Problem 46 - Algebra 2 and Trigonometry

Question

In \(44-51,\) locate the point that corresponds to each of the given complex numbers. $$ 4-2 i $$

Step-by-Step Solution

Answer

Plot the point at (4, -2) on the complex plane.

Step 1: Understand the Components of a Complex Number

A complex number is represented in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. For the complex number given, \(4 - 2i\), the real part \(a\) is 4 and the imaginary part \(b\) is -2.

Step 2: Identify the Coordinate System

Complex numbers are plotted on a complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. The point is plotted with coordinates \((a, b)\).

Step 3: Plot the Complex Number

To locate the point for the complex number \(4-2i\), we use the coordinates \((4, -2)\). Move 4 units along the real axis (x-axis) and then move down 2 units along the imaginary axis (y-axis) to find the point.

Key Concepts

Real Part

The real part of a complex number is one of its fundamental components. When we express a complex number in the form \(a + bi\), the real part is represented by \(a\). It essentially corresponds to the number line we know from basic arithmetic. In the complex number \(4 - 2i\), the real part is \(4\).

• The real part lies along the horizontal axis (x-axis) of the complex plane.
• It indicates the "real-world" portion of the number, shedding light on values we're familiar with in non-complex scenarios.

When dealing with complex numbers, always separate the real component from the imaginary. This helps in identifying precise values on the complex plane. By isolating and focusing on the real part initially, you lay the groundwork for understanding more sophisticated operations involving complex numbers.

Imaginary Part

The imaginary part of a complex number introduces us to numbers that may not fit into the traditional "real" framework we're accustomed to. In the standard form \(a + bi\), the imaginary part is represented by \(bi\). In our specific example of \(4 - 2i\), the imaginary part is \(-2i\).

• The imaginary part is plotted along the vertical axis (y-axis) on the complex plane.
• It gives us a way to work with problems that can’t be solved with only real numbers, such as square roots of negative numbers.

Understanding the imaginary part is key to mastering complex numbers. Think of it as an "extension" beyond the real. This imaginary component allows computations with otherwise "unsolvable" problems, opening a door to more advanced mathematical concepts.

Complex Plane

The complex plane is a two-dimensional plane used to graphically represent complex numbers. Much like the Cartesian coordinate system, it utilizes two axes:

• The horizontal x-axis for the real part.
• The vertical y-axis for the imaginary part.

In plotting the complex number \(4 - 2i\) on the complex plane, you move 4 units along the x-axis to accommodate the real part and \(-2\) units along the y-axis to address the imaginary part. This results in the coordinate point \((4, -2)\).
Visualizing complex numbers on the complex plane aids in comprehending their behavior in addition to providing insights into higher-level mathematical concepts such as vector spaces and signal processing. Embracing this visual representation is crucial for anyone learning about complex numbers.