%5E36.jpeg "(1+i)^36")
Exploring the Power of Complex Numbers: (1 + i)^36
In the realm of mathematics, complex numbers offer a rich tapestry of fascinating properties. One such intriguing problem involves calculating the value of (1 + i)^36. Let's delve into the intricacies of this expression and unravel its secrets.
Understanding Complex Numbers
Complex numbers are numbers that extend the real number system by incorporating the imaginary unit "i," where i^2 = -1. They are typically expressed in the form a + bi, where a and b are real numbers.
De Moivre's Theorem: A Powerful Tool
To tackle (1 + i)^36, we employ De Moivre's Theorem, a fundamental principle in complex number theory. This 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)
Converting to Polar Form
First, we need to express (1 + i) in polar form. This involves finding its magnitude (r) and argument (θ).
- Magnitude (r): r = √(1² + 1²) = √2
- Argument (θ): θ = tan⁻¹(1/1) = π/4 (since (1 + i) lies in the first quadrant)
Therefore, (1 + i) = √2(cos π/4 + i sin π/4).
Applying De Moivre's Theorem
Now, we can apply De Moivre's theorem:
(1 + i)^36 = (√2(cos π/4 + i sin π/4))^36
= (√2)^36 (cos (36 * π/4) + i sin (36 * π/4))
= 2^18 (cos 9π + i sin 9π)
= 2^18 (-1 + 0i)
= -2^18
The Final Result
Therefore, (1 + i)^36 equals -2^18. This seemingly complex calculation was made possible by understanding complex numbers, utilizing De Moivre's theorem, and performing some trigonometric manipulations.
This exploration highlights the power and elegance of complex number theory, revealing hidden patterns and simplifying intricate expressions.