Question

Do the following by calculator. Round to three significant digits, where necessary. Write each complex number in polar form. $$-3-7 i$$

Short answer

|z| = 7.62, \(\theta\) must be computed with a calculator and adjusted for the angle being in the third quadrant before rounding.

Step-by-step solution

Find Magnitude

Calculate the magnitude (also called modulus) of the complex number \(-3 - 7i\) using the formula \(|z| = \sqrt{a^2 + b^2}\), where \(a\) is the real part and \(b\) is the imaginary part. Thus, \(|z| = \sqrt{(-3)^2 + (-7)^2} = \sqrt{9 + 49} = \sqrt{58}\).

Calculate Argument

Find the argument (angle) \(\theta\) of the complex number, which is the angle made with the positive real axis. Use the formula \(\text{arg}(z) = \tan^{-1}\left(\frac{b}{a}\right)\). Since our complex number is in the third quadrant, we must add \(\pi\) to our angle to get the correct argument in radians. Thus \(\theta = \tan^{-1}\left(\frac{-7}{-3}\right) + \pi\).

Round Calculation

Round the resulting calculations to three significant digits. Magnitude: \(|z| \approx 7.62\). Argument: Calculate \(\theta\) using a calculator and add \(\pi\) to find the angle in the third quadrant, then round to three significant digits.

Writing Polar Form

Express the complex number in polar form using the magnitude and argument: \(z = |z|(\cos(\theta) + i\sin(\theta))\). Substitute the rounded magnitude and argument into the equation.

Key concepts

Magnitude of Complex Number

Understanding the magnitude of a complex number is a fundamental concept in complex analysis. The magnitude, also known as the modulus, of a complex number is a measure of its distance from the origin in the complex plane. Imagine a two-dimensional graph where the x-axis represents the real part and the y-axis represents the imaginary part of a complex number. The complex number \( -3 - 7i \) for instance, can be visualized as a point located 3 units left and 7 units down from the origin.

The formula to calculate the magnitude of a complex number \( z = a + bi \) is \( |z| = \sqrt{a^2 + b^2} \). This equation is derived from the Pythagorean theorem, which relates the sides of a right triangle. Therefore, in our example, the magnitude is \( \sqrt{(-3)^2 + (-7)^2} = \sqrt{58} \), which is approximately 7.62 when rounded to three significant digits.

Argument of Complex Number

The argument of a complex number is just as crucial as its magnitude. It defines the direction of the vector representing the complex number in the complex plane, similar to how a compass direction works. By convention, the argument is measured as the angle from the positive x-axis (real axis) to the vector representing the complex number, moving counterclockwise.

To find the argument of a complex number \( z = a + bi \), we use the formula \( \text{arg}(z) = \tan^{-1}\left(\frac{b}{a}\right) \). For the example \( -3 - 7i \), the computed angle would place it in the third quadrant since both the real and imaginary parts are negative. However, the arctangent function \( \tan^{-1} \) only gives us the principal value, which is between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \). To correct for this and find the true argument, we have to add \( \pi \) to the result of \( \tan^{-1}\left(\frac{-7}{-3}\right) \) to get the angle in the third quadrant.

Once we've added \( \pi \) to adjust for the quadrant, the argument needs to be rounded to three significant digits for our purposes. Knowing how to correctly calculate the argument allows us to succinctly describe the position and orientation of any complex number within its plane.

Polar Coordinates

Polar coordinates offer a different perspective from the traditional Cartesian (x, y) coordinate system for representing points on a plane, particularly useful for complex numbers and certain types of engineering and physics problems. Rather than using perpendicular axes, the position of a point in polar coordinates is determined by its distance from the origin (the radius or magnitude) and the angle (argument) measured counterclockwise from the positive x-axis.

A complex number in polar form can be expressed as \( z = r(\cos(\theta) + i\sin(\theta)) \), where \( r \) is the magnitude and \( \theta \) is the argument. This form is particularly helpful for tasks such as multiplication or division of complex numbers and converting between different forms.

Returning to the example \( -3 - 7i \), the polar form is written by substituting the previously calculated magnitude and argument into this equation. It encapsulates the essence of the complex number's position using just two values. Polar coordinates make certain complex number operations more intuitive and, when visualized, can help in understanding the properties of functions that involve complex variables.