Problem 10 - Precalculus: Mathematics for Calculus

Question

Find the real and imaginary parts of the complex number. $$\frac{4+7 i}{2}$$

Step-by-Step Solution

Answer

Real part: 2, Imaginary part: 3.5i.

Step 1: Identify the Complex Number

Given the complex number \( \frac{4+7i}{2} \). We need to find its real and imaginary parts.

Step 2: Separate the Real and Imaginary Parts

The expression \( \frac{4+7i}{2} \) can be rewritten as \( \frac{4}{2} + \frac{7i}{2} \). This separates the complex number into its real and imaginary components.

Step 3: Simplify the Real Part

Calculate \( \frac{4}{2} = 2 \). So, the real part is 2.

Step 4: Simplify the Imaginary Part

Calculate \( \frac{7}{2} = 3.5 \). Therefore, the imaginary part is 3.5i.

Key Concepts

Real Part of a Complex Number

The real part of a complex number is the component that does not involve the imaginary unit \( i \). Every complex number can be expressed in its standard form as \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit with the property that \( i^2 = -1 \). In this expression, \( a \) represents the real part of the complex number.

For example, in the complex number \( \frac{4+7i}{2} \), to determine the real part, we first divide each component by 2, resulting in \( \frac{4}{2} + \frac{7i}{2} \). Here, \( \frac{4}{2} \) simplifies to 2. Thus, the real part of this complex number is 2.

When working with complex numbers, always focus on the component without the \( i \) as your real part. Understanding the meaning of the real part will help in many mathematical contexts, such as in solving equations or analyzing complex functions.

Imaginary Part of a Complex Number

The imaginary part of a complex number is the component that is multiplied by the imaginary unit \( i \). In the standard form \( a + bi \), the term \( bi \) represents the imaginary part. The coefficient \( b \) is what denotes the magnitude of the imaginary part, and it is important to remember that \( i \) itself is a square root of \( -1 \).

Considering the complex number \( \frac{4+7i}{2} \), separating the parts by dividing each term gives us \( \frac{4}{2} + \frac{7i}{2} \). The term \( \frac{7i}{2} \) includes \( 7 \) as the coefficient of \( i \), indicating the imaginary part is 3.5 (since \( \frac{7}{2} = 3.5 \)). Therefore, the imaginary component of this number is \( 3.5i \).

Focusing on distinguishing the imaginary portion helps in understanding how complex numbers behave differently from real numbers and opens up many important insights in fields like engineering and physics.

Separating Components of Complex Numbers

To efficiently handle complex numbers, understanding how to separate their components is crucial. This involves dividing the number into its real and imaginary parts.

Consider the given complex number \( \frac{4+7i}{2} \). The task is to express it in a form where each part can be clearly identified. Start by breaking down the fraction: treat \( 4 \) and \( 7i \) as separate entities and divide each by 2. Thus, \( \frac{4}{2} \) becomes the real part, and \( \frac{7i}{2} \) becomes the imaginary part.

This results in \( 2 + 3.5i \), a form where the real and imaginary parts are plainly visible.

• The term \( 2 \) stands alone and represents the real component.
• The term \( 3.5i \) is clearly the imaginary component, marked by its connection to \( i \).

Separation of components aids in many calculations and analyses. It allows easy handling of complex numbers, facilitating operations such as addition, subtraction, and multiplication, and also providing clarity in their application in various mathematical problems.