Graphing in the Complex Plane

The Complex Plane

We've now learned that Complex Numbers take the form of $z=a+bi$. Our next step is to learn how they appear on the complex plane.

The complex plane is very similar to the Cartesian plane that you are used to working in. The only difference is that the $x$-axis is replaced with the real ($Re$) axis, and the $y$-axis is replaced with the imaginary ($Im$) axis.

Before we dive into graphing complex numbers, let's think about how we graph points in the Cartesian plane. Every point on the Cartesian plane is written as $(x,y)$, where $x$ is the horizontal component, and $y$ is the vertical component. For example, the point $(3,2)$ is $3$ units to the right and $2$ units above the origin. Now, let's look at a complex number and break it down a little further. $$z=a+bi$$ Remember that $a$ is the real component, and $bi$ is the imaginary component. This means that when we graph this number as a point on the complex plane, it's horizontal position is $a$ units away from the origin, and its vertical position is $b$ units away from the origin.

Here is the complex number $z=-2+3i$ on the complex plane:

The point is placed $2$ units to the left of the origin and $3$ units above the origin. Just to sure up our understanding, here is the same point with each of the components labeled.

The horizontal component is $-2$, the vertical component is $3i$, and the diagonal is the magnitude of the complex number, which you'll learn about later.

This number would be $4$ units to the right and $5$ units below the origin.

The big takeaway here is that graphing points in the complex plane isn't very different than graphing points in the Cartesian plane. Remember that with the form $z=a+bi$, your horizontal component will be $a$, and your vertical component will be $b$.

What if we wanted to graph the number $z$, where

$z=$ $5$$+$$3i$

Well $z$ is a complex number, so we cannot graph it on the regular Cartesian plane. We must graph it on the $complex$ $plane$

Instead of a y and x axis, the complex plane has axis that are for imaginary numbers and real numbers. Where the y axis is on a Cartesian plane is where the imaginary numbers go, and where the x would go the real numbers lie.

So if we were to plot our number $z,$ we would go to the real numbers and go across five and go to the imaginary numbers and go up three, since we have $3i.$

The Complex Plane and Vector Analogy

In this explanation, we'll extend imaginary numbers to the set of complex numbers (not as "complex" as they seem!) and we'll look at the geometric interpretation of them.

Complex Numbers

We can define a complex number to be a number of the form $a+bi$ for real numbers $a,b$. As you might notice, this is just a real number added to an imaginary number!

Note that all real and imaginary numbers are also complex numbers, as you can let $a=0$ or $b=0$.

We can add complex numbers by adding their real and imaginary parts together; we can subtract them by subtracting their respective parts together. $$(a+bi) + (c+di) = (a+c) + (b+d)i$$ $$(a+bi) - (c+di) = (a-c) + (b-d)i$$ We also denote $\text{Re}(z)$ to be the real part of $z$ and $\text{Im}(z)$ to be the imaginary part of $z$ (NOTE: this does not include the $i$).

The Complex Plane

So far, we've only seen complex numbers in algebra. There's still a lot more (after all, this is the geometry part). We want to find a geometric representation of complex numbers, so we take a page from the euclidean plane and we define the complex plane. This plane has horizontal axis called the real axis and vertical axis called the imaginary axis. To plot a complex number in this plane, we take its real and imaginary parts and take locations on both axes corresponding to them.

Ex:

Here, the points graphed are $z = 3+5i$, $w=-7+2i$, $v=-5-5i$.

Vector Analogy

Let's try to add two complex numbers and see what the result is in the complex plane. $$z = 2+3i$$ $$w = 5+2i$$ $$z + w = v = 7 + 5i$$

Now let's connect some points:

Right! If we add two complex numbers together, they'll behave like vectors when added. The diagram above is the same diagram of vector addition. Thus, for example, we can express the complex number $a+bi$ as the vector $\left(\begin{array}{c}a\\b\end{array}\right)$.

That's all there is to this explanation of complex numbers. This is just the tip of the iceberg, and more will be explained later. Thank's for reading!

Here is an awesome video:

The original page is at BetterExplained.com.

Let $x,y\in\mathbb{R}$. ($\mathbb{R}=$ the set all real numbers). Then, $z=x+iy$ is a complex number. Now, how do we graphically represent this? The answer lies in treating $z$ as a pair of real numbers, namely $(x,y)$. Then we simply represent this ($z$) as a point in the plane $\mathbb{R}^2$, just like we plot a pair of real numbers. (When we plot complex numbers, we call the plane as Argand plane or Gaussian plane.)