Problem 7 - Precalculus Mathematics for Calculus

Question

Find the real and imaginary parts of the complex number. $$\frac{-2-5 i}{3}$$

Step-by-Step Solution

Answer

Real: \(-\frac{2}{3}\), Imaginary: \(-\frac{5}{3}\)

Step 1: Identify the Complex Number Format

The given complex number is expressed in the form \( \frac{-2-5i}{3} \). To find the real and imaginary parts, we'll first need to separate the expression into these two parts.

Step 2: Divide the Real Component

The real component of the given complex number is \(-2\). We divide it by 3 to find the real part of the complex number: \( \frac{-2}{3} \).

Step 3: Divide the Imaginary Component

The imaginary component of the given complex number is \(-5i\). We divide the coefficient of \(i\), which is \(-5\), by 3 to find the imaginary part of the complex number: \( \frac{-5}{3} \). Thus, the imaginary part is \( \frac{-5}{3}i\).

Key Concepts

Real Part

Every complex number can be expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. The real part of a complex number is simply the "\( a \)" term, which is the non-imaginary component. In the expression given, \( \frac{-2 - 5i}{3} \), the real part is derived from dividing the real portion of the numerator, which is \(-2\), by the denominator.
Thus, the expression becomes divided explicitly between its components as \( \frac{-2}{3} - \frac{5i}{3} \). The real part is plainly \( \frac{-2}{3} \), which shows how we can separate it from the complex expression and underline how real numbers can exist independently yet as part of a complex structure.

Imaginary Part

The imaginary part of a complex number is associated with the term of the number that includes the imaginary unit, \( i \). Imaginary numbers are constructed around the concept that \( i \) is the square root of negative one \( (\sqrt{-1}) \). In the complex expression \( \frac{-2 - 5i}{3} \), to extract the imaginary part, we look at the coefficient of \( i \) which is \(-5\). This coefficient is then divided by 3, giving us the imaginary part \( \frac{-5}{3}i \).
The presence of \( i \) signifies its imaginary nature, ensuring it stays distinct from the real fraction component.

Division in Complex Numbers

When dividing complex numbers, each component – real and imaginary – must be divided separately by the denominator. In some cases, complex numbers in the denominator require more detailed handling, but in simple expressions like \( \frac{-2 - 5i}{3} \), direct division suffices. Here’s how to handle such division:

• Identify each part: Real (\(-2\)) and Imaginary (\(-5i\)).
• Divide each by the denominator (3 in this case), leading to \( \frac{-2}{3} - \frac{5i}{3} \).
• The result is a simplified form where the division is applied separately to both parts.

This approach illustrates the separation within complex numbers due to the real and imaginary nature, simplifying the arithmetic involved.