Problem 5 - Intermediate Algebra

Question

Fill in the blanks. a. \(i=\text{____}\) b. \(i^{2}=\text{____}\) c. \(i^{3}=\text{____}\) d. \(i^{4}=\text{____}\) e. In general, the powers of \(i\) cycle through ______ possible outcomes.

Step-by-Step Solution

Answer

a. \(i\), b. \(-1\), c. \(-i\), d. \(1\), e. 4 outcomes.

Step 1: Understand the Meaning of i

The symbol \(i\) represents the imaginary unit in mathematics. By definition, \(i\) is the square root of \(-1\). Therefore, \(i = \sqrt{-1}\).

Step 2: Compute i squared

To find \(i^{2}\), square the defining expression of \(i\). \[ i^{2} = (\sqrt{-1})^2 = -1 \]So, \(i^{2} = -1\).

Step 3: Calculate i cubed

The expression \(i^{3}\) can be rewritten as \(i^{2} \times i\). Since \(i^{2} = -1\), we have:\[ i^{3} = i^{2} \times i = -1 \times i = -i \]Thus, \(i^{3} = -i\).

Step 4: Determine i to the Fourth

\(i^{4}\) can be expressed as \(i^{2} \times i^{2}\), so:\[ i^{4} = (-1) \times (-1) = 1 \]Hence, \(i^{4} = 1\).

Step 5: Identify the Cycle of i's Powers

From the calculations above, the powers of \(i\) cycle every four powers as follows: \(i, -1, -i, 1\). After that, the cycle repeats. Therefore, the powers of \(i\) cycle through 4 possible outcomes.

Key Concepts

Imaginary Unit

The imaginary unit, often denoted as \(i\), serves a crucial role in complex numbers, helping us work with square roots of negative numbers. In mathematics, \(i\) is defined so that \(i = \sqrt{-1}\).
This allows us to solve equations that don't have real solutions. For instance, the equation \(x^2 + 1 = 0\) has no real solution because no real number squared equals -1.

By using \(i\), we can express the solutions as \(x = i\) and \(x = -i\). This expands our number system from just real numbers to what we call complex numbers, encompassing both real and imaginary parts.

• **Real part:** the usual numbers like -3, 0, 7.
• **Imaginary part:** involves \(i\), like \(5i\), \(-2i\).

Understanding \(i\)'s definition is fundamental to grasping more complex mathematical concepts.

Powers of i

The powers of the imaginary unit \(i\) follow a set pattern that simplifies calculations involving complex numbers. Starting with \(i^1 = i\), each successive power can be determined as follows:

• \(i^2 = -1\) because squaring \(i\) gives the square root of a negative number squared, which simplifies to just -1.
• \(i^3 = i^2 \times i = -1 \times i = -i\)
• \(i^4 = i^2 \times i^2 = (-1) \times (-1) = 1\)

After \(i^4 = 1\), the cycle restarts, so \(i^5 = i\) again, and so on.
Recognizing this pattern is incredibly helpful, enabling quick calculation of any power of \(i\) without repetitive multiplication. It also ensures that powers of \(i\) are always one of these four outcomes: \((i, -1, -i, 1)\).
This concept seamlessly ties into the topic of cyclic patterns in mathematics, illustrating how seemingly complex sequences can simplify.

Cyclic Patterns in Mathematics

Cyclic patterns vividly appear when examining the powers of the imaginary unit \(i\). As seen, the powers of \(i\) repeat their values in a consistent pattern: \(i, -1, -i, 1\). This pattern is called a cycle and consists of four distinct outcomes.
These cyclic patterns are relevant beyond complex numbers. They emerge in various areas, such as:

• **Modular arithmetic:** where numbers "wrap around" after reaching a certain value.
• **Trigonometry:** cycles appear in periodic functions like sine and cosine.
• **Computer science:** cyclic patterns are used in algorithms that repeat steps over time.

Recognizing and applying these patterns can simplify complex problem-solving and enhance prediction accuracy. In mathematics, understanding cyclic behavior allows us to make fast calculations and predictions, taking advantage of the natural repetitions present in many mathematical processes.