Problem 8 - Algebra and Trigonometry

Question

Graph the complex number and find its modulus. $$ 6 $$

Step-by-Step Solution

Answer

The graph is a point at (6,0) on the complex plane, and the modulus is 6.

Step 1: Identify the Complex Number Components

A complex number typically has the form \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. In this case, the given number is simply \( 6 \), which can be rewritten as \( 6 + 0i \). Therefore, the real part \( a = 6 \) and the imaginary part \( b = 0 \).

Step 2: Graph the Complex Number

To graph the complex number \( z = 6 + 0i \), plot the point corresponding to \( 6 + 0i \) on the complex plane. The horizontal axis (real axis) represents the real part of the complex number, while the vertical axis (imaginary axis) represents the imaginary part. So, plot the point at (6,0).

Step 3: Determine the Modulus of the Complex Number

The modulus of a complex number \( z = a + bi \) is given by the formula \( |z| = \sqrt{a^2 + b^2} \). For our complex number, this becomes \( |6 + 0i| = \sqrt{6^2 + 0^2} = \sqrt{36} = 6 \). Thus, the modulus of the complex number is 6.

Key Concepts

Graphing Complex Numbers

Graphing complex numbers is similar to graphing points on a Cartesian coordinate system, but instead, it involves the complex plane. The complex plane is a two-dimensional plane consisting of a real axis and an imaginary axis. These axes are like the x and y axes you're familiar with in regular graphing, but here they describe different aspects of a complex number.

To graph a complex number, you identify its real and imaginary parts. For the complex number \( z = 6 + 0i \), as in our example, the real part is 6 and the imaginary part is 0. On the complex plane:

• The real part (6) is plotted along the horizontal axis.
• The imaginary part (0) is plotted along the vertical axis.

Here, you simply plot the point (6,0). This is your complex number's position on the complex plane, illustrating its real and imaginary components.

Modulus of a Complex Number

The modulus of a complex number measures its distance from the origin (0, 0) on the complex plane. It gives a sense of the 'size' or 'magnitude' of the complex number. The formula to compute the modulus of a complex number \( z = a + bi \) is:
\[ |z| = \sqrt{a^2 + b^2} \]

In our specific example, the complex number is \( 6 + 0i \). By substituting into the modulus formula, we have:
\[ |6 + 0i| = \sqrt{6^2 + 0^2} = \sqrt{36} = 6 \]
Thus, the modulus is 6, which corresponds to the distance from the origin to the point (6,0) on the complex plane. Calculating the modulus is helpful for understanding the scale or amplitude of complex numbers.

Real and Imaginary Parts

A complex number often takes the form \( z = a + bi \), where:

• \( a \) is the real part
• \( b \) is the imaginary part

These two parts combine to provide a complete picture of the complex number on the complex plane.

For a pure real number like 6, the imaginary part is 0. This makes such a number along the horizontal axis (real axis) of the complex plane. Even though it may seem like any real number isn't complex, in the context of complex numbers, a real number like 6 is indeed considered a complex number with an imaginary part of zero. This interpretation helps in broadening the general concept of numbers to include both real and imaginary dimensions, making complex calculations and transformations more versatile.