22 Find The Least Non-zero Integral Value Of K For Which (1+i/1-i)^k= I
Introduction
In this article, we will delve into the world of complex numbers and explore the concept of raising a complex number to a power. We will specifically focus on finding the least non-zero integral value of k for which the expression (1+i/1-i)^k equals i. This problem requires a deep understanding of complex numbers, their properties, and the rules of exponents.
Understanding Complex Numbers
Before we proceed, let's briefly review the basics of complex numbers. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies the equation i^2 = -1. The complex number a + bi can be represented graphically on a complex plane, with the real part a on the x-axis and the imaginary part b on the y-axis.
Simplifying the Expression
To simplify the expression (1+i/1-i)^k, we can start by rationalizing the denominator. We can do this by multiplying the numerator and denominator by the conjugate of the denominator, which is 1+i.
(1+i/1-i) = (1+i)(1+i) / (1-i)(1+i)
Using the difference of squares formula, we can simplify the expression further:
(1+i/1-i) = (1+2i+i^2) / (1-i^2)
Since i^2 = -1, we can substitute this value into the expression:
(1+i/1-i) = (1+2i-1) / (1+1)
Simplifying the numerator, we get:
(1+i/1-i) = 2i / 2
Which simplifies to:
(1+i/1-i) = i
Finding the Least Non-Zero Integral Value of k
Now that we have simplified the expression, we can set it equal to i and solve for k:
(i)^k = i
Since i^2 = -1, we can raise both sides of the equation to the power of 2:
(i^2)^k = i^2
Which simplifies to:
(-1)^k = -1
This equation is true when k is an odd integer. Therefore, the least non-zero integral value of k for which (1+i/1-i)^k = i is k = 1.
Conclusion
In this article, we have explored the concept of raising a complex number to a power and found the least non-zero integral value of k for which the expression (1+i/1-i)^k equals i. We have simplified the expression using the rules of exponents and the properties of complex numbers. Our final answer is k = 1, which is the least non-zero integral value of k that satisfies the given equation.
Additional Examples
Here are a few additional examples of complex numbers raised to a power:
- (1+i)^2 = 2i
- (1+i)^3 = -2+2i
- (1+i)^4 = -4
These examples demonstrate the properties of complex numbers and the rules of exponents.
Applications of Complex Numbers
Complex numbers have numerous applications in mathematics, physics, and engineering. Some of the key applications include:
- Electrical Engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
- Signal Processing: Complex numbers are used to represent signals and analyze their frequency content.
- Control Systems: Complex numbers are used to analyze the stability and behavior of control systems.
- Quantum Mechanics: Complex numbers are used to represent wave functions and analyze the behavior of particles.
These applications demonstrate the importance and relevance of complex numbers in various fields.
Final Thoughts
In conclusion, complex numbers are a fundamental concept in mathematics, and their properties and rules of exponents are essential for solving problems involving complex numbers. The least non-zero integral value of k for which (1+i/1-i)^k = i is k = 1, which is a critical result in the field of complex analysis. We hope that this article has provided a comprehensive overview of the topic and has inspired readers to explore the world of complex numbers further.
Introduction
In our previous article, we explored the concept of raising a complex number to a power and found the least non-zero integral value of k for which the expression (1+i/1-i)^k equals i. In this article, we will provide a Q&A section to address some of the common questions and doubts that readers may have.
Q: What is the significance of the expression (1+i/1-i)^k?
A: The expression (1+i/1-i)^k is a complex number raised to a power. It is a fundamental concept in mathematics, and its properties and rules of exponents are essential for solving problems involving complex numbers.
Q: How do you simplify the expression (1+i/1-i)?
A: To simplify the expression (1+i/1-i), we can start by rationalizing the denominator. We can do this by multiplying the numerator and denominator by the conjugate of the denominator, which is 1+i.
Q: What is the final simplified form of the expression (1+i/1-i)?
A: The final simplified form of the expression (1+i/1-i) is i.
