Finding the root of a complex number generally involves converting the complex number to polar form, applying De Moivre's theorem, and then converting back to rectangular form if needed. Let's break down the process:
Steps to Find the Root of a Complex Number
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Convert to Polar Form: Express the complex number, say z = a + bi, in polar form z = r(cos θ + i sin θ), where:
- r is the modulus (or absolute value) of z, calculated as r = √(a² + b²).
- θ is the argument of z, calculated as θ = arctan(b/a). Be mindful of the quadrant in which a + bi lies to get the correct angle. You might need to add π or 2π to the arctangent result.
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Apply De Moivre's Theorem: To find the n-th root of z, we use the formula:
- z(1/n) = r(1/n) [cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)], where k = 0, 1, 2, ..., n-1. This formula provides n distinct roots.
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Calculate the Roots: Substitute the values of r, θ, n, and each value of k (from 0 to n-1) into the formula to find the n distinct roots.
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Convert Back to Rectangular Form (Optional): If desired, convert each root back to rectangular form x + yi using the formulas:
- x = r(1/n) cos((θ + 2πk)/n)
- y = r(1/n) sin((θ + 2πk)/n)
Example: Finding the Square Root of 1 + i
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Convert to Polar Form:
- r = √(1² + 1²) = √2
- θ = arctan(1/1) = π/4
- So, 1 + i = √2 (cos(π/4) + i sin(π/4))
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Apply De Moivre's Theorem: We want the square root, so n = 2.
- (1 + i)(1/2) = (√2)(1/2) [cos((π/4 + 2πk)/2) + i sin((π/4 + 2πk)/2)]
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Calculate the Roots:
- For k = 0:
- (√2)(1/2) [cos(π/8) + i sin(π/8)] ≈ 1.099 + 0.455i
- For k = 1:
- (√2)(1/2) [cos(9π/8) + i sin(9π/8)] ≈ -1.099 - 0.455i
- For k = 0:
Therefore, the square roots of 1 + i are approximately 1.099 + 0.455i and -1.099 - 0.455i.
Summary
Finding the root of a complex number involves converting it to polar form, using De Moivre's theorem to extract the roots, and optionally converting back to rectangular form. The process yields n distinct roots for the n-th root of the complex number.
