Select The Correct Answer.What Is The Value Of $i^n$ If The Remainder Of $\frac{n}{4}$ Is 2?A. $i$ B. $1$ C. $-i$ D. $-1$

Understanding the Value of $i^n$

When it comes to complex numbers, particularly those involving the imaginary unit $i$, understanding the pattern of their powers is crucial. The imaginary unit $i$ is defined as the square root of $-1$, denoted by $i = \sqrt{-1}$. This means that $i^2 = -1$. In this article, we will explore the value of $i^n$ when the remainder of $\frac{n}{4}$ is 2.

The Pattern of Powers of $i$

To begin with, let's examine the pattern of powers of $i$. We know that $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$. This pattern repeats every four powers of $i$. In other words, the powers of $i$ follow a cyclic pattern: $i, -1, -i, 1$.

The Remainder of $\frac{n}{4}$

Now, let's consider the condition that the remainder of $\frac{n}{4}$ is 2. This means that $n$ can be expressed as $4k + 2$, where $k$ is an integer. In other words, $n$ is of the form $2, 6, 10, 14, \ldots$.

Evaluating $i^n$

Using the pattern of powers of $i$, we can evaluate $i^n$ for $n = 2, 6, 10, 14, \ldots$. Since the remainder of $\frac{n}{4}$ is 2, we know that $i^n = i^2 = -1$.

Conclusion

In conclusion, when the remainder of $\frac{n}{4}$ is 2, the value of $i^n$ is $-1$. This is because the powers of $i$ follow a cyclic pattern, and $i^2 = -1$. Therefore, the correct answer is D. $-1$.

Example Use Cases

Here are a few example use cases where understanding the value of $i^n$ is crucial:

• Electrical Engineering: In electrical engineering, complex numbers are used to represent AC circuits. The value of $i^n$ is used to calculate the impedance of a circuit.
• Signal Processing: In signal processing, complex numbers are used to represent signals. The value of $i^n$ is used to filter out noise from a signal.
• Computer Graphics: In computer graphics, complex numbers are used to represent 3D transformations. The value of $i^n$ is used to rotate objects in 3D space.

Common Mistakes

Here are a few common mistakes to avoid when evaluating $i^n$:

• Not considering the remainder of $\frac{n}{4}$: Failing to consider the remainder of $\frac{n}{4}$ can lead to incorrect results.
• Not using the correct pattern of powers of $i$: Using the wrong pattern of powers of $i$ can lead to incorrect results.
• Not checking for special cases: Failing to check for special cases, such as $n = 0$, can lead to incorrect results.

Conclusion

In conclusion, understanding the value of $i^n$ is crucial in various fields, including electrical engineering, signal processing, and computer graphics. By considering the remainder of $\frac{n}{4}$ and using the correct pattern of powers of $i$, we can evaluate $i^n$ accurately.
Frequently Asked Questions (FAQs) about $i^n$

In this article, we will address some of the most frequently asked questions about $i^n$. Whether you are a student, a teacher, or a professional, these FAQs will help you understand the value of $i^n$ and its applications.

Q: What is the value of $i^n$ if the remainder of $\frac{n}{4}$ is 1?

A: If the remainder of $\frac{n}{4}$ is 1, then $i^n = i$. This is because the powers of $i$ follow a cyclic pattern, and $i^1 = i$.

Q: What is the value of $i^n$ if the remainder of $\frac{n}{4}$ is 3?

A: If the remainder of $\frac{n}{4}$ is 3, then $i^n = -i$. This is because the powers of $i$ follow a cyclic pattern, and $i^3 = -i$.

Q: What is the value of $i^n$ if the remainder of $\frac{n}{4}$ is 0?

A: If the remainder of $\frac{n}{4}$ is 0, then $i^n = 1$. This is because the powers of $i$ follow a cyclic pattern, and $i^4 = 1$.

Q: Can I use the value of $i^n$ to solve real-world problems?

A: Yes, the value of $i^n$ can be used to solve real-world problems in various fields, including electrical engineering, signal processing, and computer graphics.

Q: How do I calculate the value of $i^n$?

A: To calculate the value of $i^n$, you need to consider the remainder of $\frac{n}{4}$ and use the correct pattern of powers of $i$. You can use a calculator or a computer program to calculate the value of $i^n$.

Q: What are some common applications of $i^n$?

A: Some common applications of $i^n$ include:

• Electrical Engineering: In electrical engineering, complex numbers are used to represent AC circuits. The value of $i^n$ is used to calculate the impedance of a circuit.
• Signal Processing: In signal processing, complex numbers are used to represent signals. The value of $i^n$ is used to filter out noise from a signal.
• Computer Graphics: In computer graphics, complex numbers are used to represent 3D transformations. The value of $i^n$ is used to rotate objects in 3D space.

Q: Can I use the value of $i^n$ to solve problems in other fields?

A: Yes, the value of $i^n$ can be used to solve problems in other fields, including physics, engineering, and computer science.

Q: How do I determine the remainder of $\frac{n}{4}$?

A: To determine the remainder of $\frac{n}{4}$, you can use the modulo operator (%). For example, if $n = 10$, then the remainder of $\frac{n}{4}$ is $10 \mod 4 = 2$.

Q: What are some common mistakes to avoid when working with $i^n$?

A: Some common mistakes to avoid when working with $i^n$ include:

• Not considering the remainder of $\frac{n}{4}$: Failing to consider the remainder of $\frac{n}{4}$ can lead to incorrect results.
• Not using the correct pattern of powers of $i$: Using the wrong pattern of powers of $i$ can lead to incorrect results.
• Not checking for special cases: Failing to check for special cases, such as $n = 0$, can lead to incorrect results.

Conclusion

In conclusion, understanding the value of $i^n$ is crucial in various fields, including electrical engineering, signal processing, and computer graphics. By considering the remainder of $\frac{n}{4}$ and using the correct pattern of powers of $i$, we can evaluate $i^n$ accurately.