%5E2023.jpeg "(1-i)^2023")
Demystifying (1-i)^2023
The expression (1-i)^2023 might look intimidating at first glance, but it's actually quite manageable with the right approach. Let's break it down step by step:
Understanding Complex Numbers
First, we need to understand that (1-i) is a complex number. A complex number is a number that can be expressed in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit, defined as the square root of -1 (i.e., i^2 = -1).
Utilizing Polar Form
To simplify the power of a complex number, it's often easier to work with its polar form. The polar form of a complex number represents it as a magnitude (distance from the origin) and an angle (direction from the positive real axis).
To convert (1-i) to polar form, we first find its magnitude:
- |1-i| = √(1^2 + (-1)^2) = √2
Next, we find the angle:
- θ = arctan(-1/1) = -45° (or -π/4 radians)
Therefore, the polar form of (1-i) is √2 * cis(-π/4), where 'cis' stands for cos(θ) + i sin(θ).
Applying De Moivre's Theorem
Now, we can utilize De Moivre's Theorem to simplify (1-i)^2023. De Moivre's Theorem states that for any complex number in polar form, (r * cis(θ))^n = r^n * cis(nθ).
Applying this to our problem:
(1-i)^2023 = (√2 * cis(-π/4))^2023 = (√2)^2023 * cis(-2023π/4)
Simplifying the Result
- (√2)^2023 can be simplified by noticing that 2023 is odd, so the result is still √2 raised to an odd power, leaving it as √2.
- The angle -2023π/4 is a multiple of π, meaning it will correspond to either 0° or 180° on the complex plane. To find the exact angle, we can divide -2023 by 4 and get -505.75. We can then add a multiple of 2π to this angle to get an equivalent angle in the range of 0 to 2π. Adding 10π (or 20π/2) gives us -0.75π, which is equivalent to 1.25π radians or 225°.
Therefore, (1-i)^2023 = √2 * cis(225°) in polar form.
We can convert this back to rectangular form:
√2 * cis(225°) = √2 * (cos(225°) + i sin(225°)) = √2 * (-√2/2 - i√2/2) = -1 - i
Conclusion
By using polar form and De Moivre's Theorem, we successfully simplified (1-i)^2023 to -1 - i. This process demonstrates the power of using different representations of complex numbers to solve seemingly complicated problems.