Problem 18 - Complex numbers from A to...Z

Question

Let \(z_{1}, z_{2}, z_{3}, z_{4}\) be distinct complex numbers such that $$ \operatorname{Re} \frac{z_{2}-z_{1}}{z_{4}-z_{1}}=\operatorname{Re} \frac{z_{2}-z_{3}}{z_{4}-z_{3}}=0 $$ a) Find all real numbers \(x\) such that $$ \left|z_{1}-z_{2}\right|^{x}+\left|z_{1}-z_{4}\right|^{x} \leq\left|z_{2}-z_{4}\right|^{x} \leq\left|z_{2}-z_{3}\right|^{x}+\left|z_{4}-z_{3}\right|^{x} . $$ b) Prove that \(\left|z_{3}-z_{1}\right| \leq\left|z_{4}-z_{2}\right|\).

Step-by-Step Solution

Answer

#Answer_part_a# The range of \(x\) that satisfies the given condition is \(x \in [0, 1]\). #Answer_part_b# We have proven that \(|z_3 - z_1| \le |z_4 - z_2|\).

Step 1: Utilize the information given

We are given that \(\operatorname{Re}\frac{z_2-z_1}{z_4-z_1} = \operatorname{Re}\frac{z_2-z_3}{z_4-z_3}=0\). This means that the two fractions have their real parts equal to zero. We can rewrite these equations as: $$ \frac{z_2-z_1}{z_4-z_1} = ia, \quad \text{and} \quad \frac{z_2-z_3}{z_4-z_3} = ib $$ where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.

Step 2: Solve part (a) - Use triangle inequality

To solve the inequalities in part (a), we need to use the Triangle Inequality Theorem which states that the sum of the lengths of any two sides of a triangle is greater than or equal to the length of the third side. $$ |z_1 - z_2| + |z_1 - z_4| \ge |z_2 - z_4| \ge |z_2 - z_3| + |z_4 - z_3| $$ Now, raise all three sides of the inequalities to the power of \(x\), so we get: $$ |z_1 - z_2|^x + |z_1 - z_4|^x \ge |z_2 - z_4|^x \ge |z_2 - z_3|^x + |z_4 - z_3|^x $$ For the given conditions to hold, both inequalities should be true. The first inequality is always true due to the triangle inequality as mentioned above. And as \(x>0\) in our case, so the second inequality holds only if \(x \in [0, 1]\).

Step 3: Solve part (b) - Find relationship between magnitudes

We are required to prove that \(|z_3 - z_1| \le |z_4 - z_2|\). To do this, we will utilize the information from Step 1. Multiplying the two equations, we get: $$ \frac{(z_2-z_1)(z_4-z_1)}{(z_4-z_3)(z_2-z_3)} = -ab $$ Taking magnitude on both sides, $$ \frac{|(z_2-z_1)(z_4-z_1)|}{|(z_4-z_3)(z_2-z_3)|} = |ab| $$ Now, use the properties of magnitude \(|z_1z_2|=|z_1||z_2|\), we get: $$ \frac{|z_2-z_1||z_4-z_1|}{|z_4-z_3||z_2-z_3|} = |ab| $$ We can rewrite it as: $$ \frac{|z_2-z_1||z_4-z_1|}{|z_3-z_1||z_3-z_4|} = |ab| $$ Since the magnitudes of complex numbers are always non-negative, we can say that \(|z_3-z_1| \le |z_4-z_2|\) as required to prove.

Key Concepts

Triangle Inequality

The triangle inequality is a fundamental property of complex numbers that mirrors the familiar geometrical theorem stating that in any triangle, the length of one side is less than or equal to the sum of the lengths of the other two sides. When dealing with complex numbers, the 'sides' are interpreted as the magnitudes of differences between numbers.

For any complex numbers, let's call them \(z_1\), \(z_2\), and \(z_3\), the triangle inequality can be expressed as \(\left|z_1 - z_2\right| + \left|z_1 - z_3\right| \geq \left|z_2 - z_3\right|\). This signifies that the sum of the magnitudes of the differences between the complex numbers \(z_1\) and \(z_2\), and \(z_1\) and \(z_3\) is at least as large as the magnitude of the difference between \(z_2\) and \(z_3\).

In the context of our exercise, the triangle inequality helps to establish the relationships between the magnitudes of the differences mentioned in part (a). Understanding this principle is key to grasping the nature of inequalities involving complex numbers.

Magnitude of Complex Numbers

The magnitude of a complex number, also referred to as its absolute value or modulus, is a measure of how far the number is from the origin in the complex plane. For a complex number \(z = a + bi\), its magnitude is calculated using the formula \(\left|z\right| = \sqrt{a^2 + b^2}\).

Calculating Magnitude

For instance, if you have \(z_1 = 3 + 4i\), the magnitude \(\left|z_1\right|\) would be \(\sqrt{3^2 + 4^2} = 5\). This corresponds to the length of the vector from the origin to the point \(3 + 4i\) on the complex plane.

Magnitudes are pivotal when dealing with the geometric interpretation of complex numbers and in solving inequalities. They play a crucial role in the simplification of complex expressions and the comparison of complex numbers' sizes, as required in part (b) of our exercise.

Properties of Complex Numbers

Complex numbers have several unique properties that are essential in both theoretical and applied mathematics. Understanding these properties enables students to handle complex arithmetic and algebraic operations.

Some noteworthy properties of complex numbers include:

• Closure: The sum, difference, product, or quotient (except by zero) of two complex numbers is another complex number.
• Conjugates: For any complex number \(z = a + bi\), its conjugate is \(\bar{z} = a - bi\). Conjugates are useful in simplifying the division of complex numbers.
• Modulus Multiplication: The magnitude of a product of two complex numbers is the product of their magnitudes, essentially \(\left|z_1z_2\right|=\left|z_1\right|\left|z_2\right|\), which was used to simplify the expression in step 3 of our solution.
• Distributive Law: Complex numbers obey the distributive law, just like real numbers: \(z_1(z_2 + z_3) = z_1z_2 + z_1z_3\).

These properties are leveraged to manipulate and solve complex equations, as demonstrated in our exercise solution. When students grasp these concepts, they can efficiently address a wide array of problems involving complex numbers.