Q: How do you find the least non-zero integral value of k for which (1+i/1-i)^k = i?
A: To find the least non-zero integral value of k for which (1+i/1-i)^k = i, we can set the expression equal to i and solve for k. Since i^2 = -1, we can raise both sides of the equation to the power of 2 and simplify the expression.
Q: What is the least non-zero integral value of k for which (1+i/1-i)^k = i?
A: The least non-zero integral value of k for which (1+i/1-i)^k = i is k = 1.
Q: What are some of the applications of complex numbers?
A: Complex numbers have numerous applications in mathematics, physics, and engineering. Some of the key applications include electrical engineering, signal processing, control systems, and quantum mechanics.
Q: Why are complex numbers important in electrical engineering?
A: Complex numbers are used to represent AC circuits and analyze their behavior. They are essential for designing and analyzing electrical circuits, including filters, amplifiers, and oscillators.
Q: How do complex numbers relate to signal processing?
A: Complex numbers are used to represent signals and analyze their frequency content. They are essential for designing and analyzing filters, including low-pass, high-pass, and band-pass filters.
Q: What is the significance of complex numbers in control systems?
A: Complex numbers are used to analyze the stability and behavior of control systems. They are essential for designing and analyzing control systems, including feedback control systems and state-space control systems.
Q: How do complex numbers relate to quantum mechanics?
A: Complex numbers are used to represent wave functions and analyze the behavior of particles. They are essential for understanding the principles of quantum mechanics, including wave-particle duality and superposition.
Q: What are some of the key properties of complex numbers?
A: Some of the key properties of complex numbers include:
- Addition: Complex numbers can be added using the formula (a + bi) + (c + di) = (a + c) + (b + d)i.
- Subtraction: Complex numbers can be subtracted using the formula (a + bi) - (c + di) = (a - c) + (b - d)i.
- Multiplication: Complex numbers can be multiplied using the formula (a + bi)(c + di) = (ac - bd) + (ad + bc)i.
- Division: Complex numbers can be divided using the formula (a + bi)/(c + di) = ((ac + bd)/(c^2 + d^2)) + ((bc - ad)/(c^2 + d^2))i.
Q: What are some of the key rules of exponents for complex numbers?
A: Some of the key rules of exponents for complex numbers include:
- Power Rule: (a + bi)^n = (a^2 + b2)((n-1)/2) + (ab)^((n-1)/2)i
- Product Rule: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
- Quotient Rule: (a + bi)/(c + di) = ((ac + bd)/(c^2 + d^2)) + ((bc - ad)/(c^2 + d^2))i
Q: What are some of the key applications of complex numbers in mathematics?
A: Complex numbers have numerous applications in mathematics, including:
- Algebra: Complex numbers are used to solve polynomial equations and analyze the behavior of functions.
- Geometry: Complex numbers are used to represent geometric shapes and analyze their properties.
- Analysis: Complex numbers are used to analyze the behavior of functions and sequences.
Q: What are some of the key applications of complex numbers in physics?
A: Complex numbers have numerous applications in physics, including:
- Electromagnetism: Complex numbers are used to represent electromagnetic waves and analyze their behavior.
- Quantum Mechanics: Complex numbers are used to represent wave functions and analyze the behavior of particles.
- Relativity: Complex numbers are used to represent spacetime and analyze the behavior of objects in curved spacetime.
Q: What are some of the key applications of complex numbers in engineering?
A: Complex numbers have numerous applications in engineering, including:
- Electrical Engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
- Signal Processing: Complex numbers are used to represent signals and analyze their frequency content.
- Control Systems: Complex numbers are used to analyze the stability and behavior of control systems.
Conclusion
In this Q&A article, we have addressed some of the common questions and doubts that readers may have about the concept of raising a complex number to a power and finding the least non-zero integral value of k for which the expression (1+i/1-i)^k equals i. We hope that this article has provided a comprehensive overview of the topic and has inspired readers to explore the world of complex numbers further.