Question

Use $z=a+b i$ to show that $z+\bar{z}=2 a$ and $z-\bar{z}=2 b i$

   Use $z=a+b i$ to show that $z+\bar{z}=2 a$ and $z-\bar{z}=2 b i$

College Algebra

College Algebra

Michael Sullivan 10th Edition

Chapter 1, Problem 89

Instant Answer

Solved by Expert Ankit Gupta

04/20/2020

Step 1

Step 1: First, we are given that $z=a+b i$ and we know that the conjugate of $z$ is $\bar{z}=a-b i$. Show more…

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Use $z=a+b i$ to show that $z+\bar{z}=2 a$ and $z-\bar{z}=2 b i$

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Key Concepts

Complex Numbers

Complex numbers are numbers expressed in the form a + bi, where 'a' represents the real part and 'b' represents the imaginary part. They are fundamental in mathematics for representing quantities that have both magnitude and direction, combining the real number system with the imaginary unit i, which satisfies i² = -1.

Complex Conjugate

The complex conjugate of a complex number z = a + bi is given by a - bi. This concept is essential in simplifying expressions and solving equations in complex analysis because it allows for the elimination of the imaginary part when needed, as seen in the sum and difference of a complex number and its conjugate.

Addition and Subtraction of Complex Numbers

When adding or subtracting complex numbers, the real parts are combined with real parts and the imaginary parts with imaginary parts. This operation is crucial as it underpins the method by which the sum of a complex number and its conjugate yields 2a (eliminating the imaginary components), and their difference yields 2bi (eliminating the real parts).

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