%5E1%2F3.jpeg "(1+i)^1/3")
Exploring the Cube Root of (1 + i)
The expression (1 + i)^(1/3) represents the cube root of the complex number (1 + i). This exploration will delve into finding the solutions to this expression and understanding their geometric interpretation.
Finding the Cube Roots
To find the cube roots of (1 + i), we can use the following steps:
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Convert to polar form:
- Find the magnitude (modulus) of (1 + i): |1 + i| = √(1² + 1²) = √2
- Find the angle (argument) of (1 + i): θ = arctan(1/1) = π/4
- Therefore, (1 + i) = √2 * (cos(π/4) + i sin(π/4))
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Apply De Moivre's Theorem:
- De Moivre's Theorem states that for any complex number in polar form, z = r(cosθ + i sinθ), and any integer n: z^n = r^n (cos(nθ) + i sin(nθ))
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Calculate the cube root:
- We need to find z^(1/3), where n = 1/3.
- Applying De Moivre's Theorem: z^(1/3) = (√2)^(1/3) * (cos((π/4 + 2πk)/3) + i sin((π/4 + 2πk)/3))
- Where k = 0, 1, 2 to obtain the three distinct cube roots.
The Three Cube Roots
- k = 0:
- z^(1/3) = (√2)^(1/3) * (cos(π/12) + i sin(π/12))
- k = 1:
- z^(1/3) = (√2)^(1/3) * (cos(9π/12) + i sin(9π/12)) = (√2)^(1/3) * (cos(3π/4) + i sin(3π/4))
- k = 2:
- z^(1/3) = (√2)^(1/3) * (cos(17π/12) + i sin(17π/12))
Geometric Interpretation
The three cube roots of (1 + i) are located on a circle with radius (√2)^(1/3) in the complex plane. The roots are evenly spaced at 120 degrees from each other. This follows a general principle that for any complex number, its nth roots will be equally spaced on a circle in the complex plane.
Conclusion
Finding the cube roots of (1 + i) involves utilizing De Moivre's Theorem and understanding the geometric representation of complex numbers. This process reveals three distinct roots, which are evenly distributed on a circle in the complex plane. This concept extends to finding the nth roots of any complex number, providing valuable insights into the properties of complex numbers and their geometric relationships.