%5E30.jpeg "(1-i)^30")
Exploring the Power of Complex Numbers: (1-i)^30
This article will delve into the fascinating world of complex numbers by exploring the simplification of the expression (1-i)^30. We'll utilize the concept of De Moivre's Theorem to tackle this problem, and along the way, we'll gain a deeper understanding of how complex numbers behave under exponentiation.
Understanding Complex Numbers
Before we embark on the journey of simplifying the expression, let's establish a foundation in complex numbers. Complex numbers are numbers that extend the real number system by introducing the imaginary unit i, defined as the square root of -1. A complex number is generally represented as a + bi, where a and b are real numbers, and i is the imaginary unit.
The Power of De Moivre's Theorem
De Moivre's Theorem provides a powerful tool for simplifying complex numbers raised to a power. It states that for any complex number in polar form, z = r(cos θ + i sin θ), and any integer n:
(z)^n = r^n (cos nθ + i sin nθ)
This theorem essentially tells us that to raise a complex number to a power, we raise its modulus (magnitude) to that power and multiply its argument (angle) by that power.
Applying De Moivre's Theorem to (1-i)^30
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Convert (1-i) to polar form:
- Modulus (r): |1-i| = √(1² + (-1)²) = √2
- Argument (θ): θ = arctan(-1/1) = -π/4 (Since (1-i) lies in the fourth quadrant)
Therefore, (1-i) = √2 (cos(-π/4) + i sin(-π/4))
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Apply De Moivre's Theorem:
- (1-i)^30 = (√2)^30 (cos (-30π/4) + i sin (-30π/4))
- Simplify the expressions:
- (√2)^30 = 2^15
- cos(-30π/4) = cos(15π/2) = 0
- sin(-30π/4) = sin(15π/2) = -1
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Final Result:
- (1-i)^30 = 2^15 (0 - i) = -2^15 i
Therefore, the simplified form of (1-i)^30 is -2^15 i.
Conclusion
Through the application of De Moivre's Theorem, we were able to efficiently simplify the expression (1-i)^30. This exercise highlights the elegance and power of complex number operations and demonstrates how De Moivre's Theorem can be utilized to handle exponentiation of complex numbers.