Question

Find the real and imaginary parts of the complex number.

$\frac{-2-5i}{3}$

Short answer

The real part is $-\frac{2}{3}$and the imaginary part is $-\frac{5}{3}$.

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. Example for complex number.

In the complex number $2+5i$, the real part is 2 and the complex part is 5.

The imaginary number i is defined as the square root of negative one. That is, $i=\sqrt{-1}$.

Step 3. Find real and imaginary parts.

To find the real and imaginary parts in the complex number $\frac{-2-5i}{3}$, first write it in standard form by distributing the denominator to numerator as

$$\begin{gathered} \frac{-2-5i}{3}=\frac{-2}{3}-\frac{5i}{3} \\ =-\frac{2}{3}-\frac{5}{3}i \end{gathered}$$

Observe that the number $-\frac{2}{3}$ is separate without imaginary number i with it, so $-\frac{2}{3}$is the real part.

And the number $-\frac{5}{3}$ is associated with the imaginary number i so it is the imaginary part of the complex number.

Thus, the real part of the given complex number is $-\frac{2}{3}$and the imaginary part is $-\frac{5}{3}$.