Problem 49 - Algebra 2 and Trigonometry

Question

In \(44-51,\) locate the point that corresponds to each of the given complex numbers. $$ 0+3 i $$

Step-by-Step Solution

Answer

The point for the complex number \(0 + 3i\) is located at \((0, 3)\) on the complex plane.

Step 1: Understand the Complex Number Format

A complex number is generally expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. In this case, you are given the complex number \(0 + 3i\), where the real part \(a = 0\) and the imaginary part \(b = 3\).

Step 2: Identify the Axis on the Complex Plane

In a complex plane, the x-axis (horizontal axis) represents the real part of the complex number, while the y-axis (vertical axis) represents the imaginary part of the complex number.

Step 3: Locate the Point on the Complex Plane

To locate the number \(0 + 3i\) on the complex plane, start at the origin \((0,0)\). Since the real part is \(0\), there is no movement along the x-axis. Then, move 3 units up on the y-axis to account for the imaginary part \(3i\).

Step 4: Plot the Point

Plot the point on the complex plane at the coordinates \((0, 3)\), where \(0\) is on the x-axis and \(3\) is on the y-axis.

Key Concepts

Complex Plane

When dealing with complex numbers, it's important to understand the concept of the complex plane. This is a two-dimensional plane where each point represents a complex number. The complex plane has two axes:

• Real Axis: This is the horizontal axis. It represents the real part of the complex number.
• Imaginary Axis: This is the vertical axis. It represents the imaginary part of the complex number.

For a given complex number like \(0 + 3i\), you can locate it by identifying its coordinates on this plane. In this example, the real part is 0, so there is no movement along the real axis (horizontal). Instead, you move upward along the imaginary axis by 3 units. This result in the point being plotted at (0, 3) on the complex plane. The complex plane allows us to visualize the two components, real and imaginary, in a straightforward manner.

Real Part

The real part of a complex number is an integral component that helps define its position on the complex plane. In a complex number represented by \(a + bi\), the real part is denoted by \(a\). Consider the complex number \(0 + 3i\); here, the real part \(a\) is 0.

• This means the number is located directly along the imaginary axis when plotted on the complex plane.
• The representation as 0 on the real axis reflects no horizontal movement from the origin.

The real part is critical when analyzing and visualizing complex numbers, as it signifies how far along the horizontal axis the number will be placed from the origin. Remember, if the real part is zero, the complex number lies entirely on the imaginary axis, showing no horizontal displacement.

Imaginary Part

The imaginary part of a complex number distinguishes it as more than just a point on the number line. Represented in the form \(a + bi\), the imaginary part is \(b\). In the case of the complex number \(0 + 3i\), the imaginary part is 3.

• This dictates movement vertically along the imaginary axis on the complex plane.
• The value of 3 indicates an upward shift from the horizontal origin point by 3 units.

Imaginary parts add a layer of complexity by introducing a new dimension to numbers, beyond the real number line. This vertical component enriches how we understand and utilize these numbers in various mathematical fields. Observing its placement helps in grasping conceptually rich ideas originating from complex number theory.