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Complex Conjugate Definition

Description

This equation shows how to find the complex conjugate (\( \htmlClass{sdt-0000000130}{^{*}} \)) of any given complex number.

\[(\htmlClass{sdt-0000000121}{a} + \htmlClass{sdt-0000000033}{b}\htmlClass{sdt-0000000129}{j})\htmlClass{sdt-0000000130}{^{*}} = \htmlClass{sdt-0000000121}{a} - \htmlClass{sdt-0000000033}{b}\htmlClass{sdt-0000000129}{j}\]

Symbols Used:

This is a symbol for any secondary generic constant. It can hold any numerical value

\( a \)

This is a symbol for any generic constant. It can hold any numerical value

\( j \)

This symbol represents the imaginary unit, which is defined as the square root of \(-1\). \( j = \sqrt{-1}\). It is the most fundamental unit in the field of complex numbers, allowing for the expression of numbers that cannot be represented on the real number line.

\( ^{*} \)

This is the symbol for the complex conjugate, representing the reflection of any complex number along the real axis. If \(Z\) is a complex number, then \(Z^{*}\) is it's conjugate.

Example

Find the complex conjugate of: \(\htmlClass{sdt-0000000006}{Z} = 3 + 2\htmlClass{sdt-0000000129}{j}\)

values:

From here, we can plug these values into the equation to get the complex conjugate:

\[ \htmlClass{sdt-0000000006}{Z} \htmlClass{sdt-0000000130}{^{*}} = 3 - 2\htmlClass{sdt-0000000129}{j} \]