Problem 7 - 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: Convert Complex Number to Standard Form

First, we need to express the given complex number \( \frac{-2-5i}{3} \) in the standard complex number form, which is generally given by \( a + bi \). The expression can be rewritten as two separate fractions: \(-\frac{2}{3} - \frac{5}{3}i\). This step ensures both the real and the imaginary components are clearly identifiable.

Step 2: Identify the Real Part

The real part of a complex number in the form \( a + bi \) is simply the \( a \) value, which is the coefficient of the non-imaginary term. In the expression \(-\frac{2}{3} - \frac{5}{3}i\), the real part is \(-\frac{2}{3}\). This part of the number does not include the imaginary unit \( i \).

Step 3: Identify the Imaginary Part

The imaginary part of a complex number in the form \( a + bi \) is the coefficient \( b \) in front of \( i \). In the expression \(-\frac{2}{3} - \frac{5}{3}i\), the imaginary part is \(-\frac{5}{3}\). This is the value that is multiplied by the imaginary unit \( i \).

Key Concepts

real part

To better understand complex numbers, it's vital to comprehend the concept of the real part. A complex number is generally written in the form \( a + bi \), where \( a \) represents the real part and \( bi \) represents the imaginary part. The real part is simply the constant number without the imaginary unit \( i \).

Let's take an example from the exercise: \(-\frac{2}{3} - \frac{5}{3}i\). In this case, the real part is \(-\frac{2}{3}\). This part indicates the position of the number along the real axis of a complex plane.

Recognizing the real part is straightforward; just look for the term that does not involve \( i \).

• Real Part: \(-\frac{2}{3}\)
• Displayed as: Constant Term
• Positioned along: Real Axis

imaginary part

The imaginary part of a complex number is just as critical as the real part. In the notation \( a + bi \), the imaginary part is represented by the term \( bi \), where \( b \) is a real number that multiplies the imaginary unit \( i \).

To identify the imaginary part, you focus on the coefficient of \( i \). From our earlier example, \(-\frac{2}{3} - \frac{5}{3}i\), the imaginary part is the number that is directly in front of \( i \).

For this expression, the imaginary part is \(-\frac{5}{3}\). This number helps position the complex number in the imaginary direction on a complex plane.

• Imaginary Part: \(-\frac{5}{3}\)
• Multiplied by: Imaginary Unit \( i \)
• Positioned along: Imaginary Axis

standard form of complex numbers

The standard form of a complex number is a crucial way to present these numbers in a recognizable manner, given by \( a + bi \). Here, \( a \) is the real part, and \( bi \) is the imaginary part.

Achieving the correct standard form is particularly important when you're simplifying expressions. Rewriting complex numbers ensures clarity, and it makes it easier to perform mathematical operations. For example, take the expression \(\frac{-2-5i}{3}\).

When expressed in standard form, it becomes \(-\frac{2}{3} - \frac{5}{3}i\).

• Standard Form Example: \(-\frac{2}{3} - \frac{5}{3}i\)
• Real part (\(a\)): \(-\frac{2}{3}\)
• Imaginary part (\(bi\)): \(-\frac{5}{3}i\)

Achieving this form makes complex numbers intuitive and easy to work with, as each component can be directly identified and manipulated.