%5E7.jpeg "(1+i)^7")
Exploring the Power of Complex Numbers: (1+i)^7
This article delves into the intriguing world of complex numbers, specifically exploring the expression (1+i)^7. While this might seem like a straightforward calculation, it unveils the elegance and power of complex numbers.
Understanding Complex Numbers
Complex numbers, denoted as a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit (√-1), extend the real number system. They play a crucial role in various fields, including mathematics, physics, and engineering.
De Moivre's Theorem: A Powerful Tool
To simplify (1+i)^7, we utilize De Moivre's Theorem. This theorem states that for any complex number in polar form (r(cos θ + i sin θ)) raised to a power 'n':
(r(cos θ + i sin θ))^n = r^n(cos(nθ) + i sin(nθ))
Converting to Polar Form
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Magnitude (r): The magnitude of (1+i) is √(1^2 + 1^2) = √2.
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Angle (θ): The angle θ is the arctangent of the imaginary part divided by the real part, i.e., tan^-1(1/1) = 45 degrees 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, using 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)) = 8√2 (cos(7π/4) + i sin(7π/4))
Simplifying the Result
The angle 7π/4 is equivalent to -π/4 in the principal argument. Therefore:
(1+i)^7 = 8√2 (cos(-π/4) + i sin(-π/4)) = 8√2 (√2/2 - i√2/2) = 8 - 8i
Conclusion
Through the application of De Moivre's Theorem, we effectively simplified (1+i)^7 to 8 - 8i. This demonstrates how complex numbers can be manipulated using elegant mathematical tools, revealing the beauty and power hidden within these seemingly abstract concepts.