Problem 31 - Precalculus Mathematics for Calculus

Question

Evaluate the expression and write the result in the form \(a+b i\) $$i^{3}$$

Step-by-Step Solution

Answer

The value of \(i^3\) is \(-i\), which is \(0 - 1i\).

Step 1: Understanding the Problem

We're asked to evaluate the expression \(i^3\) and express the result in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit, defined as \(i = \sqrt{-1}\).

Step 2: Recall Properties of Imaginary Numbers

The imaginary unit \(i\) has a pattern in its powers: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). This pattern repeats every four powers.

Step 3: Apply the Power Cycle to Evaluate

Using the circular property of powers of \(i\), we identify that \(i^3\) corresponds directly to \(-i\) since \(i^3 = -i\) in the cyclic pattern of imaginary units.

Step 4: Express in Standard Form

We express \(-i\) in the form \(a + bi\). In this case, \(a = 0\) and \(b = -1\), so \(-i = 0 - 1i\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined by the property \(i = \sqrt{-1}\). This definition may seem abstract, as no real number squared will result in a negative number. However, in the field of complex numbers, \(i\) allows us to extend the real number system to solve equations that do not have solutions in the real numbers alone.

The imaginary unit helps us explore mathematical concepts beyond the limitations of real numbers. It forms the basis for complex numbers, which are numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers.

• The concept of an imaginary number was first encountered within the context of finding solutions to quadratic equations.
• Imaginary numbers can combine with real numbers, forming what we call complex numbers.
• The notation \(i\) is universally used to represent the square root of minus one, setting the foundation for complex operations.

Powers of i

The powers of \(i\) follow a predictable pattern, which is cyclic and repeats every four powers. Understanding this cycle is important for evaluating expressions like \(i^3\). Here is a breakdown of the pattern:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

After \(i^4\), the cycle repeats, meaning:

• \(i^5 = i\)
• \(i^6 = -1\)
• \(i^7 = -i\)
• \(i^8 = 1\)

Recognizing this cycle is key to simplifying powers of \(i\). For \(i^3\), we observe that it simplifies to \(-i\).

Any power of \(i\) can be reduced to one of these four basic forms by dividing the exponent by 4 and looking at the remainder. The remainder determines which part of the cycle the power corresponds to. If, for example, you are given \(i^{23}\), you divide 23 by 4, getting a remainder of 3; thus, \(i^{23} = i^3 = -i\). This cyclical property greatly simplifies working with large powers of \(i\).

Standard Form a+bi

Complex numbers are typically written in what is called the "standard form," which is expressed as \(a + bi\).

In this form, \(a\) is the real part of the number, and \(bi\) is the imaginary part. While real numbers have no imaginary component, complex numbers encompass both real and imaginary parts, providing a richer number framework.

• For example, the number \(3 + 4i\) has a real part of 3 and an imaginary part of 4.
• Conversely, the number \(-7\) can also be seen as \(-7 + 0i\), indicating it is purely real.
• Imaginary numbers such as \(-i\) would be written as \(0 - 1i\). Here \(a = 0\) and \(b = -1\).

The standard form is useful because it provides a clear, uniform way of working with and simplifying complex numbers. By keeping numbers in this form, we can easily add, subtract, multiply, and divide complex numbers with consistent results. It also makes it easy to spot the real and imaginary portions of a complex number, which aids in further mathematical operations or transformations.