Problem 5 - 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. $$3+7 i$$

Step-by-Step Solution

Answer

Real part: 3, Imaginary part: 7, Nonreal complex number.

Step 1: Identify the Real Part

In a complex number of the form \( a + bi \), \( a \) represents the real part. Here, the complex number is \( 3 + 7i \), so the real part \( a = 3 \).

Step 2: Identify the Imaginary Part

For a complex number \( a + bi \), \( b \) is the coefficient of the imaginary component \( i \). In the number \( 3 + 7i \), the imaginary part \( b = 7 \).

Step 3: Classify the Complex Number

A complex number can be classified as one or more of the following: real, pure imaginary, or nonreal complex. Since the number \( 3 + 7i \) has both a non-zero real part and a non-zero imaginary part, it is classified as a nonreal complex number. It is neither purely real (since the imaginary part is not zero) nor purely imaginary (since the real part is not zero).

Key Concepts

Real Part of a Complex Number

Complex numbers are usually expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property \( i^2 = -1 \). The real part of a complex number is simply the portion that does not involve the imaginary unit \( i \).
It's the "\( a \)" in the expression \( a + bi \). For example, in the complex number \( 3 + 7i \), the real part is \( 3 \). This means if you were to plot this number on the complex plane, you'd move 3 units along the real axis.

• The real part shows the horizontal position of the complex number on the complex plane.
• This is crucial for performing operations such as addition or subtraction of complex numbers.

Identifying the real part is straightforward but vital. It provides foundational insight into understanding and manipulating complex numbers.

Imaginary Part of a Complex Number

In a complex number represented as \( a + bi \), the imaginary part is the component that is multiplied by \( i \). This part is key to the number's identity as a complex rather than a mere real number.
The term "imaginary part" refers to the "\( b \)" in \( a + bi \). For instance, in the number \( 3 + 7i \), the imaginary part is \( 7 \). This signifies how far you move along the imaginary axis when visualizing the complex number on the complex plane.

• The imaginary unit, \( i \), is defined such that \( i^2 = -1 \).
• This part is essential in calculations and transformations involving complex numbers.

Grasping the imaginary part can help in understanding complex conjugates, roots of complex equations, and numerous applications in engineering and physics.

Classification of Complex Numbers

To classify a complex number, it helps to look at both the real and imaginary parts. Complex numbers fall into three main categories: real, pure imaginary, and nonreal complex numbers.

• Real Numbers: These complex numbers have no imaginary part; \( b = 0 \). For example, the number 4 is a real number because it can be expressed as \( 4 + 0i \).
• Pure Imaginary Numbers: These have no real part; \( a = 0 \). An example is \( 0 + 5i \), where it only exists along the imaginary axis.
• Nonreal Complex Numbers: These numbers have both a real and non-zero imaginary part. The number \( 3 + 7i \) is a nonreal complex number because both 3 and 7 are non-zero.

Understanding how to classify complex numbers helps in determining the nature of the solution to complex equations and is foundational in fields such as complex analysis and signal processing.