%5E18.jpeg "(1+i)^18")
Demystifying (1 + i)^18: A Journey Through Complex Numbers
The expression (1 + i)^18 might seem daunting at first glance, but with the right tools and understanding, it becomes surprisingly manageable. Let's embark on this journey to unravel the mystery.
Understanding Complex Numbers
At the heart of this problem lies the concept of complex numbers. Complex numbers extend the real number system by introducing the imaginary unit, i, defined as the square root of -1. A complex number takes the form a + bi, where 'a' and 'b' are real numbers.
Polar Form: A Powerful Tool
To efficiently handle powers of complex numbers, we often utilize their polar form. This form expresses a complex number in terms of its magnitude (r) and angle (θ). The conversion from rectangular form (a + bi) to polar form (r(cos θ + i sin θ)) is achieved using:
- r = √(a² + b²)
- θ = arctan(b/a)
Applying the Power of De Moivre's Theorem
De Moivre's Theorem provides a shortcut for calculating powers of complex numbers in polar form. It states:
[r(cos θ + i sin θ)]^n = r^n (cos (nθ) + i sin (nθ))
This elegant theorem simplifies the process immensely!
Solving for (1 + i)^18
Let's apply our newfound knowledge to (1 + i)^18:
-
Convert to Polar Form:
- r = √(1² + 1²) = √2
- θ = arctan(1/1) = π/4
-
Apply De Moivre's Theorem:
- (1 + i)^18 = (√2(cos π/4 + i sin π/4))^18
- = (√2)^18 (cos (18 * π/4) + i sin (18 * π/4))
-
Simplify:
- = 2^9 (cos (9π/2) + i sin (9π/2))
- = 512 (cos (π/2) + i sin (π/2)) (since 9π/2 is coterminal with π/2)
- = 512 (0 + i)
- = 512i
Conclusion
By leveraging the power of complex number representation and De Moivre's Theorem, we've successfully simplified (1 + i)^18 to 512i. This journey highlights the elegance and efficiency of working with complex numbers in their polar form. Understanding these concepts opens doors to tackling more complex calculations and exploring the fascinating world of complex analysis.