Problem 75 - Algebra A Combined Function

Question

Find each power of \(i\). $$ i^{11} $$

Step-by-Step Solution

Answer

The value of \(i^{11}\) is \(-i\).

Step 1: Understanding the Basics

Recall that the imaginary unit, represented by the symbol \(i\), is defined as \(i = \sqrt{-1}\). The powers of \(i\) cycle in a pattern: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). This cycle repeats every four powers. Therefore, to solve \(i^{11}\), we need to use this cyclical pattern.

Step 2: Determine the Cycle Position

The cycle of powers of \(i\) repeats every four terms, so we can find the position within the cycle by taking the power modulo 4. Calculate \(11 \mod 4\). Dividing 11 by 4 gives a quotient of 2 and a remainder of 3. Thus, \(i^{11}\) corresponds to the same value as \(i^3\).

Step 3: Identify the Value from the Cycle

Using the pattern of powers of \(i\), remember that \(i^3 = -i\). Since we've determined that \(i^{11}\) corresponds to \(i^3\) in the cycle, it follows that \(i^{11} = -i\).

Key Concepts

Imaginary Unit

In mathematics, the imaginary unit is a unique number denoted by the symbol \( i \). It is defined by the equation \( i = \sqrt{-1} \). Understanding the imaginary unit is crucial when dealing with complex numbers. A complex number is often expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers. The imaginary unit allows us to extend the concept of numbers beyond the real number line to include values that involve the square roots of negative numbers.
Here are a few key points about the imaginary unit:

• \( i^2 = -1 \) serves as a foundational identity. This is because the square of the imaginary unit is what transforms a real number into its imaginary form.
• As you work with powers of \( i \), these powers create a pattern that repeats in cycles. The first four values are: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \).
• This cyclical nature of \( i \) is essential to simplify computations involving powers of \( i \). By understanding how these powers cycle, calculations become much more manageable.

Cyclical Pattern

The cyclical pattern of powers of \( i \) is a fascinating and essential concept to understand when working with imaginary numbers. As mentioned, the powers of \( i \) repeat every four terms:

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

This cycle then begins again with \( i^5 \) equivalent to \( i^1 \), and so forth. Recognizing this repeating pattern simplifies the process of finding any power of \( i \). For instance, when tasked with finding \( i^{11} \), rather than multiplying \( i \) eleven times, you can use the cyclical pattern. This will help identify which term in the cycle corresponds to the power in question. By using the calculation \( 11 \mod 4 \), you determine that \( i^{11} \) is equivalent to \( i^3 \), which is \(-i\).

Modular Arithmetic

Modular arithmetic is a handy mathematical tool that helps determine the equivalence of numbers within a cyclical range, often referred to as 'clock arithmetic'. It becomes especially useful when working with powers of imaginary numbers, such as the imaginary unit \( i \). The key operation in modular arithmetic is finding the remainder of a division.
In the context of powers of \( i \), modular arithmetic allows us to simplify computations. Since the powers of \( i \) repeat every four terms, finding a power such as \( i^{11} \) uses modular arithmetic as follows: Calculate \( 11 \mod 4 \). Divide 11 by 4, which gives a quotient (which we don’t actually use) and a remainder of 3. This remainder indicates where we are within the cyclical pattern, thus allowing us to determine the equivalent power of \( i \) corresponding to that point in the cycle.
Using modular arithmetic in this way reduces what might initially seem like a complex problem into a straightforward one.