Problem 3 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Complex Number

Write the complex number in its standard form as \(\frac{-2}{3} + \frac{-5}{3}i\). This separates the real part from the imaginary part, allowing us to identify each component individually.

Step 2: Identify the Real Part

The real part of a complex number is the part that does not involve the imaginary unit \(i\). From \(\frac{-2}{3} + \frac{-5}{3}i\), identify the real part as \(\frac{-2}{3}\).

Step 3: Identify the Imaginary Part

The imaginary part of a complex number is the coefficient of \(i\), the imaginary unit. From \(\frac{-2}{3} + \frac{-5}{3}i\), identify the imaginary part as \(\frac{-5}{3}\).

Key Concepts

Real Part

In a complex number, the real part is the number that you find in front, without the imaginary unit \(i\). It represents a standard real number that we encounter in everyday mathematics. Consider the complex number given in our exercise: \(\frac{-2}{3} + \frac{-5}{3}i\). Here, the real part is \(\frac{-2}{3}\). This segment, \(\frac{-2}{3}\), does not involve the imaginary unit at all.

• The real part is always a standalone number.
• It behaves just like any normal number in calculations that only involve real numbers.

Recognizing the real part of a complex number is crucial because it allows one to interpret the number partly in the context of the real number line.

Imaginary Part

The imaginary part of a complex number is all about the presence of \(i\). This part places the complex number in a different dimension than typical real numbers. Essentially, it gives complex numbers their unique properties. Referring again to our example: \(\frac{-2}{3} + \frac{-5}{3}i\), the imaginary part is determined by the coefficient in front of \(i\). Here, it is \(\frac{-5}{3}\).

• The imaginary part always comes with the letter \(i\), signifying the involvement of the imaginary unit.
• This part helps visualize numbers in a way that is not possible on the traditional number line.

While the real part draws from standard numbers, the imaginary part expands our ability to solve equations and perform calculations beyond the real realm.

Standard Form

Writing a complex number in standard form requires arranging it as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. This format is helpful because it clearly separates the number into parts that are easy to identify and work with. In our exercise, the complex number \(\frac{-2}{3} + \frac{-5}{3}i\) is presented in this standard form.

• The part with no \(i\) (\(\frac{-2}{3}\)) is the real part, \(a\).
• The part with \(i\) (\(\frac{-5}{3}i\)) is the imaginary part, \(bi\).

Using the standard form allows for greater clarity and understanding when performing arithmetic operations with complex numbers. It acts as the base for adding, subtracting, multiplying, and dividing complex numbers efficiently.