- #1
juliany
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Homework Statement
Find the five values of (1+i√3)^(3/5)
This question was from my recent end of year exam, I hadn't come across a question like it in my revision, does it mean find the five roots of (1+i√3)^(3/5) ?
What are the five values of (1+i√3)^(3/5)?
- Thread starter juliany
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In summary, the conversation is about finding the five 5th roots of (1+i√3)^3 or solving the equation X^5 = (1+i√3)^3 = -8. One method suggested is to apply D'Moivre's theorem with n=3/5. It is also noted that the third power of (1+i√3) is equal to -8, which is a real number. Some participants in the conversation discuss the possibility of 1+i√3 being equal to -2.
Find the five values of (1+i√3)^(3/5)
This question was from my recent end of year exam, I hadn't come across a question like it in my revision, does it mean find the five roots of (1+i√3)^(3/5) ?
- #2
vela
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You want to find the five 5th roots of (1+i√3)3.
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you want to find all the solutions of the equation X5 = (1 + i√3)3 = -8.
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HallsofIvy
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It's remarkable that the third power is equal to -8!
Yes, you can find the five fifth roots of [itex](1+i\sqrt{3})^3[/itex] or just apply D'Moivre's theorem to [itex]1+ i\sqrt{3}[/itex] with n= 3/5.
Yes, you can find the five fifth roots of [itex](1+i\sqrt{3})^3[/itex] or just apply D'Moivre's theorem to [itex]1+ i\sqrt{3}[/itex] with n= 3/5.
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why?
- #6
Mentallic
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Because it's a real number. (-2)3=-8 as well so some might be reluctant enough to say that [tex]1+i\sqrt{3}=-2[/tex] :tongue:
1. What are complex numbers and how do they differ from real numbers?
Complex numbers are numbers that contain both a real part and an imaginary part. The imaginary part is represented by the letter "i", which stands for the square root of -1. Real numbers, on the other hand, do not have an imaginary component. In other words, complex numbers are an extension of real numbers that allow for the representation of numbers on a two-dimensional plane.
2. How do you perform arithmetic operations with complex numbers?
To add or subtract complex numbers, you simply combine the real parts and the imaginary parts separately. For multiplication, you use the FOIL method, where you multiply the first terms, then the outer terms, then the inner terms, and finally the last terms. To divide complex numbers, you use the conjugate of the denominator to eliminate the imaginary part and then simplify the expression.
3. What is the geometric interpretation of complex numbers?
Complex numbers can be represented on a two-dimensional plane called the complex plane. The real part of a complex number is plotted on the x-axis, and the imaginary part is plotted on the y-axis. This allows for the visualization of complex numbers as points on a plane, making it easier to understand operations and relationships between them.
4. How are complex numbers used in real-world applications?
Complex numbers have various applications in fields such as engineering, physics, and economics. They are used to model electrical circuits, analyze waveforms, and predict stock market trends, among many other things. In essence, complex numbers provide a powerful tool for representing and solving problems in multiple dimensions.
5. Can complex numbers have a real-world meaning?
While the concept of imaginary numbers may seem abstract, they do have real-world meaning and applications. For example, the square root of -1 can represent the phase difference between two alternating currents in an electrical circuit. Complex numbers can also be used to describe the motion of objects in two dimensions, making them a useful tool in physics and engineering.
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