%5E120.jpeg "(1+i)^120")
Exploring the Power of Complex Numbers: (1 + i)^120
In the realm of mathematics, complex numbers hold a fascinating power, allowing us to explore intricate relationships and delve into the depths of mathematical structures. One such intriguing problem involves calculating the value of (1 + i)^120, where 'i' represents the imaginary unit, defined as the square root of -1.
Let's embark on a journey to decipher this complex expression.
De Moivre's Theorem: A Key to Unlocking the Power
To tackle this problem, we can leverage a powerful tool in complex number analysis: De Moivre's Theorem. This theorem states that for any complex number in polar form, represented as z = r(cosθ + isinθ), its nth power can be calculated as:
z^n = r^n (cos(nθ) + isin(nθ))
Applying De Moivre's Theorem to (1 + i)
Firstly, we need to express (1 + i) in polar form. To do this, we can visualize (1 + i) in the complex plane. It lies in the first quadrant with a magnitude of √2 and an angle of 45 degrees (π/4 radians).
Therefore, (1 + i) can be expressed in polar form as:
(1 + i) = √2 (cos(π/4) + isin(π/4))
Now, applying De Moivre's theorem, we get:
(1 + i)^120 = (√2)^120 (cos(120π/4) + isin(120π/4))
Simplifying the expression:
(1 + i)^120 = 2^60 (cos(30π) + isin(30π))
Since cos(30π) = 1 and sin(30π) = 0, we finally arrive at:
(1 + i)^120 = 2^60
Conclusion: A Striking Result
Therefore, the value of (1 + i)^120 is a remarkably large real number, 2^60. This result highlights the incredible power of complex numbers and the elegance of De Moivre's theorem in simplifying complex expressions.
By understanding the fundamental principles of complex numbers and applying appropriate mathematical tools, we can uncover hidden patterns and unlock the intricacies of the mathematical world.