Problem 4 - Precalculus

Question

For Exercises specify the real and imaginary parts of each complex number. (a) \(-2+\sqrt{7} i\) (b) \(1+5^{1 / 3} i\) (c) \(-3 i\) (d) 0

Step-by-Step Solution

Answer

(a) Real: -2, Imaginary: \(\sqrt{7}\); (b) Real: 1, Imaginary: \(5^{1/3}\); (c) Real: 0, Imaginary: -3; (d) Real: 0, Imaginary: 0.

Step 1: Identify Real and Imaginary Parts of (a)

The given complex number is \(-2 + \sqrt{7} i\). A complex number is generally of the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Here, \(-2\) is the real part and \(\sqrt{7}\) is the coefficient of \(i\), making it the imaginary part.

Step 2: Identify Real and Imaginary Parts of (b)

The complex number is \(1 + 5^{1/3} i\). Following the form \(a + bi\), here \(1\) is the real part. The value \(5^{1/3}\) is the coefficient of \(i\), making it the imaginary part.

Step 3: Identify Real and Imaginary Parts of (c)

The complex number is \(-3i\). Expressed as \(a + bi\), where \(a\) is 0 because there is no additional constant term, and \(-3\) is the imaginary part.

Step 4: Identify Real and Imaginary Parts of (d)

The number 0 can be represented as \(0 + 0i\), thus the real part is \(0\) and the imaginary part is also \(0\).

Key Concepts

Real Part

In the realm of complex numbers, the real part is a crucial component represented by the term without the imaginary unit "i". Generally, a complex number takes the form of \(a + bi\), where "\(a\)" is the real part. Let's delve into some examples from the exercise:

• For the complex number \(-2 + \sqrt{7} i\), the real part is \(-2\).
• In the complex number \(1 + 5^{1/3} i\), the real part is \(1\).
• With \(-3i\), even though it might not seem obvious, the real part is \(0\) since there is no standalone constant term.
• When the number is just \(0\), it can be written as \(0 + 0i\). Here the real part is \(0\).

Understanding which term is real helps evaluate or plot complex numbers, as the real part can be visualized along the horizontal axis of the complex plane.

Imaginary Part

The imaginary part of a complex number is tied to the "\(bi\)" component, where "\(b\)"—the coefficient of \(i\)—represents this part. In a complex number \(a + bi\), \(b\) is the imaginary part and influences its vertical positioning on the complex plane:

• In \(-2 + \sqrt{7} i\), \(\sqrt{7}\) serves as the imaginary part, paired with \(i\).
• For \(1 + 5^{1/3} i\), \(5^{1/3}\) is the imaginary part that is multiplied by \(i\).
• When it comes to \(-3i\), the imaginary part is straightforward— it is \(-3\).
• In the expression \(0 + 0i\), the imaginary part is \(0\) since there is no contribution from the imaginary unit \(i\).

Understanding the imaginary part is crucial for discerning the presence of the imaginary component in calculations and visualizations on a graph.

Complex Number Form

Complex numbers form a unique and elegant structure in mathematics, typically denoted by \(a + bi\). In this format, "\(a\)" represents the real portion, while "\(bi\)" signifies the imaginary section:

• A complex number like \(-2 + \sqrt{7} i\) confirms this form, as {\(-2\)} acts as the real part and {\(\sqrt{7} i\)} as the imaginary.
• The structure \(1 + 5^{1/3} i\) adheres to this standardized format, making it easier to identify components.
• The example \(-3i\) can be reinterpreted as \(0 - 3i\), clarifying its conformity through zero as the real component.
• Even a solitary \(0\) adheres to the form when viewed as \(0 + 0i\), illustrating both principal parts simply as zero.

Understanding the structure of complex numbers allows for easy manipulation in algebraic computations and visual interpretations in a two-dimensional complex plane.