%5E20.jpeg "(1-i)^20")
Exploring the Power of Complex Numbers: (1 - i)^20
This article delves into the fascinating world of complex numbers by exploring the seemingly daunting expression (1 - i)^20. We will unravel its hidden structure and arrive at a surprisingly simple result.
Understanding Complex Numbers
Complex numbers extend the realm of real numbers by introducing the imaginary unit, i, defined as the square root of -1. This opens up a new dimension in mathematics, allowing us to represent and manipulate quantities that go beyond the confines of the real number line.
A complex number takes the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit. For example, 2 + 3i is a complex number with a real part of 2 and an imaginary part of 3.
De Moivre's Theorem: A Powerful Tool
To tackle (1 - i)^20 efficiently, we employ a powerful tool called De Moivre's Theorem. This theorem states:
(cos θ + i sin θ)^n = cos (nθ) + i sin (nθ)
This theorem allows us to raise complex numbers in polar form to any power with relative ease.
Transforming (1 - i) into Polar Form
First, we need to express (1 - i) in polar form. This involves finding its magnitude (or modulus) and its angle (or argument).
- Magnitude: The magnitude of (1 - i) is calculated as √(1² + (-1)²) = √2.
- Angle: The angle of (1 - i) is found by considering its position in the complex plane. Since it lies in the fourth quadrant with equal real and imaginary parts, its angle is -45° or -π/4 radians.
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 to raise (1 - i) to the power of 20:
(1 - i)^20 = [√2 (cos(-π/4) + i sin(-π/4))]^20
= (√2)^20 (cos(-20π/4) + i sin(-20π/4))
= 2^10 (cos(-5π) + i sin(-5π))
Simplifying the Result
Since the cosine and sine functions have a period of 2π, cos(-5π) = cos(-π) = -1 and sin(-5π) = sin(-π) = 0.
Therefore, (1 - i)^20 = 2^10 (-1 + 0i) = -1024.
Conclusion
Despite its seemingly intimidating appearance, (1 - i)^20 simplifies to a real number, -1024, with the help of De Moivre's Theorem and basic trigonometric identities. This example demonstrates the power and elegance of complex numbers and the tools used to manipulate them.