Question

If \(z_{i}, \ldots, z_{n}\) are complex, prove that $$ \left|z_{1}+z_{2}+\cdots+z_{n}\right| \leq\left|z_{1}\right|+\left|z_{2}\right|+\cdots+\left|z_{n}\right| . $$

Short answer

Answer: The triangle inequality theorem states that the absolute value of the sum of complex numbers is less than or equal to the sum of their absolute values, or formally: $$|z_1 + z_2| \le |z_1| + |z_2|$$.

Step-by-step solution

Understand the triangle inequality theorem

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle is always greater than or equal to the length of the third side. In the context of complex numbers, it means that the absolute value of the sum of two complex numbers is less than or equal to the sum of their absolute values, or formally: $$|z_1 + z_2| \le |z_1| + |z_2|$$

Apply the theorem on each pair of the sum of complex numbers

We can build a sequence of sums of adjacent complex numbers and apply the triangle inequality theorem on each pair. Let's start with the first two complex numbers: $$|z_1 + z_2| \le |z_1| + |z_2|$$ Now, consider the sum \((z_1 + z_2) + z_3\), and apply the triangle inequality theorem to evaluate the sum. We can use the previous result for \((z_1+z_2)\): $$|(z_1 + z_2) + z_3| \le |z_1 + z_2| + |z_3| \le (|z_1| + |z_2|) + |z_3|$$ Repeating the process for the remaining terms in the sum, we can generalize this result for any \(n\) complex numbers: $$|z_1 + z_2 + \cdots + z_n| \le |z_1| + |z_2| + \cdots + |z_n|$$

Conclusion

Applying the triangle inequality theorem to each pair of the sum of complex numbers \(z_1, \ldots, z_n\), we can prove the given exercise $$|z_1 + z_2 + \cdots + z_n| \le |z_1| + |z_2| + \cdots + |z_n|$$

Key concepts

Complex Numbers

Complex numbers are one of the most fascinating concepts in mathematics, extending the familiar real number system to include the imaginary unit, denoted as 'i', where i^2 = -1. A complex number is typically written in the form a + bi, where a and b are real numbers. The real part, a, and the imaginary part, b, combine to form a multidimensional number.

When dealing with complex numbers, it's important to understand their geometric representation. They can be visualized as points or vectors in a two-dimensional plane, known as the complex plane, where the x-axis corresponds to the real part and the y-axis corresponds to the imaginary part. This allows for a profound connection between algebra and geometry via complex numbers, making it easier to perform operations like addition, subtraction, and even multiplication by considering the geometric implications. Understanding this can aid students in grasping more abstract concepts such as the problem given, where multiple complex numbers are added together.

Absolute Value

The absolute value of a real number is a measure of its magnitude without considering its direction or sign; for a non-negative number, it is the number itself, and for a negative number, it is the number multiplied by -1. But when we extend this concept to complex numbers, absolute value—or modulus—measures the distance from the origin to the point represented by the complex number in the complex plane.

More formally, if we have a complex number z = a + bi, its absolute value is given by |z| = \(\sqrt{a^2 + b^2}\). This can be understood as the hypotenuse of a right-angled triangle with sides of lengths a and b in the complex plane. It's crucial in mathematical analysis since it translates complex operations into manageable real-value expressions and sets the stage for exploring properties like the triangle inequality in the context of complex numbers. This property asserts that the absolute value of the sum of two complex numbers is always less than or equal to the sum of their absolute values, a cornerstone for solving the problem at hand.

Mathematical Analysis

Mathematical analysis is a branch of mathematics that deals with limits, sequences, derivatives, integrals, and infinite series. Its theorems and properties are instrumental in evaluating the behavior and properties of real-valued and complex-valued functions. A key component of mathematical analysis is understanding how to navigate inequalities, such as the triangle inequality theorem.

This theorem is not just a geometric fact about the sides of a triangle; it’s also a fundamental principle in analysis that underpins the stability of numerical operations. The proof exercise involving the sum of complex numbers demonstrates the power of mathematical analysis to discern patterns and general rules that apply to a wide range of problems. For instance, ensuring that each step of the inequality proof adheres to the principles of analysis, students not only solve the exercise but also strengthen their analytical skills, enabling them to tackle more sophisticated problems in mathematics and beyond.