91Ó°ÊÓ

Complex numbers have two parts: the real part and the imaginary part. The imaginary axis is the vertical axis on the complex plane, used to represent the imaginary part of a complex number.

The horizontal axis is called the real axis and represents the real part. In a complex number written as \(a + bi\), \(a\) is the real part, and \(bi\) is the imaginary part. For example, in the complex number \(-3i\), there is no real part (\(a = 0\)), and the imaginary part is \(-3i\).

You plot the imaginary part on the vertical axis of the complex plane. This means for \(-3i\), you would place a point at -3 on the imaginary axis.

Visualize it like this:
- The point \(1i\) would be at 1 on the vertical imaginary axis.
- \(-2i\) would be at -2 on the vertical imaginary axis.
Understanding the imaginary axis can make it easier to graph complex numbers and solve problems involving them.

Plotting complex numbers on the complex plane involves using both the real and imaginary parts. Let's break down the steps:

1. Identify the real part and the imaginary part of your complex number. For instance, in \(-3i\), the real part is \(0\) and the imaginary part is \(-3\).

2. Find the real part’s position on the horizontal axis (real axis). Since our real part is zero, we'll be right at the origin for this dimension.

3. Find the imaginary part’s position on the vertical axis (imaginary axis). For \(-3i\), we move -3 units down on the imaginary axis.

4. Plot the point where these two values intersect.

Following these steps will place \(-3i\) precisely at the point corresponding to zero on the real axis and -3 on the imaginary axis. Whenever you graph a complex number, you're essentially combining these two separate dimensions into one visual representation.

The absolute value (or magnitude) of a complex number represents its distance from the origin (0, 0) on the complex plane.

The formula for finding the absolute value of a complex number \(a + bi\) is \(\sqrt{a^2 + b^2}\). For example, let’s calculate the absolute value of \(-3i\):

1. Identify the real part \(a\) and the imaginary part \(b\): Here, \(a = 0\) and \(b = -3\).

2. Square both parts: \(0^2 = 0\) and \((-3)^2 = 9\).

3. Add these squares: \(0 + 9 = 9\).

4. Take the square root of the result: \(\sqrt{9} = 3\).

Thus, the absolute value of \(-3i\) is 3.

This means the point -3i is 3 units away from the origin on the complex plane. Understanding this helps to grasp how far a complex number is from the origin, giving a sense of its size or magnitude.