Problem 62 - Algebra and Trigonometry

Question

In Exercises \(53-64,\) use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. $$ (1-i)^{5} $$

Step-by-Step Solution

Answer

Thus, \((1-i)^{5} = 2sqrt{2} - 2sqrt{2}i\).

Step 1: Convert to Polar Form

Conversion from rectangular form (\(a+bi\)) to polar form (\(r(cos\theta + isin\theta))\), where \(r= \sqrt{a^2 + b^2}\) (the absolute value of the complex number) and \(\theta = atan2(b,a)\) (the argument of the complex number, using two-argument arctan function). For the complex number (1-i), \(r = \sqrt{1^2 + (-1)^2} = \sqrt{2}\) and \(\theta = atan2(-1,1) = -π/4\). So in polar form, the number is \(\sqrt{2}(cos(-π/4) + isin(-π/4))\).

Step 2: Apply DeMoivre's Theorem

By DeMoivre's Theorem, \((r(cos\theta + isin\theta))^n = r^n (cos(n\theta) + isin(n\theta)).\) Therefore, \((\sqrt{2}(cos(-π/4) + isin(-π/4)))^5 = 2^{5/2} (cos(5*(-π/4)) + isin(5*(-π/4))) = 4sqrt{2} (cos(-5π/4) + isin(-5π/4)).\)

Step 3: Convert back to Rectangular Form

Conversion back to rectangular form (\(a+bi\)), uses the identities \(a = r cos\theta\) and \(b = r sin\theta\). For our previous result, \(a = 4sqrt{2} cos(-5π/4) = 2sqrt{2}\) and \(b = 4sqrt{2} sin(-5π/4) = - 2sqrt{2}\). Hence, \(4sqrt{2} (cos(-5π/4) + isin(-5π/4))\) is equivalent to \(2sqrt{2} - 2sqrt{2}i\) in rectangular form.

Key Concepts

Complex Numbers

Complex numbers are a type of number that expand upon the traditional idea of numbers being 'real'. A complex number is written in the form of \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, and \(i\) represents the square root of -1. The beauty of complex numbers lies in their ability to precisely represent all aspects of quantities having both magnitude and direction, such as in electrical engineering and quantum physics.

In simpler terms, while real numbers can be visualized as points on a one-dimensional number line, complex numbers require a two-dimensional plane, known as the complex plane. Here, the x-axis represents the real component while the y-axis stands for the imaginary component. This is pivotal for understanding the behavior of certain functions and for solving equations that would otherwise have no solution within the realm of real numbers only.

Polar Form Conversion

When working with complex numbers, converting between rectangular form (\(a+bi\)) and polar form (\(r(cos\theta + isin\theta)\)) is a powerful tool, especially when dealing with multiplication or raising powers, as with DeMoivre's Theorem.

To convert a given complex number from rectangular to polar form, you find the magnitude \(r\) and angle \(\theta\). The magnitude is obtained by taking the square root of the sum of the squares of the real and imaginary parts (\(r= \sqrt{a^2 + b^2}\)), representing the distance from the origin in the complex plane. The angle \(\theta\), often called the argument, measures the direction of the number and is given by the arctangent of \(b/a\), or more accurately, by the two-argument arctangent function \(atan2(b,a)\), which takes into account the signs of \(a\) and \(b\) to find the correct quadrant of \(\theta\).

Thus, a complex number in polar form captures the essence of a point in terms of how far it is from the origin and in which direction, which simplifies many operations.

Rectangular Form

After performing certain operations in polar form, such as using DeMoivre's Theorem, it can be practical to revert back to the rectangular form, \(a+bi\). This form directly represents the horizontal (real) and vertical (imaginary) displacements from the origin, making it straightforward to understand and use in addition and subtraction operations over complex numbers.

The process involves converting the polar coordinates back into their rectangular counterparts by using the relationships \(a = r cos\theta\) for the real part, and \(b = r sin\theta\) for the imaginary part. This conversion utilizes the original magnitude \(r\) and the angle \(\theta\) to re-establish the complex number in a familiar x + yi format.

Understanding rectangular form is key to working efficiently with complex numbers in various fields, such as signal processing and control systems, where this representation is often used.