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Chapter 8: Problem 104

What are conjugates? Give an example with your explanation.

Short Answer

Expert verified
Conjugate of a complex number is obtained by switching the sign of its imaginary part. For the complex number \(3 + 4i\), the conjugate would be \(3 - 4i\).

Step by step solution

01

Understanding Complex Numbers

A complex number is a number that can be expressed in the form \(a + bi\), where 'a' and 'b' are real numbers, and 'i' is an imaginary unit given by \(i = \sqrt{-1}\). 'a' is called the real part and 'b' is called the imaginary part of the complex number.
02

Defining the Conjugate

The conjugate of a complex number is obtained by changing the sign of its imaginary part. For example, the conjugate of the complex number \(a + bi\) is \(a - bi\). Similarly, the conjugate of \(a - bi\) is \(a + bi\). So in general, if you have a complex number \(z = a + bi\), the conjugate of 'z', denoted by \(\overline{z}\), is \(a - bi\).
03

Example

For example, consider the complex number \(z = 3 + 4i\). The conjugate of \(z\) can be determined by changing the sign of its imaginary part. So, the conjugate of \(z\) would be \(a - bi = 3 - 4i\). Thus, \(\overline{z} = 3 - 4i\) is the conjugate of the complex number \(z = 3 + 4i\).

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