2 z_2 + z_1 z_2^2 - z_2^3 &= 0 \end{align*}

Step 5: Factor the equation

We will now factor the equation, we can notice that this equation can be factored: \begin{align*} (z_1^3 - z_2^3) + (z_1^2 z_2 + z_1 z_2^2) &= 0 \\ (z_1 - z_2)(z_1^2 + z_1 z_2 + z_2^2) + z_1 z_2(z_1 + z_2) &= 0 \end{align*} We also know that \(z_1 + z_2 + z_3= 0\) or \(z_1 + z_2 = -z_3\), we can substitute this into the equation to simplify: $$ (z_1 - z_2)(z_1^2 + z_1 z_2 + z_2^2) - z_1 z_2 z_3 = 0$$

Step 6: Analyze equality conditions

The above equation is true if one of the following conditions is met: 1. \(z_1 = z_2\) 2. \((z_1^2 + z_1 z_2 + z_2^2) - z_1 z_2 z_3 = 0\) Suppose \(z_1 = z_2\). Then, from equation (1), \(z_3 = -2z_1 = -2z_2\). Consequently, \(\left|z_1\right| = \left|z_2\right| = \left|z_3\right|\). On the other hand, if condition 2 holds, then we can write the equation in terms of magnitudes: $$\left|z_1\right|^2 + \left|z_1\right| \left|z_2\right| e^{i\theta} + \left|z_2\right|^2 = \left|z_1\right| \left|z_2\right| \left|z_3\right| $$ Where \(\theta\) is the argument of \(z_1z_2\) (we can do this because a complex number and its magnitude differ only by an angle). Here, we have assumed that \(z_1\) and \(z_2\) are non-zero, otherwise \(\left|z_1\right| = \left|z_2\right| = \left|z_3\right|\) trivially. By symmetry, we get two more equations: $$\left|z_2\right|^2 + \left|z_2\right| \left|z_3\right| e^{i\phi} + \left|z_3\right|^2 = \left|z_1\right| \left|z_2\right| \left|z_3\right| $$ $$\left|z_3\right|^2 + \left|z_3\right| \left|z_1\right| e^{i\gamma} + \left|z_1\right|^2 = \left|z_1\right| \left|z_2\right| \left|z_3\right| $$ Where \(\phi\) is the argument of \(z_2z_3\) and \(\gamma\) is the argument of \(z_1z_3\). Now, adding these three equations and utilizing the identity \(e^{i\theta} + e^{i\phi} + e^{i\gamma} = 0\), we get: $$\left|z_1\right|^2 + \left|z_2\right|^2 + \left|z_3\right|^2 = \left|z_1\right| \left|z_2\right| \left|z_3\right| \\ \left(\left|z_1\right| + \left|z_2\right| + \left|z_3\right|\right)^2 = 3\left(\left|z_1\right|^2 + \left|z_2\right|^2 + \left|z_3\right|^2\right)$$ Since \(\left|z_1\right| + \left|z_2\right| + \left|z_3\right| > 0 \), it must be true that \(\left|z_1\right| = \left|z_2\right| = \left|z_3\right|\). Thus, in any case, we can conclude that \(\left|z_1\right| = \left|z_2\right| = \left|z_3\right|\).

Key Concepts

Triangular Inequality

The triangular inequality is a fundamental concept in complex numbers and inequalities. It states that for any complex numbers \(a\) and \(b\), the magnitude of their sum is less than or equal to the sum of their magnitudes. Symbolically, it is expressed as:

• \(|a + b| \leq |a| + |b|\)

This inequality reflects the idea that the direct path between two points is the shortest, so the total magnitude of a detour (sum of magnitudes) is always greater or equal. This principle can be seen as analogous to certain properties in vector spaces.

### Application in Our ProblemIn the given complex number problem, the triangular inequality can guide judgments about magnitudes among \(z_1, z_2,\) and \(z_3\). Since magnetudes (absolute values) align with definitions from the inequality, this property hints toward equality when exploring setups where the sum and product of these numbers equal zero. This suggests symmetry in their magnitude values, helping us show \(|z_1| = |z_2| = |z_3|\). It implies that in certain balanced conditions described by the exercise's form, numbers can't stretch further than their combined maximum length, reinforcing equal magnitude conclusions.

Magnitude Equality

Magnitude equality is another key point concerning complex numbers. In mathematical terms, when two complex numbers have the same magnitude, it means:

• \(|a| = |b|\)

This relationship conveys that both numbers exist at the same 'distance' from the origin on the complex plane but might not be identical (possibly rotated).
### Tying Magnitude Equality to Problem ConstraintsIn the exercise, proving magnitude equality among \(z_1\), \(z_2\), and \(z_3\) is central. Given constraints allow transformation and factoring leading to potential proofs through conditions like \(z_1 = z_2\) or factoring patterns reveal equality. The nature and solution behavior showcase a potential inherent symmetry among our variables leading ultimately to the conclusion that:

• \(|z_1| = |z_2| = |z_3|\)

Demonstrating that these magnitudes must maintain a balance, dictated by the predefined structural relationships in the exercise's equations.

Complex Conjugates

Complex conjugates play a critical role in understanding and manipulating expressions involving complex numbers. For a complex number \(z=a+bi\), its complex conjugate is \(\bar{z} = a - bi\). This concept offers several benefits:

• When a number is multiplied by its conjugate, it yields a non-negative real number: \(|z|^2 = z\bar{z}\).
• It allows for easier division of complex numbers by eliminating imaginary parts in the denominator.

### Influence in Solution ApproachesUsing complex conjugates can sometimes help in simplifying expressions and equations, clarifying magnitude-related properties among complex variables.In the solution to our problem, while the explicit use of conjugates isn't directly stated, recognizing their properties aids in manipulating terms effectively during proof steps and could back certain simplifications or findings if assessing magnitudes directly from other identity pathways or symmetry considerations. Comprehending conjugates expands understanding of magnitude negotiations within complex number interactions.