Problem 39 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\(-i = 0 - i\)

Step 1: Understanding Imaginary Unit

The imaginary unit is denoted as \(i\) and is defined as the square root of \(-1\). This means \(i^2 = -1\).

Step 2: Simplifying Higher Powers of i

To evaluate expressions like \(i^3\), we need to simplify higher powers of \(i\) using the fact that \(i^2 = -1\). Recall that powers of \(i\) follow a pattern every four exponents: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then it repeats.

Step 3: Calculate \(i^3\)

Using the pattern of powers of \(i\), recognize that \(i^3 = i^{2+1} = (i^2) \cdot i = (-1) \cdot i = -i\).

Key Concepts

Imaginary Unit

The imaginary unit, symbolized by \(i\), is a fundamental concept in complex numbers. It's defined as \(i = \sqrt{-1}\). This may seem strange at first, but the imaginary unit allows us to work with numbers that are impossible to handle on the real number line.
By definition, multiplying the imaginary unit by itself, that is \(i \cdot i\), gives \(i^2 = -1\).
This pivotal property of \(i\) serves as a building block for dealing with complex numbers and simplifies mathematical expressions that involve negative square roots.

• \(i\) is the imaginary unit.
• \(i^2 = -1\).
• The imaginary unit helps solve equations like \(x^2 = -1\) by making \(x = i\) a solution.

Powers of i

Understanding the powers of \(i\) is crucial as they form a predictable cycle. This cycle repeats every four exponents, a pattern that simplifies working with higher powers of \(i\).
The cycle is as follows:

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

After reaching \(i^4 = 1\), the cycle repeats itself, similar to the hands of a clock coming full circle. This pattern makes it easier to simplify expressions by reducing any power of \(i\) to one of these four core values.
For example, if you have \(i^7\), you can think of it as four full cycles plus three additional steps: \(i^7 = i^4 \cdot i^3 = 1 \cdot (-i) = -i\). This technique simplifies complex calculations and is powerful when evaluating expressions involving \(i\).

Simplification of Expressions

Simplifying expressions with the imaginary unit involves using the cyclical nature of \(i\). Once you grasp this pattern, evaluating expressions becomes much less daunting.
Consider the expression \(i^3\) as an example. Here’s how you simplify it:

• With \(i^3 = i^{2+1}\), recognize this as the product of \(i^2\) and \(i\).
• Knowing \(i^2 = -1\), substitute this in: \((-1) \cdot i = -i\).

Thus, \(i^3 = -i\), an elegant result.
Such simplification techniques are fundamental when writing complex numbers in their standard form \(a + bi\), where \(a\) and \(b\) are real numbers. It ensures clarity and eases further mathematical operations, which are essential in calculations involving electrical engineering, fluid dynamics, and other fields.