Problem 72 - Algebra A Combined Function

Question

Find each power of \(i\). $$ i^{10} $$

Step-by-Step Solution

Answer

The value of \( i^{10} \) is \(-1\).

Step 1: Understand Powers of i

The imaginary unit, represented by \( i \), has a cyclical pattern when raised to different powers. It follows this cycle: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). After \( i^4 \), the cycle repeats. Recognizing this pattern helps to simplify powers of \( i \) beyond \( i^4 \).

Step 2: Simplify the Power Exponent

To simplify \( i^{10} \), note that any power of \( i \) can be reduced based on its congruence modulo 4. Calculate \( 10 \mod 4 \), which is equal to 2. Therefore, \( i^{10} \) simplifies to \( i^2 \).

Step 3: Compute the Result Using the Cycle

Use the cyclic pattern established in Step 1. Since \( i^2 = -1 \), plug that value to find \( i^{10} = i^2 = -1 \). Thus, the power of \( i \) evaluated at 10 results in \(-1\).

Key Concepts

Imaginary Unit

The imaginary unit, commonly denoted by the symbol \(i\), is a fundamental concept in complex numbers. It is defined in such a way that \(i^2 = -1\). This definition seems simple but opens up a new dimension for solving equations that don't have solutions in real numbers, like \(x^2 = -1\). Because no real number squared gives a negative result, mathematicians introduced \(i\) for this purpose.

Some key points about \(i\):

• \(i^0 = 1\) because any number raised to the power of zero is one.
• \(i^1 = i\), by definition.
• \(i^2 = -1\), as intended by its definition to solve equations like \(x^2 + 1 = 0\).
• \(i^3 = -i\), derived from multiplying \(i^2\) by \(i\).
• \(i^4 = 1\), which completes the cycle since \(i^4 = i^2 \times i^2 = (-1) \times (-1) = 1\).

This cyclical nature makes it easy to handle powers of \(i\), especially when simplifying complex expressions.

Modular Arithmetic

Modular arithmetic is a fundamental tool in number theory, often referred to as "clock" arithmetic because it deals with numbers wrapping around after reaching a certain value—the modulus. In dealing with powers of the imaginary unit \(i\), modular arithmetic helps in simplifying complex computations by reducing the powers of \(i\) using the modulus of 4.

Here's why it's useful:

• When you compute powers like \(i^{10}\), you can simplify it by finding \(10 \mod 4\). This equates to 2, so \(i^{10} = i^2\).
• The cycle of powers of \(i\) is 4, i.e., after every 4 terms, the sequence repeats. So finding the remainder when dividing by 4 simplifies the calculation.
• This cyclical pattern means you only need to compute powers \(i^0\) to \(i^3\) to find any power of \(i\), based on the remainder of the exponent mod 4.

Modular arithmetic thus clarifies the cyclic repetition inherent in powers of \(i\) and can simplify complex arithmetic significantly.

Complex Numbers

Complex numbers are an extension of the real number system, represented as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. They vastly enlarge the possibilities for solving equations and modeling phenomena beyond real numbers.

Characteristics of complex numbers include:

• The real part, \(a\), which is a standard real number.
• The imaginary part, \(bi\), which incorporates the imaginary unit \(i\).
• They can be visualized on a plane with the real part on the x-axis and the imaginary part on the y-axis.
• Operations such as addition, subtraction, multiplication, and division of complex numbers follow algebraic rules similar to real numbers, with attention to \(i\)'s properties.
• They are particularly useful in fields like engineering, physics, and applied mathematics because they can represent scenarios where real numbers alone aren't sufficient.

Understanding complex numbers involves comprehending the imaginary unit \(i\) and incorporating it into larger numeric concepts, allowing for solutions that would otherwise remain unexplored.