Question

Do the following by calculator. Round to three significant digits, where necessary. Write each complex number in polar form. $$-5-2 i$$

Short answer

\( -5 - 2i \) in polar form is \( \sqrt{29}(\cos(-2.677) + i\sin(-2.677)) \) or \( \sqrt{29}(\cos(3.606) + i\sin(3.606)) \), where the angle is in radians.

Step-by-step solution

Calculate the magnitude of the complex number

Find the magnitude (or modulus) of the complex number, which is the distance from the origin to the point \( -5, -2 \) in the complex plane. The magnitude is given by the formula \( r = \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the real and imaginary parts respectively. For \( -5 - 2i \) we get \( r = \sqrt{(-5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29} \).

Determine the argument of the complex number

To get the argument \( \theta \) of the complex number, which is the angle in polar coordinates, use the arctangent function of the imaginary part over the real part: \( \theta = \arctan(\frac{b}{a}) \). However, since the complex number lies in the third quadrant, we must add \( \pi \) to the result of \( \arctan(\frac{-2}{-5}) \) to get the right angle. Calculate \( \theta = \arctan(\frac{-2}{-5}) + \pi \) using a calculator and round to three significant digits.

Write the complex number in polar form

Using the magnitude \( r \) and the argument \( \theta \) found in the previous steps, write the complex number in polar form as \( r(\cos(\theta) + i\sin(\theta)) \). Use the values from steps 1 and 2 to express \( -5 - 2i \) in polar form, rounded to three significant digits.

Key concepts

Magnitude of Complex Numbers

Understanding the magnitude of a complex number is essential for visualizing it in the complex plane. Think of it as the distance from the origin (where the real and imaginary axes meet) to the given complex number point. It's like finding how far a point is from the center of a coordinate system.

To find the magnitude, we use the Pythagorean theorem. If our complex number is expressed as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, the magnitude (often denoted as \( r \)) is \( r = \sqrt{a^2 + b^2} \).

Using the Pythagorean theorem, for the complex number \( -5 - 2i \), the magnitude is calculated as \( \sqrt{(-5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29} \). This value represents the 'length' of the vector pointing to the complex number on the plane.

Argument of Complex Numbers

The argument of a complex number defines the direction of the vector from the origin to the point in the complex plane. It's given as the angle \( \theta \) that the line makes with the positive real axis, measured in a counterclockwise direction.

For any complex number \( a + bi \), the argument is calculated using the arctangent function: \( \theta = \arctan(\frac{b}{a}) \). But there's a twist! Since our angle must correctly represent the quadrant in which the complex number lies, we sometimes need to adjust our result. For \( -5 - 2i \), our number is in the third quadrant where both \( a \) and \( b \) are negative. Therefore, we must add \( \pi \) radians to the arctangent of the ratio of imaginary to real parts.

We get the adjusted argument: \( \theta = \arctan(\frac{-2}{-5}) + \pi \). By using a calculator to find \( \arctan(\frac{-2}{-5}) \) and adding \( \pi \), you can determine the precise direction of the vector representing our complex number.

Complex Number Conversion

Complex number conversion is about changing how we represent complex numbers. There are mainly two forms: rectangular (or Cartesian) form, \( a + bi \), and polar form, \( r(\cos(\theta) + i\sin(\theta)) \). The polar form is especially useful when multiplying or dividing complex numbers as it simplifies the processes to straightforward multiplication or division of their magnitudes and adding or subtracting their arguments.

To convert a complex number from rectangular to polar form, we first need the magnitude \( r \) and the argument \( \theta \), as determined in the earlier steps. For our example, \( -5 - 2i \), we have the magnitude \( \sqrt{29} \) and the argument obtained by calculating \( \arctan(\frac{-2}{-5}) + \pi \).

We then express the number in polar form by substituting these values into the polar notation, creating \( \sqrt{29}(\cos(\theta) + i\sin(\theta)) \), where \( \theta \) is the calculated argument rounded to three significant figures. This conversion provides a different perspective for understanding and working with complex numbers in mathematical and practical applications.