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Exploring the Power of Complex Numbers: (1+i)^5
In the realm of mathematics, complex numbers often present intriguing challenges, and the expression (1+i)^5 is no exception. This article delves into the process of calculating this power and explores some key concepts related to complex numbers.
Understanding Complex Numbers
Before tackling the power, let's understand the basics. A complex number is expressed in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit, defined as √-1.
De Moivre's Theorem: A Powerful Tool
De Moivre's theorem provides a elegant solution for calculating powers of complex numbers in polar form. It states that:
(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)
To apply this theorem, we need to convert (1+i) from rectangular form (a + bi) to polar form (r(cos θ + i sin θ)).
Converting to Polar Form
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Find the magnitude (r): r = |1+i| = √(1^2 + 1^2) = √2
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Find the angle (θ): θ = arctan(b/a) = arctan(1/1) = π/4 radians (or 45 degrees)
Therefore, (1+i) in polar form is √2(cos(π/4) + i sin(π/4)).
Applying De Moivre's Theorem
Now, we can use De Moivre's theorem to calculate (1+i)^5:
(1+i)^5 = [√2(cos(π/4) + i sin(π/4))]^5 = (√2)^5 * (cos(5π/4) + i sin(5π/4)) = 4√2 * (-√2/2 - i√2/2) = -4 - 4i
Conclusion
By leveraging De Moivre's theorem and understanding the conversion between rectangular and polar forms, we've successfully calculated (1+i)^5. This process highlights the power of complex numbers and the elegance of mathematical tools used to manipulate them.