Chapter 3, Problem 1MC

To determine

Whether the statement “The product of a complex number and its conjugate is a real number” is true or false.

Answer to Problem 1MC

The given statement is true.

Explanation of Solution

Definitions used:

The number i:

The number i is defined such that $i=\sqrt{-1}$ and $i^2=-1$.

To express roots of negative numbers in terms of i, we can use the fact that $\sqrt{-p}=\sqrt{-1\cdot p}=\sqrt{-1}\cdot \sqrt{p}=i\sqrt{p}$, where p is a positive real number.

Conjugate of a complex number:

The conjugate of a complex number $a+bi$ is $a-bi$. The numbers $a+bi$ and $a-bi$ are complex conjugates.

Calculation:

Consider the complex number $a+bi$ where a and b are any real numbers.

The conjugate of $a+bi$ is $a-bi$.

Evaluate the product as follows.

$\begin{array}{c}\left(a+bi\right)\left(a-bi\right)=a^2-{\left(ib\right)}^2\\ =a^2-i^2{b}^2\\ =a^2-\left(-1\right)b^2\left(\because i^2=-1\right)\\ =a^2+b^2\end{array}$

Here, $a^2+b^2$ is a pure real number and it is true for any real values of a and b.

Thus, the product of a complex number and its conjugate is a real number.

Therefore, the given statement is true.