%5E5.jpeg "(1-i)^5")
Exploring the Power of Complex Numbers: (1-i)^5
This article delves into the intriguing world of complex numbers, specifically investigating the expansion of (1 - i)^5. We'll utilize De Moivre's Theorem and explore the fascinating patterns that emerge.
Understanding Complex Numbers
Complex numbers are a powerful extension of the real number system, encompassing numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1. This seemingly simple extension opens up a whole new world of mathematical possibilities.
De Moivre's Theorem: The Key to Complex Powers
De Moivre's Theorem provides a powerful tool for calculating powers of complex numbers expressed in polar form. It states that for any complex number z = r(cos θ + i sin θ) and any integer n,
z^n = r^n (cos nθ + i sin nθ).
Applying De Moivre's Theorem to (1-i)^5
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Polar Form: First, we need to express (1-i) in polar form. The magnitude r is found using the Pythagorean theorem: r = √(1^2 + (-1)^2) = √2.
The angle θ is determined by considering the location of (1-i) in the complex plane. It lies in the fourth quadrant, and its angle with the positive real axis can be found using the arctangent function: θ = arctan(-1/1) = -π/4.
Therefore, (1-i) in polar form is: √2 (cos(-π/4) + i sin(-π/4))
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Applying De Moivre's Theorem: Now, we raise this polar form to the power of 5 using De Moivre's Theorem:
(1 - i)^5 = [√2 (cos(-π/4) + i sin(-π/4))]^5 = (√2)^5 (cos(-5π/4) + i sin(-5π/4))
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Simplifying: Finally, we simplify the result:
(1 - i)^5 = 4√2 (cos(3π/4) + i sin(3π/4)) = 4√2 (-√2/2 + i √2/2) = -4 + 4i
Conclusion
Through the use of De Moivre's Theorem, we have successfully determined that (1-i)^5 = -4 + 4i. This exploration demonstrates the elegant power of complex numbers and how De Moivre's Theorem simplifies calculations involving complex powers. It also highlights the captivating interplay between geometry and algebra in the complex plane.