Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
If z, w E C, prove that z+w=z+w.

If z, w E C, prove that z+w=z+w.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
100%

In latex please.

### Complex Numbers and Conjugates **Problem Statement:** If \( z, w \in \mathbb{C} \), prove that \( \overline{z + w} = \overline{z} + \overline{w} \). **Explanation:** To prove this property of complex numbers, we will use the definition of the complex conjugate. **Step-by-Step Solution:** 1. **Let \( z \) and \( w \) be complex numbers:** - \( z = a + bi \) - \( w = c + di \) where \( a, b, c, d \) are real numbers and \( i \) is the imaginary unit. 2. **Sum of \( z \) and \( w \):** - \( z + w = (a + bi) + (c + di) = (a + c) + (b + d)i \) 3. **Complex conjugate of the sum \( z + w \):** - \( \overline{z + w} = \overline{(a + c) + (b + d)i} = (a + c) - (b + d)i \) 4. **Complex conjugate of \( z \) and \( w \) individually:** - \( \overline{z} = \overline{a + bi} = a - bi \) - \( \overline{w} = \overline{c + di} = c - di \) 5. **Sum of the individual conjugates:** - \( \overline{z} + \overline{w} = (a - bi) + (c - di) = (a + c) - (b + d)i \) 6. **Conclusion:** - We observe that \( \overline{z + w} = (a + c) - (b + d)i \) - And \( \overline{z} + \overline{w} = (a + c) - (b + d)i \) Therefore, \( \overline{z + w} = \overline{z} + \overline{w} \), as required. This proof demonstrates the linearity of the complex conjugate operation, meaning the conjugate of a sum is the sum of the conjugates.
Transcribed Image Text:### Complex Numbers and Conjugates **Problem Statement:** If \( z, w \in \mathbb{C} \), prove that \( \overline{z + w} = \overline{z} + \overline{w} \). **Explanation:** To prove this property of complex numbers, we will use the definition of the complex conjugate. **Step-by-Step Solution:** 1. **Let \( z \) and \( w \) be complex numbers:** - \( z = a + bi \) - \( w = c + di \) where \( a, b, c, d \) are real numbers and \( i \) is the imaginary unit. 2. **Sum of \( z \) and \( w \):** - \( z + w = (a + bi) + (c + di) = (a + c) + (b + d)i \) 3. **Complex conjugate of the sum \( z + w \):** - \( \overline{z + w} = \overline{(a + c) + (b + d)i} = (a + c) - (b + d)i \) 4. **Complex conjugate of \( z \) and \( w \) individually:** - \( \overline{z} = \overline{a + bi} = a - bi \) - \( \overline{w} = \overline{c + di} = c - di \) 5. **Sum of the individual conjugates:** - \( \overline{z} + \overline{w} = (a - bi) + (c - di) = (a + c) - (b + d)i \) 6. **Conclusion:** - We observe that \( \overline{z + w} = (a + c) - (b + d)i \) - And \( \overline{z} + \overline{w} = (a + c) - (b + d)i \) Therefore, \( \overline{z + w} = \overline{z} + \overline{w} \), as required. This proof demonstrates the linearity of the complex conjugate operation, meaning the conjugate of a sum is the sum of the conjugates.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS