%5E30.jpeg "(1+i)^30")
Exploring the Power of Complex Numbers: (1 + i)^30
The expression (1 + i)^30 might seem daunting at first glance, but with the right tools, it becomes a fascinating journey into the world of complex numbers. Let's dive in and uncover the secrets hidden within this seemingly complex calculation.
Understanding Complex Numbers
Before tackling the exponent, it's essential to understand what we're dealing with. Complex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.
1 + i is a complex number with both a real component (1) and an imaginary component (1). Visualizing complex numbers on a plane (known as the complex plane) can make their manipulation easier.
De Moivre's Theorem: The Key to Unlocking the Power
To efficiently calculate (1 + i)^30, we turn to De Moivre's Theorem, a powerful tool for dealing with powers of complex numbers. This theorem states:
(cos θ + i sin θ)^n = cos (nθ) + i sin (nθ)
But how do we apply this to (1 + i)? We need to convert 1 + i into its polar form, which involves its magnitude (or modulus) and angle (or argument).
Finding the Modulus and Argument
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Modulus (r): The modulus is the distance from the origin to the complex number on the complex plane. We can calculate it using the Pythagorean theorem: r = √(a^2 + b^2) = √(1^2 + 1^2) = √2.
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Argument (θ): The argument is the angle between the positive real axis and the line connecting the origin to the complex number. We can find it using the arctangent function: θ = arctan(b/a) = arctan(1/1) = π/4.
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 find (1 + i)^30:
(√2(cos(π/4) + i sin(π/4)))^30 = √2^30 (cos(30 * π/4) + i sin(30 * π/4))
Simplifying further:
2^15 (cos(15π/2) + i sin(15π/2))
Remember that the cosine and sine functions have a period of 2π. We can simplify the argument:
15π/2 = 7π + π/2
Therefore:
2^15 (cos(π/2) + i sin(π/2))
Final Result
The cosine of π/2 is 0, and the sine of π/2 is 1. So, our final answer is:
(1 + i)^30 = 2^15 * i
This result demonstrates that even seemingly complex expressions involving powers of complex numbers can be simplified using the right tools and understanding of the underlying principles.