Problem 39 - Precalculus with Limits

Question

In Exercises 33-42, find the standard form of the complex number. Then represent the complex number graphically. \(7(\cos\ 0 + i\ \sin\ 0)\)

Step-by-Step Solution

Answer

The standard form of the complex number is \(7 + 0i\), and it is represented graphically as a single point at (7,0) on the complex plane.

Step 1: Convert from Polar to Rectangular Form

To convert a number from polar to rectangular form, you may use the formulas: \[a = r \cos \theta \]\[b = r \sin \theta \]The standard form of a complex number is a + bi. Substituting the value 7 for r and 0 for \(\theta\), we get:\[a = 7 \cos (0) \]\[b = 7 \sin (0) \]

Step 2: Calculate a and b

The value of \(\cos (0)\) and \(\sin (0)\) are 1 and 0 respectively. Substituting these values, we get:\[a = 7 * 1 = 7\]\[b = 7*0 = 0\]

Step 3: Write Complex Number in Standard Form

The standard form of the complex number is:\[a + bi\]Therefore, our complex number in standard form is:\[7 + 0i\]

Step 4: Graph Representation

When represented graphically, a complex number \(a+bi\) corresponds to a point \(a,b\) in the complex plane. Since a=7 and b=0, we place a dot at the point (7,0) on the plane.

Key Concepts

Polar to Rectangular Form Conversion

Understanding how to switch between polar and rectangular forms of complex numbers is essential for both mathematical theory and its applications.

The polar form of a complex number is expressed as \(r(\cos \theta + i\sin \theta)\), where \(r\) is the magnitude of the complex number, and \(\theta\) is the angle it makes with the positive x-axis.
To convert a complex number from polar to rectangular form, you apply the trigonometric identities:

• \(\cos(\theta)\) for the real part
• \(\sin(\theta)\) for the imaginary part

and multiply these by the magnitude \(r\).

In a practical example, if you have \(7(\cos 0 + i \sin 0)\), as given in the textbook exercise, you would use the conversion formulas:

• \(a = r \cos \theta\)
• \(b = r \sin \theta\)

Since \(\cos(0) = 1\) and \(\sin(0) = 0\), the conversion from polar to rectangular form leads to the complex number in the standard form \(7 + 0i\), or simply \(7\).

Standard Form of a Complex Number

The standard form of a complex number is crucial for its manipulation in algebra. It is written as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, and \(i\) is the imaginary unit such that \(i^2 = -1\).

When we convert from polar to rectangular form, the resulting standard form provides a straightforward expression of the complex number. As we did in the previous example, \(a\) is calculated by multiplying the radius \(r\) with \(\cos(\theta)\), and \(b\) is obtained by multiplying \(r\) with \(\sin(\theta)\). If \(\theta\) is zero degrees, the \(\sin\) term vanishes, leaving us with \(a\) as the radius and \(b\) as zero, resulting in the simple standard form.

Real and Imaginary Components

In our example, the real component \(a\) is 7, and the imaginary component \(b\) is 0. Thus, we write the complex number as \(7 + 0i\) or just \(7\), emphasizing that sometimes the imaginary part can be non-existent, leading to a real number.

Graphical Representation of Complex Numbers

Graphical representation is a powerful tool for visualizing complex numbers. Each complex number corresponds to a unique point on the complex plane, also known as the Argand diagram. This plane is similar to the Cartesian coordinate system, with the horizontal axis representing the real part and the vertical axis representing the imaginary part of the complex number.

A point on the complex plane has coordinates \(a, b\), where \(a\) is the real part and \(b\) is the imaginary part of the complex number \(a + bi\). If you were to graph \(7 + 0i\), as obtained in our earlier step-by-step solution, you would place a dot at \(7, 0\) on the complex plane.

Why Use Graphical Representation?

It aids in understanding operations such as addition, subtraction, multiplication, and division of complex numbers. Moreover, it can reveal details such as the magnitude and the argument (angle) of the complex number, which are not immediately apparent in the standard algebraic form. The complex number from the exercise corresponds to a point on the real axis, reaffirming that it is, in fact, a real number.