Problem 48 - Algebra 2 and Trigonometry

Question

In \(44-51,\) locate the point that corresponds to each of the given complex numbers. $$ \frac{1}{2}+4 i $$

Step-by-Step Solution

Answer

The point corresponding to \(\frac{1}{2} + 4i\) is \((\frac{1}{2}, 4)\) on the complex plane.

Step 1: Understand the complex number format

A complex number is written in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. In the complex number \(\frac{1}{2} + 4i\), \(\frac{1}{2}\) is the real part and \(4i\) is the imaginary part.

Step 2: Identify the real part

The real part of the complex number \(\frac{1}{2} + 4i\) is \(\frac{1}{2}\). This value corresponds to the x-coordinate on the complex plane.

Step 3: Identify the imaginary part

The imaginary part of the complex number \(\frac{1}{2} + 4i\) is \(4\). This value corresponds to the y-coordinate on the complex plane.

Step 4: Locate the point on the complex plane

To locate \(\frac{1}{2} + 4i\) on the complex plane, move \(\frac{1}{2}\) units along the real axis (horizontal) from the origin, and then move \(4\) units up along the imaginary axis (vertical). This gives you the point \((\frac{1}{2}, 4)\).

Key Concepts

Real Part of a Complex Number

The real part of a complex number is the component that does not involve the imaginary unit, denoted by the symbol \(i\). When a complex number is expressed in the general form \(a + bi\), \(a\) represents the real part.

To visualize this component, imagine plotting it on the x-axis of a graph, called the real axis, which forms part of the complex plane.

• For example, in the complex number \(\frac{1}{2} + 4i\), the real part is \(\frac{1}{2}\).
• This means we would start at the origin of the graph and move \(\frac{1}{2}\) units along the horizontal x-axis.

The real part is crucial for locating the exact position of a complex number on a two-dimensional plane.

Imaginary Part of a Complex Number

The imaginary part of a complex number is linked to the coefficient of \(i\). In our example, the complex number \(\frac{1}{2} + 4i\) has an imaginary part of \(4\).

This component is portrayed along the y-axis, known as the imaginary axis, on the complex plane.

• While the real part determines the x-coordinate, the imaginary part sets the y-coordinate.
• For the number \(\frac{1}{2} + 4i\), the imaginary part is \(4\), which tells us to move up \(4\) units along the vertical y-axis.

The imaginary part is essential because, in conjunction with the real part, it fully specifies the number's position in the complex plane.

Complex Plane

The complex plane is a two-dimensional graphical representation where complex numbers, such as \(a + bi\), are visualized. This plane is a valuable tool for understanding the structure and behavior of complex numbers.

• The horizontal axis, or "real axis," displays the real part of the complex numbers.
• The vertical axis, or "imaginary axis," exhibits their imaginary parts.

To locate a complex number like \(\frac{1}{2} + 4i\), you follow two steps:

• Move \(\frac{1}{2}\) units from the origin along the real axis.
• Then ascend \(4\) units up along the imaginary axis.

This results in the point \((\frac{1}{2}, 4)\), giving us a clear, visual indication of its real and imaginary constituents on the complex plane. Consequently, the complex plane not only aids in calculating but also intuitively understanding and predicting the relationships between complex numbers.