Problem 7 - A Graphical Approach to College Algebra

Question

For each complex number, (a) state the real part, (b) state the imaginary part, and (c) identify the number as one or more of the following: real, pure imaginary, or nonreal complex. $$i \sqrt{7}$$

Step-by-Step Solution

Answer

Real: 0, Imaginary: \(\sqrt{7}\), Pure imaginary.

Step 1: Express the Complex Number

The complex number given is \( i \sqrt{7} \). The standard form for a complex number is \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. Here, \( a = 0 \) and \( b = \sqrt{7} \).

Step 2: Identify the Real Part

In the complex number \( i \sqrt{7} \), the real part \( a \) is 0.

Step 3: Identify the Imaginary Part

The imaginary part \( b \) of the complex number \( i \sqrt{7} \) is \( \sqrt{7} \).

Step 4: Classify the Number

Since the real part is 0 (\( a = 0 \)) and the imaginary part is non-zero, \( i \sqrt{7} \) is classified as a pure imaginary number.

Key Concepts

Real Part

Complex numbers are made up of two parts: the real part and the imaginary part. The real part of a complex number is the component that does not involve the imaginary unit \( i \). When a complex number is expressed in the standard form \( a + bi \), the real part is the coefficient \( a \). For example, in the complex number \( i \sqrt{7} \), the term \( a \) is missing. Hence, the real part is simply \( 0 \). This means there is no shift along the real number line.
In any complex number, understanding the real part helps one to focus on addition, subtraction, and comparison with other real numbers. It's essential to always identify the real part when working with complex numbers.

Imaginary Part

The imaginary part of a complex number is what sets it apart from real numbers. In the form \( a + bi \), the imaginary part is represented by \( b \), the coefficient of \( i \). For \( i \sqrt{7} \), the imaginary part is \( \sqrt{7} \) because it's the number paired with \( i \).
It's crucial to remember that the imaginary part gives complex numbers their unique capabilities, allowing them to extend beyond the real number line. When identifying the imaginary part, check the coefficient of \( i \) in the expression. This insight is critical for operations like multiplication and division in the realm of complex numbers. Imaginary parts help in representing rotations and oscillations in various scientific fields.

Pure Imaginary Number

A pure imaginary number is a special type of complex number where the real part is \( 0 \) and the imaginary part is non-zero. This means the entire value lies along the imaginary number line. For the given complex number \( i \sqrt{7} \), since the real part is \( 0 \) and the imaginary part is \( \sqrt{7} \), it is classified as a pure imaginary number.
Recognizing pure imaginary numbers is important when examining elements that cannot exist in the real number field alone, like certain types of roots and oscillations. Such numbers are crucial in circuits, signal processing, and wave mechanics. They serve as powerful tools in mathematical calculations, offering different perspectives when analyzing complex equations.