What Is The Value Of $i^{97} - I$?A. − I -i − I B. 0C. − 2 I -2i − 2 I D. Β 96 \beta^{96} Β 96
Introduction to Complex Numbers
Complex numbers are mathematical expressions that consist of a real number part and an imaginary number part. The imaginary unit, denoted by i, is defined as the square root of -1. In other words, i is a number that, when multiplied by itself, gives -1. This concept is crucial in mathematics, particularly in algebra, geometry, and calculus.
Properties of the Imaginary Unit
The imaginary unit i has several important properties that are essential to understanding complex numbers. One of the most significant properties is that i is a root of the equation x^2 + 1 = 0. This means that i satisfies the equation i^2 + 1 = 0, where i^2 is equal to -1.
Simplifying the Expression $i^{97} - i$
To simplify the expression $i^{97} - i$, we need to understand the pattern of powers of i. When we raise i to different powers, we get a repeating cycle of four values: i, -1, -i, and 1. This cycle repeats every four powers of i.
Understanding the Pattern of Powers of i
Let's examine the pattern of powers of i:
- i^1 = i
- i^2 = -1
- i^3 = -i
- i^4 = 1
- i^5 = i (since i^4 * i = 1 * i = i)
- i^6 = -1 (since i^4 * i^2 = 1 * -1 = -1)
- i^7 = -i (since i^4 * i^3 = 1 * -i = -i)
- i^8 = 1 (since i^4 * i^4 = 1 * 1 = 1)
As we can see, the pattern of powers of i repeats every four powers.
Applying the Pattern to $i^{97} - i$
Now that we understand the pattern of powers of i, we can apply it to the expression $i^{97} - i$. Since 97 is not a multiple of 4, we need to find the remainder when 97 is divided by 4. The remainder is 1, which means that i^97 is equal to i^1, which is i.
Simplifying the Expression
Now that we know that i^97 is equal to i, we can simplify the expression $i^{97} - i$:
Conclusion
In conclusion, the value of $i^{97} - i$ is 0. This is because i^97 is equal to i, and when we subtract i from i, we get 0.
Final Answer
The final answer is B. 0.
Introduction
In our previous article, we explored the value of $i^{97} - i$ and concluded that it is equal to 0. However, we understand that there may be some questions and doubts regarding this concept. In this article, we will address some of the frequently asked questions related to the value of $i^{97} - i$.
Q: What is the significance of the imaginary unit i?
A: The imaginary unit i is a fundamental concept in mathematics, particularly in algebra and geometry. It is defined as the square root of -1, and it is used to extend the real number system to the complex number system.
Q: How does the pattern of powers of i help in simplifying the expression $i^{97} - i$?
A: The pattern of powers of i is crucial in simplifying the expression $i^{97} - i$. Since 97 is not a multiple of 4, we need to find the remainder when 97 is divided by 4. The remainder is 1, which means that i^97 is equal to i^1, which is i.
Q: Why does the expression $i^{97} - i$ equal 0?
A: The expression $i^{97} - i$ equals 0 because i^97 is equal to i, and when we subtract i from i, we get 0. This is due to the properties of the imaginary unit i and the pattern of its powers.
Q: Can you provide more examples of simplifying expressions involving powers of i?
A: Yes, here are a few examples:
- i^3 - i = -i - i = -2i
- i^5 + i = i + i = 2i
- i^7 - i^3 = -i - (-i) = 0
Q: How does the concept of complex numbers relate to the value of $i^{97} - i$?
A: Complex numbers are mathematical expressions that consist of a real number part and an imaginary number part. The imaginary unit i is a fundamental component of complex numbers, and its properties are essential in understanding the value of $i^{97} - i$.
Q: Can you provide more information on the properties of the imaginary unit i?
A: Yes, here are some key properties of the imaginary unit i:
- i^2 = -1
- i^3 = -i
- i^4 = 1
- i^5 = i
- i^6 = -1
- i^7 = -i
- i^8 = 1
Conclusion
In conclusion, the value of $i^{97} - i$ is 0, and it is due to the properties of the imaginary unit i and the pattern of its powers. We hope that this Q&A article has provided a better understanding of this concept and has addressed some of the frequently asked questions related to it.
Final Answer
The final answer is B. 0.