%5E7.jpeg "(1-i)^7")
Exploring the Power of Complex Numbers: (1-i)^7
This article explores the calculation of (1-i)^7, demonstrating how to effectively work with complex numbers raised to a power.
Understanding Complex Numbers
Complex numbers are numbers 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
De Moivre's Theorem is a powerful tool for simplifying the process of raising complex numbers to a power. It states that:
(cos θ + i sin θ) ^ n = cos (nθ) + i sin (nθ)
where 'n' is any integer.
Applying De Moivre's Theorem to (1-i)^7
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Express (1-i) in polar form:
- To do this, we need to find the magnitude (r) and angle (θ) of (1-i).
- Magnitude: |1 - i| = √(1^2 + (-1)^2) = √2
- Angle: θ = arctan(-1/1) = -π/4 (Note: We choose the angle in the fourth quadrant since the real part is positive and the imaginary part is negative).
- Therefore, (1-i) = √2 (cos(-π/4) + i sin(-π/4))
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Apply De Moivre's Theorem:
- (1-i)^7 = [√2 (cos(-π/4) + i sin(-π/4))]^7
- = (√2)^7 (cos(-7π/4) + i sin(-7π/4))
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Simplify:
- (√2)^7 = 8√2
- cos(-7π/4) = cos(π/4) = √2/2
- sin(-7π/4) = sin(π/4) = √2/2
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Final Result:
- (1-i)^7 = 8√2 (√2/2 + i √2/2)
- = 8 + 8i
Conclusion
By applying De Moivre's Theorem, we successfully calculated (1-i)^7, expressing the result in its standard form: 8 + 8i. This process demonstrates the versatility of complex numbers and the usefulness of De Moivre's Theorem for simplifying complex number calculations.