%5E4-is-equal-to.jpeg "(1-i)^4 Is Equal To")
Demystifying (1-i)^4
In the realm of complex numbers, exploring the power of a complex number can be an intriguing journey. Let's delve into the intriguing calculation of (1-i)^4.
Understanding the Fundamentals
- 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.
- De Moivre's Theorem: This theorem provides a powerful tool to calculate powers of complex numbers in polar form. It states that for any complex number z = r(cos θ + i sin θ) and integer n, we have: z^n = r^n(cos nθ + i sin nθ).
Calculating (1-i)^4
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Polar Form: We begin by converting (1-i) into its polar form. The modulus of (1-i) is √(1² + (-1)²) = √2. The argument (angle) can be found using the arctangent function: θ = arctan(-1/1) = -π/4. Since (1-i) lies in the fourth quadrant, we add 2π to the angle, giving us θ = 7π/4. Therefore, (1-i) = √2(cos(7π/4) + i sin(7π/4)).
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De Moivre's Theorem: Applying De Moivre's theorem, we have: (1-i)^4 = (√2)^4(cos(4 * 7π/4) + i sin(4 * 7π/4))
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Simplification: Simplifying the expression: (1-i)^4 = 4(cos(7π) + i sin(7π)) (1-i)^4 = 4(-1 + 0i) (1-i)^4 = -4
Conclusion
The calculation reveals that (1-i)^4 equals -4. This exercise demonstrates the elegant interplay between complex numbers and their powers. De Moivre's theorem provides a convenient pathway to calculate powers of complex numbers, often simplifying complex calculations.