Problem 1 - Precalculus Mathematics for Calculus

Question

A complex number \(z=a+b i\) has two parts: \(a\) is the _____ part, and \(b\) is the _____. To graph \(a+b i\) we graph the ordered pair \((\square, \square)\) in the complex plane.

Step-by-Step Solution

Answer

The real part is \(a\), the imaginary part is \(b\), and the ordered pair is \((a, b)\).

Step 1: Identify Parts of Complex Number

A complex number is of the form \(z = a + bi\), where \(a\) and \(b\) are real numbers. The number \(a\) is called the real part of the complex number, and \(b\) is called the imaginary part.

Step 2: Determine Ordered Pair for Graphing

To graph a complex number \(a + bi\), we use the complex plane, which is similar to a coordinate grid. The real part \(a\) is plotted on the x-axis and the imaginary part \(b\) is plotted on the y-axis. Therefore, the ordered pair is \((a, b)\).

Key Concepts

Real Part

The real part of a complex number is quite simple. For a complex number written as \( z = a + bi \), the "real part" is the value \( a \). This part is not associated with the imaginary unit \( i \) and behaves just like regular numbers you are used to. It is the component of the complex number that we plot on the x-axis of a complex plane.

• The real part is always a real number.
• It determines how "far" the complex number is along the horizontal axis.

So, whenever you are asked to find the real part, look for the number without the \( i \). If your complex number looks like \( 3 + 4i \), the real part is \( 3 \). It's like the "longitude" of your complex number's location on the complex plane. Remember, this value can be positive, negative, or zero.

Imaginary Part

Next, let's talk about the imaginary part. Just like the real part, the imaginary part holds a specific value with a key twist—it includes the imaginary unit. In the expression \( z = a + bi \), you find the imaginary part as \( b \), but it is often expressed with the \( i \), like \( bi \). This part is what gives complex numbers their unique characteristic.

• Remember, \( i \) represents the square root of -1.
• The imaginary part, \( b \), is plotted on the y-axis of the complex plane.

This means when you are graphing, \( b \) determines how far above or below the x-axis your complex number sits. It’s like the "latitude" of your position on the complex plane. Using our example \( 3 + 4i \): the imaginary part is the number \( 4 \), and it dictates the vertical position. Whether positive or negative, it shows the "up or down" position on the graph.

Complex Plane

The complex plane is an essential tool for visualizing complex numbers. Imagine it like a regular coordinate grid you’ve used in geometry, but with a twist. Instead of just real numbers, this plane encompasses the full picture of complex numbers.

• It has a horizontal axis called the "real axis," where you plot the real parts.
• The vertical axis is known as the "imaginary axis," where you plot the imaginary parts.

Think of plotting a complex number like \( 3 + 4i \) on this plane. You would move "3" units along the real axis and then "4" units up on the imaginary axis. This creates a point that directly corresponds to the ordered pair \((3, 4)\). The complex plane helps you see relationships and patterns among complex numbers, much like how we use regular graphs to understand real numbers. It’s your "map" where you can journey with both real and imaginary companions!