Problem 8 - 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 is 2; imaginary part is \( \frac{7}{2} \).

Step 1: Identify the Complex Number Format

The given expression is \( \frac{4+7i}{2} \). This complex number is in the form \( a + bi \), where \( a = 4 \) is the real part, and \( bi = 7i \) is the imaginary part.

Step 2: Simplify the Complex Number

Divide both the real part and the imaginary part by 2. This means you compute \( \frac{4}{2} + \frac{7i}{2} \).

Step 3: Compute the Real Part

The real part of the complex number is \( \frac{4}{2} = 2 \).

Step 4: Compute the Imaginary Part

The imaginary part of the complex number is \( \frac{7}{2}i \), or \( \frac{7}{2} \) when expressed as a multiplier of \( i \).

Key Concepts

real part

When dealing with complex numbers, it's essential to understand their components. A complex number is usually written in the form \( a + bi \), where \( a \) and \( b \) are real numbers. The term \( a \) represents the "real part" of the complex number.
To find the real part of a complex number given in the expression \( \frac{4 + 7i}{2} \), we look at the numerator, which is \( 4 + 7i \). The part without the \( i \), the real part, is \( 4 \). However, since the entire expression \( \frac{4 + 7i}{2} \) is being divided by 2, we need to divide the real part \( 4 \) by 2 as well.
This computation leads us to:

• The expression for real part: \( \frac{4}{2} \)
• The real part is: 2

imaginary part

The imaginary part of a complex number is based on the coefficient of \( i \), a symbol signifying the square root of -1. In our expression \( \frac{4 + 7i}{2} \), the term \( 7i \) is where we find our imaginary part. The coefficient \( 7 \) is key, as the imaginary part centers on this number.
To extract the imaginary part from the expression \( \frac{4 + 7i}{2} \), we must likewise divide its coefficient \( 7 \) by 2. The solution breaks down as follows:

• Imaginary part from \( 7i \): \( \frac{7}{2}i \)
• Simplified imaginary part: \( \frac{7}{2} \) times \( i \)

Visualizing this helps make the concept clear: whenever you have a complex number, look to the \( bi \) part, and the coefficient of \( i \) tells you about the imaginary part.

simplifying complex numbers

Simplifying complex numbers can make them more digestible both for performing calculations and for proper understanding. The expression \( \frac{4 + 7i}{2} \) invites us to simplify it by distributing the division across the real and imaginary components separately.
Here's a quick step-by-step approach to simplify it:

• Break down the original expression: \( \frac{4}{2} + \frac{7i}{2} \)
• Simplify each part separately.
• Real part simplified to 2
• Imaginary part simplified to \( \frac{7}{2}i \)

The final result of these operations is \( 2 + \frac{7}{2}i \). By separately addressing each component, we render the complex number in a more manageable form. This method also helps to ensure accuracy in further mathematical operations with the complex number.