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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
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### 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.
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