Prove the triangle inequality for complex numbers: if z, we C then lz + w|≤|z| + |w| |z|² = zz where z = x - iy if z = x + iy, x, y en
Prove the triangle inequality for complex numbers: if z, we C then lz + w|≤|z| + |w| |z|² = zz where z = x - iy if z = x + iy, x, y en
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Title: Understanding the Triangle Inequality for Complex Numbers**
**Objective:** Prove the triangle inequality for complex numbers: If \( z, w \in \mathbb{C} \), then \( |z + w| \leq |z| + |w| \).
**Background Concept:**
- The modulus (or magnitude) of a complex number \( z \) is represented as \( |z| \).
- For a given complex number \( z = x + iy \), its conjugate is \( \overline{z} = x - iy \) where \( x, y \in \mathbb{R} \).
**Important Formula:**
- The square of the modulus of a complex number \( z \) is given by:
\[
|z|^2 = z \overline{z}
\]
This formula helps in calculating the magnitude of a complex number as the product of the complex number and its conjugate.
**Proof Strategy:**
The triangle inequality asserts that for the sum of two complex numbers, the magnitude of the sum is less than or equal to the sum of their magnitudes. This is a fundamental property and is used extensively in complex analysis and related fields.
Understanding this inequality is crucial for students of mathematics as it lays the groundwork for exploring more complex concepts in the realm of complex numbers and their applications.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37e8ed93-7bef-4409-89ed-52264f64a27e%2F87dd028f-02e9-4643-9f4c-6c01a00dc215%2Fk3ahrar_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Understanding the Triangle Inequality for Complex Numbers**
**Objective:** Prove the triangle inequality for complex numbers: If \( z, w \in \mathbb{C} \), then \( |z + w| \leq |z| + |w| \).
**Background Concept:**
- The modulus (or magnitude) of a complex number \( z \) is represented as \( |z| \).
- For a given complex number \( z = x + iy \), its conjugate is \( \overline{z} = x - iy \) where \( x, y \in \mathbb{R} \).
**Important Formula:**
- The square of the modulus of a complex number \( z \) is given by:
\[
|z|^2 = z \overline{z}
\]
This formula helps in calculating the magnitude of a complex number as the product of the complex number and its conjugate.
**Proof Strategy:**
The triangle inequality asserts that for the sum of two complex numbers, the magnitude of the sum is less than or equal to the sum of their magnitudes. This is a fundamental property and is used extensively in complex analysis and related fields.
Understanding this inequality is crucial for students of mathematics as it lays the groundwork for exploring more complex concepts in the realm of complex numbers and their applications.
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