Question

Conjugate Recall that the symbol$\overline{z}$represents the complex conjugate of z. If$z=a+bi\text{and}w=c+di$, show that each statement is true.

$z=\overline{z}\text{ifandonlyif}z\text{isreal.}$

Short answer

It is shown that$z=\overline{z}$ if and only if z is real.

Step-by-step solution

Step 1. Concept of complex conjugate.

The complex conjugate of the complex number$a+bi$ is represented by a horizontal bar over it, that is, the complex conjugate of$a+bi$ is denoted by $\overline{a+bi}$.

The complex conjugate of any complex number can be obtained by changing the sign of the imaginary part of the complex number. The complex conjugate of $a+bi$is $a-bi$.

The product of a complex number and its complex conjugate is given by

$\left(a+bi\right)\left(a-bi\right)=a^2+b^2$

The imaginary number i is defined as the square root of negative one. That is, $i^2=-1$.

Step 2. Concept of multiplication of complex numbers.

To multiply two complex numbers, multiply them as binomial expressions and use the result of imaginary number, $i^2=-1$.

The FOIL method of multiplying binomials is

$\left(a+b\right)+\left(c+d\right)=a\cdot c+a\cdot d+b\cdot c+b\cdot d$

The product of a complex number and its complex conjugate is given by

$\left(a+bi\right)\left(a-bi\right)=a^2+b^2$

Step 3. Evaluate the expressions.

In order to show that the given statement is true, observe that while finding the complex conjugate, we only change the sign of the imaginary part. So, if the complex number z is pure real number, then its complex conjugate will be the same number because there is no imaginary part in the number.

It implies that $z=\overline{z}$ if and only if z is real.