%5E4.jpeg "(1-i)^4")
Exploring the Power of Complex Numbers: (1-i)^4
This article delves into the interesting calculation of (1-i) raised to the power of 4, showcasing how complex numbers behave under exponentiation.
Understanding Complex Numbers
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
To simplify the calculation of (1-i)^4, we can leverage De Moivre's Theorem. This theorem states that for any complex number in polar form, r(cos θ + i sin θ), and any integer 'n':
(r(cos θ + i sin θ))^n = r^n(cos(nθ) + i sin(nθ))
Applying De Moivre's Theorem
-
Convert to Polar Form: First, we need to convert (1-i) to polar form. We can find the magnitude, 'r', as:
r = √(1² + (-1)²) = √2
The angle, 'θ', can be found using the arctangent function:
θ = arctan(-1/1) = -π/4
Therefore, (1-i) = √2(cos(-π/4) + i sin(-π/4))
-
Applying De Moivre's Theorem: Now we can apply De Moivre's Theorem:
(1-i)^4 = (√2(cos(-π/4) + i sin(-π/4)))^4 = (√2)^4 (cos(-π) + i sin(-π))
-
Simplify:
(1-i)^4 = 4(cos(-π) + i sin(-π)) = 4(-1 + 0i) = -4
Conclusion
Through De Moivre's Theorem, we efficiently calculated that (1-i)^4 = -4. This demonstration showcases how complex numbers behave under exponentiation, providing insights into their intriguing properties.