%5E12.jpeg "(1+i)^12")
Exploring (1 + i)^12: A Journey into Complex Numbers
The expression (1 + i)^12 might seem daunting at first glance, but it offers a fascinating exploration into the world of complex numbers. Let's break down how to calculate this and uncover its intriguing properties.
Understanding Complex Numbers
Complex numbers are numbers that extend the real number system by including the imaginary unit "i," where i^2 = -1. A complex number is generally written in the form a + bi, where 'a' and 'b' are real numbers.
De Moivre's Theorem: Our Key
To simplify (1 + i)^12, we'll employ De Moivre's Theorem. This powerful 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))
Transforming (1 + i) into Polar Form
-
Magnitude (r): The magnitude of (1 + i) is found using the Pythagorean theorem: √(1^2 + 1^2) = √2.
-
Angle (θ): The angle θ is found using the arctangent function: arctan(1/1) = π/4 (or 45 degrees).
Therefore, (1 + i) in polar form is √2(cos(π/4) + i sin(π/4)).
Applying De Moivre's Theorem
Now, let's apply De Moivre's Theorem to (1 + i)^12:
(√2(cos(π/4) + i sin(π/4)))^12 = (√2)^12 (cos(12(π/4)) + i sin(12(π/4)))
This simplifies to: 2^6 (cos(3π) + i sin(3π))
Final Result
Simplifying further, we get:
2^6(-1 + i * 0) = -64
Therefore, (1 + i)^12 = -64.
A Surprising Outcome
It's remarkable how raising a complex number to a seemingly high power leads to a simple, real number result. This demonstrates the elegance and power of De Moivre's Theorem in simplifying complex number calculations.