Question

Round to three significant digits, where necessary, in this exercise. Write each complex number in polar form. $$-3-7 i$$

Short answer

\(\sqrt{58}\left(\cos(\arctan\frac{7}{3} + \pi) + i\sin(\arctan\frac{7}{3} + \pi)\right)\), approximately \(7.62(\cos(2.50) + i\sin(2.50))\)

Step-by-step solution

Find the Magnitude of the Complex Number

To write the complex number in polar form, first determine its magnitude, which is the distance from the origin to the point in the complex plane. The magnitude (or modulus) is given by the formula \(r = \sqrt{a^2 + b^2}\), where \(a\) is the real part, and \(b\) is the imaginary part of the complex number \(-3 - 7i\). Compute \(r = \sqrt{(-3)^2 + (-7)^2} = \sqrt{9 + 49} = \sqrt{58}\).

Calculate the Argument of the Complex Number

The argument (or angle) \(\theta\) can be found using the arctangent function: \(\theta = \arctan\left(\frac{b}{a}\right)\). Here, \(a = -3\) and \(b = -7\). Calculate \(\theta = \arctan\left(\frac{-7}{-3}\right)\). Since the complex number is in the third quadrant, add \(\pi\) radians to the result to find the correct argument.

Write the Complex Number in Polar Form

Combine the magnitude and the argument to express the complex number in polar form \(r(\cos\theta + i\sin\theta)\). Using the calculated magnitude and argument, write the final polar form, rounding to three significant digits where necessary.

Key concepts

Magnitude of Complex Number

The magnitude of a complex number is a measure of its distance from the origin on the complex plane. It is also referred to as the modulus of the complex number. For a complex number represented as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, the magnitude is calculated using the formula \( r = \sqrt{a^2 + b^2} \). Imagine plotting the point ( \(a\), \(b\) ) and forming a right triangle with the axes. The magnitude is the hypotenuse of this triangle.

Continuing from the exercise example, for the complex number -3-7i, you'd calculate the magnitude as \( \sqrt{(-3)^2 + (-7)^2} = \sqrt{9 + 49} = \sqrt{58} \), which is the straight-line distance from the origin to the point (-3, -7) on the complex plane. The process of finding the magnitude is crucial to converting the complex number to its polar form.

Argument of Complex Number

The argument of a complex number is the angle formed by the line joining the point corresponding to the complex number on the complex plane and the positive real axis. This angle is typically measured in radians. To find the argument \( \theta \) for a complex number \(a + bi\), one can use the formula \( \theta = \arctan\left(\frac{b}{a}\right) \), which gives the angle in the first or fourth quadrant. However, since the signs of \(a\) and \(b\) determine the quadrant where the point lies, it is necessary to adjust \(\theta\) accordingly to get the correct angle.

In the provided exercise, with the complex number being in the third quadrant, \(\theta\) calculated with the arctangent function yields the angle's reference value. To find the actual argument, we need to add \(\pi\) radians to ensure the angle is oriented correctly. Understanding the argument is fundamental to express the complex number correctly in polar coordinates.

Polar Coordinates

Polar coordinates represent points in a plane using a radius and an angle, unlike the Cartesian coordinates that use an x and y-axis. The two key components in polar coordinates are the magnitude, also known as the radial coordinate (usually represented as \(r\)), and the argument, or the angular coordinate ( \(\theta\) ). In the context of complex numbers, to express a number in polar coordinates means to represent it in terms of its magnitude and argument, denoted as \(r(\cos \theta + i\sin \theta)\).

This representation is extremely useful in various mathematical applications, including multiplication and division of complex numbers, as it simplifies the operations to manipulation of magnitudes and addition or subtraction of angles. The conversion to polar form is especially beneficial in fields like engineering and physics, where complex numbers frequently occur.

Complex Plane

The complex plane is a two-dimensional plane in which all complex numbers can be plotted. Here, the horizontal axis is the real axis, and the vertical axis is the imaginary axis. Each point on the complex plane corresponds to a unique complex number with a real part (corresponding to the horizontal 'x' coordinate) and an imaginary part (corresponding to the vertical 'y' coordinate).

The idea of the complex plane provides a geometrical interpretation of complex numbers, allowing for the visualization of algebraic operations as geometric transformations. For example, adding two complex numbers corresponds to vector addition on the complex plane. The ability to visualize complex operations facilitates a deeper understanding and can make otherwise challenging concepts more accessible. The value in representing complex numbers on the complex plane, particularly in polar form, lies in the ease with which complex arithmetic can be understood and computed.