Question

Yes or No? If No, give a reason.

Is every real number also a complex number?

Short answer

Yes, the sum of a complex number and its complex conjugate is a real number.

Step-by-step solution

Step 1. Concept of complex number.

A complex number is of the form $a+bi$. Where a and b are some real numbers and i is the imaginary number.

The number a is known as the real part of the complex number and b is known as imaginary part of the complex number.

Step 2. 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 $a+bi$of is $a-bi$.

Step 3. Explanation.

While adding a complex number and its complex conjugate, the imaginary part get canceled out because they are like terms with opposite signs. And we are left with the sum of the real parts of complex number and its complex conjugate.

For example, sum of the complex number $2+3i$and its complex conjugate $2-3i$is given by

localid="1644146099757" $$\begin{gathered} \left(2+3i\right)+\left(2-3i\right)=2+3i+2-3i \\ =2+2+3i-3i \\ =\left(2+2\right)+\left(3-3\right)i \\ =4+0i \\ =4 \end{gathered}$$

So Yes, the sum of a complex number and its complex conjugate is a real number.