Problem 43 - Contemporary Precalculus

Question

In Exercises \(37-52,\) express the number in polar form. $$-\sqrt{3}-\sqrt{3} i$$

Step-by-Step Solution

Answer

Question: Express the complex number \(-\sqrt{3}-\sqrt{3}i\) in polar form. Answer: The polar form of the complex number \(-\sqrt{3}-\sqrt{3}i\) is \(\sqrt{6}(\cos{\frac{5\pi}{4}} + i\sin{\frac{5\pi}{4}})\).

Step 1: Find the real part and the imaginary part of the complex number

The given complex number is \(-\sqrt{3}-\sqrt{3}i\). The real part (denoted by \(a\)) is \(-\sqrt{3}\), and the imaginary part (denoted by \(b\)) is \(-\sqrt{3}\).

Step 2: Calculate the magnitude

Formula to find the magnitude \(r\) of a complex number is: $$r = \sqrt{a^2 + b^2}$$ Substitute the values of \(a\) and \(b\): $$r = \sqrt{(-\sqrt{3})^2 + (-\sqrt{3})^2}$$ $$r = \sqrt{3+3}$$ $$r = \sqrt{6}$$

Step 3: Calculate the argument

Formula to find the argument \(\theta\) of a complex number is: $$\theta = \arctan{\frac{b}{a}}$$ Substitute the values of \(a\) and \(b\): $$\theta = \arctan{\frac{-\sqrt{3}}{-\sqrt{3}}}$$ $$\theta = \arctan{1}$$ Since both \(a\) and \(b\) are negative, the complex number lies in the third quadrant. We have to add \(\pi\) to the angle to determine the correct argument: $$\theta = \arctan{1} + \pi$$ $$\theta = \frac{\pi}{4} + \pi$$ $$\theta = \frac{5\pi}{4}$$

Step 4: Write the polar form

The polar form of a complex number is given by: $$z = r(\cos{\theta} + i\sin{\theta})$$ Substitute the values of \(r\) and \(\theta\): $$z = \sqrt{6}(\cos{\frac{5\pi}{4}} + i\sin{\frac{5\pi}{4}})$$ So, the polar form of the complex number \(-\sqrt{3}-\sqrt{3}i\) is \(\sqrt{6}(\cos{\frac{5\pi}{4}} + i\sin{\frac{5\pi}{4}})\).

Key Concepts

Complex Numbers

Complex numbers are fundamental in mathematics and engineering, and understanding them is crucial for further study in these fields. A complex number is a combination of a real part and an imaginary part. It is generally represented in the form of z = a + bi, where 'a' is the real part and 'b' represents the imaginary part, with 'i' being the imaginary unit satisfying the equation i^2 = -1.

One of the fascinating aspects of complex numbers is their ability to capture the notion of magnitude and direction, which is why they are extremely useful in fields like physics and electrical engineering where concepts like waves and oscillations are fundamental. The standard algebraic form of a complex number is convenient for basic operations such as addition and multiplication, but to talk about the magnitude and direction, we often turn to the polar form.

Magnitude of Complex Numbers

The magnitude or 'modulus' of a complex number is a measure of its 'size' or 'length' when considering the complex number as a point in a plane. It is essentially the distance of the complex number from the origin (0,0) in the complex plane, which can be visualized geometrically. To find the magnitude (or r) of the complex number z = a + bi, we use the Pythagorean theorem applied to a right-angled triangle with sides of lengths 'a' and 'b'.

The formula for the magnitude of a complex number is r = √(a^2 + b^2). This process involves squaring each part, summing the results, and taking the square root of the sum. In our exercise, the magnitude of -√3 - √3i is calculated as √(3+3), resulting in √6. The magnitude is a critical part of the complex number's identity in polar form, representing how far from the origin the number is, regardless of its direction.

Argument of Complex Numbers

The argument of a complex number is the measure of the angle it makes with the positive real axis, often denoted by θ (theta). This angle provides the directional component of a complex number in the complex plane. A principal argument is the value of θ in the interval (-π, π], which gives a unique representation for every complex number excluding the negative real axis, where arguments can differ by multiples of 2π.

To calculate the argument, we use the inverse tangent function (arctan) on the ratio of the imaginary part to the real part θ = arctan(b/a). However, this ratio alone doesn't always give the correct angle; we need to consider the quadrant in which the complex number is located. As we see in the solution, since both parts of our complex number are negative, the number is located in the third quadrant. Therefore, we must adjust the angle calculated by arctan to obtain the true argument of θ = 5π/4 for our complex number, taking into account the correct orientation in the complex plane.