Problem 7 - Algebra and Trigonometry

Question

Graph the complex number and find its modulus. $$ -2 $$

Step-by-Step Solution

Answer

Plot at -2 on the real axis; modulus is 2.

Step 1: Identify the Components

First, recognize that the complex number -2 can be written in the form a + bi, where a = -2 and b = 0. This places the number on the complex plane with a real part of -2 and an imaginary part of 0.

Step 2: Plot the Complex Number

On the complex plane, plot the complex number -2 by moving 2 units left along the real axis (since -2 is on the negative side). Do not move along the imaginary axis, since the imaginary part is 0.

Step 3: Calculate the Modulus

The modulus of a complex number a + bi is given by the formula:\[| z | = \sqrt{a^2 + b^2}\]Substitute a = -2 and b = 0:\[| z | = \sqrt{(-2)^2 + 0^2} = \sqrt{4} = 2\]Thus, the modulus of the complex number -2 is 2.

Key Concepts

Complex Plane

The complex plane, sometimes referred to as the Argand plane, is a way to visualize complex numbers. Each complex number corresponds to a unique point on this plane, making it an invaluable tool for understanding their properties.

In the complex plane:

• The horizontal axis represents the real part of complex numbers, termed the real axis.
• The vertical axis represents the imaginary part, termed the imaginary axis.

Graphing the complex number \(-2\) involves locating \(-2\) on the real axis and \(0\) on the imaginary axis. This results in a point being plotted \(2\) units to the left of the origin on the horizontal line, with no movement up or down.

Modulus

The modulus of a complex number is a measure of its magnitude or size, akin to the absolute value for real numbers. It tells you how far the number is from the origin of the complex plane.

To find the modulus of a complex number \(a + bi\), we use the formula:\[| z | = \sqrt{a^2 + b^2}\]For our example, \(-2 + 0i\), this becomes:

• \(a = -2\)
• \(b = 0\)

Thus,\[| z | = \sqrt{(-2)^2 + 0^2} = \sqrt{4} = 2\]This means the complex number \(-2\) is \(2\) units away from the origin.

Imaginary Part

The imaginary part of a complex number determines its position on the vertical axis of the complex plane. In the standard form \(a + bi\), \(b\) represents this part.

A complex number like \(-2 + 0i\) shows that the imaginary part is \(0\). This means the number does not move vertically on the complex plane. It's purely a movement along the real axis. An imaginary part of \(0\) indicates that the complex number is, in fact, a real number.

Real Part

The real part of a complex number is the component that appears on the real axis of the complex plane, denoted by \(a\) in \(a + bi\). It shows the horizontal position of the number.

For the complex number \(-2 + 0i\):

• \(a = -2\)

This signifies the placement is \(2\) units to the left of the origin, solely along the real axis. A real part like \(-2\) helps us understand how the complex number aligns horizontally, exclusive of any vertical imaginary movement.