Problem 12 - College Algebra

Question

For the following exercises, plot the complex numbers on the complex plane. $$ -2+3 i $$

Step-by-Step Solution

Answer

Plot \((-2, 3)\) on the complex plane.

Step 1: Understanding the Complex Number

The given complex number is \(-2 + 3i\), where \(-2\) is the real part and \(3i\) is the imaginary part. We will plot this point on the complex plane, which consists of a horizontal real axis and a vertical imaginary axis.

Step 2: Identify the Real and Imaginary Parts

In the complex number \(-2 + 3i\), identify the real and imaginary parts. Here, the real part is \(-2\) and the imaginary part is \(3\).

Step 3: Plot on the Complex Plane

To plot \(-2 + 3i\) on the complex plane, start at the origin (0,0). Move \(-2\) units along the horizontal axis (to the left, because it is negative) and \(3\) units along the vertical axis (upwards, because it is positive).

Step 4: Mark the Point

Mark the point corresponding to the coordinates \((-2, 3)\) on the complex plane. This represents the complex number \(-2 + 3i\).

Key Concepts

Plotting on Complex Plane

Complex numbers can be visualized on what we call the "complex plane". Think of this plane as similar to the standard Cartesian plane. However, instead of having two real number axes, it has a real axis (horizontal) and an imaginary axis (vertical).
When plotting the complex number \(-2 + 3i\), follow these steps:

• Start at the origin, which is \(0+0i\).
• Move along the real axis according to the real part of the number. For \(-2\), go 2 units to the left.
• Next, move along the imaginary axis based on the imaginary part, which is \(3i\), moving 3 units upwards.

After making these moves, you should land at the coordinates \((-2, 3)\), accurately plotting your complex number.

Real and Imaginary Parts

In a complex number \(a+bi\), \(a\) represents the real part, and \(bi\) represents the imaginary part. Let's break it down with \(-2 + 3i\):

• The real part, denoted by \(a\), is \(-2\).
• The imaginary part, represented by \(b\), is \(3\) because it's paired with \(i\), the imaginary unit.

The real part indicates how far left or right the point is plotted on the real axis. Meanwhile, the imaginary part shows the distance up or down on the imaginary axis. Understanding these components is crucial for correctly representing complex numbers on the plane.

Complex Plane Coordinates

Coordinates in the complex plane are expressed similarly to Cartesian coordinates but represent different concepts.
For complex numbers like \(-2 + 3i\):

• The first number \(-2\) corresponds to the horizontal position, based on the real part.
• The second number \(3\) corresponds to the vertical position, derived from the imaginary part.

With these coordinates \((-2, 3)\), you can precisely locate the complex number on the plane. This approach gives a clear visual representation of the number's magnitude and direction, helping to understand complex operations and interactions.