%5E2017.jpeg "(1+i)^2017")
Exploring the Power of Complex Numbers: (1 + i)^2017
The expression (1 + i)^2017, where 'i' is the imaginary unit (√-1), presents a fascinating challenge in complex number arithmetic. While it might seem daunting at first glance, we can unravel its complexity through the power of De Moivre's Theorem and the cyclical nature of complex numbers.
De Moivre's Theorem: A Powerful Tool
De Moivre's Theorem states that for any complex number in polar form (r(cos θ + i sin θ)) and any integer n:
(r(cos θ + i sin θ))^n = r^n(cos nθ + i sin nθ)
This theorem provides a concise and elegant way to calculate powers of complex numbers.
Transforming (1 + i) into Polar Form
To apply De Moivre's Theorem, we need to express (1 + i) in polar form.
- Magnitude: The magnitude (r) of (1 + i) is √(1^2 + 1^2) = √2.
- Angle: The angle (θ) 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).
Applying De Moivre's Theorem
Now, we can apply De Moivre's Theorem to calculate (1 + i)^2017:
(√2(cos π/4 + i sin π/4))^2017 = (√2)^2017 (cos (2017π/4) + i sin (2017π/4))
Understanding the Cyclic Nature
The angle 2017π/4 is a multiple of π/4, meaning it will repeat the same values of sine and cosine as smaller multiples. We can simplify it by finding its remainder when divided by 2π (a full circle).
2017π/4 = 504π + π/4
The integer multiple of 2π (504π) doesn't affect the sine and cosine values, so we only need to consider the remainder π/4.
Therefore:
(√2)^2017 (cos (2017π/4) + i sin (2017π/4)) = (√2)^2017 (cos π/4 + i sin π/4)
Final Result
The final result of (1 + i)^2017 is:
(√2)^2017 (cos π/4 + i sin π/4) = 2^1008.5 (1/√2 + i/√2) = 2^1008 (1 + i)
We have successfully calculated the power of the complex number (1 + i)^2017 using De Moivre's Theorem and understanding the cyclic nature of complex numbers.