(1-i)^5 In Rectangular Form

Jasonbradley

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Jun 16, 2024

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Finding (1 - i)^5 in Rectangular Form

This article will demonstrate how to calculate (1 - i)^5 in rectangular form.

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

De Moivre's Theorem is a powerful tool for finding powers of complex numbers. It states:

(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)

where n is an integer.

Calculating (1 - i)^5

1. Convert to Polar Form:

   • Find the modulus (magnitude) of 1 - i:
     • |1 - i| = √(1² + (-1)²) = √2
   • Find the argument (angle) of 1 - i:
     • 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))

2. Apply De Moivre's Theorem:

   (1 - i)^5 = [√2(cos(-π/4) + i sin(-π/4))]^5 = (√2)^5 * (cos(-5π/4) + i sin(-5π/4))

3. Simplify:

   = 4√2 * (cos(-5π/4) + i sin(-5π/4)) = 4√2 * (cos(3π/4) + i sin(3π/4)) = 4√2 * (-√2/2 + i√2/2) = -4 + 4i

Therefore, (1 - i)^5 in rectangular form is -4 + 4i.