Question

Do the following by calculator. Round to three significant digits, where necessary. Write each complex number in polar form. $$-4+7 i$$

Short answer

Using the magnitude and the argument, the polar form is approximately \( \sqrt{65}(\cos(2.356) + i\sin(2.356)) \) or \( 8.06 (\cos(2.356) + i\sin(2.356)) \) to three significant digits.

Step-by-step solution

Identify the Real and Imaginary Parts

For the complex number \( -4 + 7i \), identify the real part \(a\) and the imaginary part \(b\). Here, the real part is \(a = -4\) and the imaginary part is \(b = 7\).

Calculate the Magnitude

Calculate the magnitude (also known as modulus or radius) of the complex number using the formula \( r = \sqrt{a^2 + b^2} \). Plugging in the values for our complex number, we get \( r = \sqrt{(-4)^2 + (7)^2} = \sqrt{16 + 49} = \sqrt{65} \).

Calculate the Argument

Calculate the argument (also known as the angle) using the formula \( \theta = \arctan\left(\frac{b}{a}\right) \), with attention to the correct quadrant. Since the complex number is in the second quadrant (real part negative, imaginary part positive), we add \( \pi \) to the result of the \( \arctan \) function to get the proper angle. Thus, \( \theta = \arctan\left(\frac{7}{-4}\right) + \pi = \arctan(-1.75) + \pi \).

Express in Polar Form

The complex number in polar form is expressed as \( r(\cos(\theta) + i\sin(\theta)) \). Using the values from the previous steps and rounding to three significant digits, we get \( \sqrt{65}(\cos(\arctan(-1.75) + \pi) + i\sin(\arctan(-1.75) + \pi)) \), where \( r \) is rounded to three significant digits.

Complete the Polar Form

After calculating all the components, plug in the values of \( r \) and \( \theta \) into the polar form expression to get the final result. Make sure the angle is in radians and rounded to three significant digits.

Key concepts

Polar Coordinates

The concept of polar coordinates plays a fundamental role in various fields including mathematics, physics, and engineering. Unlike Cartesian coordinates which use a grid of horizontal and vertical lines to locate a point in a plane through an x (horizontal) and y (vertical) value, polar coordinates define a point based on its distance from a reference point and the angle formed with a reference direction.

In the context of complex numbers, the polar coordinates system is exceptionally handy. A complex number, usually written as \( z = a + bi \), can be represented in polar form by identifying its distance from the origin (which is called the magnitude or modulus) and the angle it forms with the positive x-axis (known as the argument or phase angle).

When converting to polar coordinates, we use the formula \( z = r(\text{cos} \theta + i\text{sin} \theta) \), where \( r \) is the magnitude and \( \theta \) is the argument. This polar form is very useful, especially when multiplying or dividing complex numbers, as it simplifies the process significantly.

Magnitude of Complex Numbers

The magnitude of a complex number, denoted as \( r \), is a measure of how far the number is from the origin (0,0) in the complex plane. To find the magnitude, we use the formula \( r = \sqrt{a^2 + b^2} \), where \( a \) represents the real part and \( b \) represents the imaginary part of the complex number.

The magnitude can be thought of as the length of the vector starting from the origin and ending at the point \( (a, b) \) represented by the complex number. It is always a non-negative real number. In our exercise, to find the magnitude for the complex number \( -4 + 7i \), we compute \( r = \sqrt{(-4)^2 + (7)^2} = \sqrt{16 + 49} = \sqrt{65} \), which gives us the 'radius' of the complex number in polar coordinates.

Importance of Magnitude

Understanding the magnitude is crucial, it tells us how 'strong' or 'intense' a complex number is. In real-life applications, this can relate to the amplitude of waves or the intensity of an electrical signal, for example.

Argument of Complex Numbers

The argument of a complex number is another crucial component when working with polar coordinates. It refers to the angle \( \theta \) that the line representing the complex number makes with the positive real axis. The argument is typically measured in radians and can be found using the formula \( \theta = \arctan(\frac{b}{a}) \), where \( a \) and \( b \) are the real and imaginary parts of the complex number, respectively.

For the complex number \( -4 + 7i \), one must be careful to consider in which quadrant of the complex plane the number lies to determine the correct angle. Since \( a < 0 \) and \( b > 0 \), the number is in the second quadrant, where the argument should be added to \( \pi \) to find its true value.

Hence, we calculate the angle using \( \theta = \arctan(\frac{7}{-4}) + \pi \), giving us the direction of the complex number relative to the positive x-axis. This angle helps us to visualize the complex number's position and is essential for vector rotation in the polar form representation.

Visualizing Arguments

Visualizing the argument of a complex number can be akin to finding the compass direction of a point from a reference point. It's a way to specify direction in the complex plane and it is indispensable for understanding the geometric interpretation of complex numbers.