If zi, …, zn are complex, prove that |z1+z2+⋯+zn| ≤|z1|+|z2|+⋯+|zn| .

Name: Problem 12, Chapter 1: Principles of Mathematical Analysis
Uploaded: 2021-03-24T17:30:41-08:00
Duration: 3 min 56 s
Channel: Chris Trentman
Description: If zi, …, zn are complex, prove that |z1+z2+⋯+zn| ≤|z1|+|z2|+⋯+|zn| .

step 1 We're asked to prove a statement about the complex numbers. The statement is that for any comple

If zi, …, zn are complex, prove that |z1+z2+⋯+zn|...

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Key Concepts

Modulus

The modulus (or absolute value) of a complex number, defined as |a + bi| = ?(a² + b²), represents the distance of the complex number from the origin in the complex plane. It provides a measure of the size or magnitude of the complex number regardless of its direction.

Complex Numbers

Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit satisfying i² = -1. They extend the real numbers and allow for a broader range of solutions to equations, particularly quadratic equations that have no real solutions.

Triangle Inequality

The triangle inequality states that for any two complex numbers, the absolute value of their sum is less than or equal to the sum of their absolute values; that is, |z + w| ? |z| + |w|. This fundamental property reflects the geometric fact that the straight-line distance between two points is the shortest path, and it generalizes to sums of more than two terms.

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite sequence of statements. In the context of the triangle inequality for n complex numbers, induction can be used by first verifying the case for two numbers and then assuming the statement holds for n – 1 terms to prove it for n terms.

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