%5E2024.jpeg "(1+i)^2024")
Exploring the Complex Power: (1 + i)^2024
The expression (1 + i)^2024 presents an intriguing challenge in complex number arithmetic. Let's break down the process of finding its value and uncover some fascinating patterns along the way.
Understanding Complex Numbers
Before diving into the calculation, it's essential to remember that complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1. This means i² = -1.
Using De Moivre's Theorem
A powerful tool for calculating powers of complex numbers is De Moivre's Theorem. It states that for any complex number in polar form, z = r(cos θ + i sin θ), and any integer n, the following holds true:
z^n = r^n(cos nθ + i sin nθ)
To apply this theorem, we need to convert our complex number (1 + i) into polar form.
Converting to Polar Form
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Find the magnitude (r): The magnitude of (1 + i) is √(1² + 1²) = √2.
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Find the angle (θ): The angle θ can be found using the arctangent function: θ = arctan(1/1) = π/4 radians (or 45 degrees).
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:
(1 + i)^2024 = (√2)^2024(cos(2024 * π/4) + i sin(2024 * π/4))
Simplifying:
(1 + i)^2024 = 2^1012(cos(506π) + i sin(506π))
Finding the Final Result
Since the cosine and sine functions have a period of 2π, we can simplify the angles:
- cos(506π) = cos(0) = 1
- sin(506π) = sin(0) = 0
Therefore:
(1 + i)^2024 = 2^1012 (1 + 0i) = 2^1012
In conclusion, the value of (1 + i)^2024 is 2 raised to the power of 1012. This result highlights the power of De Moivre's theorem in simplifying complex number calculations.