%5E100.jpeg "(1-i)^100")
Exploring the Power of Complex Numbers: (1-i)^100
The expression (1-i)^100 presents an interesting challenge in the realm of complex numbers. Let's break down how to solve this and explore the fascinating results.
Understanding Complex Numbers
Before we dive into the calculation, it's essential to recall the basics of complex numbers. A complex number is a number 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.
Utilizing the Polar Form
The easiest way to deal with powers of complex numbers is to convert them into polar form. Here's how:
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Magnitude: The magnitude of (1-i) is found using the Pythagorean theorem: √(1² + (-1)²) = √2.
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Angle: The angle (or argument) of (1-i) can be found using the arctangent function: arctan(-1/1) = -45°. Since the point (1, -1) lies in the fourth quadrant, we add 360° to get the principal argument, which is 315°.
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Polar Form: Now we can write (1-i) in polar form as √2 * cis(315°), where "cis" represents cos θ + i sin θ.
Applying DeMoivre's Theorem
DeMoivre's Theorem provides a powerful tool for simplifying powers of complex numbers in polar form:
(r cis θ)^n = r^n cis (nθ)
Applying this to our problem:
(√2 * cis(315°))^100 = (√2)^100 * cis (100 * 315°)
Simplifying further:
= 2^50 * cis(31500°)
Finding the Principal Argument
Since the angle in a complex number is cyclical, we need to find the equivalent angle within the range of 0° to 360°. We can do this by dividing 31500° by 360° and taking the remainder:
31500° mod 360° = 180°
Therefore, (1-i)^100 = 2^50 * cis(180°)
Final Result
Finally, converting back to rectangular form:
2^50 * cis(180°) = 2^50 * (cos 180° + i sin 180°) = -2^50
Conclusion
The expression (1-i)^100 simplifies to a surprisingly simple real number: -2^50. This example showcases the elegance and power of using polar form and DeMoivre's theorem to efficiently deal with complex number operations.