# Thread: The Complex Plane

1. ## The Complex Plane

Right now, I am learning of the Complex Plane, and I believe that I am having difficulty truly understanding what this means. So far, this is how I visualized it:

you have:

a + bi

and these two parts of the complex number can be graphed on the complex plane, which is a variant of the cartesian plane with an x and y axis for real and imaginary numbers respectively.

If you add two real numbers A and B, you get a value that is itself real because it's like taking an actual length and adding on another actual length.

In this case, you are adding a real number a with an imaginary number bi, which can not be defined on a real number line. When you try to add these two guys together, it's quite unlike your first case because you cannot add these lengths together in a linear fashion. Therefore you have to represent imaginary numbers on a different line. your result for adding these two together is expressed as a + bi, which is the blue line.

Is this the correct way of thinking about complex numbers and their graphing on the complex plane? If I am of or if there's something more to this concept that I need to visualize or understand, I would really like to know.

Thanks
---Richard

2. Originally Posted by Skinner
Right now, I am learning of the Complex Plane, and I believe that I am having difficulty truly understanding what this means. So far, this is how I visualized it:

you have:

a + bi

and these two parts of the complex number can be graphed on the complex plane, which is a variant of the cartesian plane with an x and y axis for real and imaginary numbers respectively.

If you add two real numbers A and B, you get a value that is itself real because it's like taking an actual length and adding on another actual length.

In this case, you are adding a real number a with an imaginary number bi, which can not be defined on a real number line. When you try to add these two guys together, it's quite unlike your first case because you cannot add these lengths together in a linear fashion. Therefore you have to represent imaginary numbers on a different line. your result for adding these two together is expressed as a + bi, which is the blue line.

Is this the correct way of thinking about complex numbers and their graphing on the complex plane? If I am of or if there's something more to this concept that I need to visualize or understand, I would really like to know.

Thanks
---Richard
There is an Algebraic way of looking at the complex number system, but for most people's purposes you have the concept down nicely.

-Dan

3. Oh yeah I know that there's the polar form of expressing complex numbers and such, but I was just making sure that I was making sense conceptually. my precal teacher told me that it was incorrect way of thinking about it but I don't really see where he's getting at anyway.

Is there anything else I should know in algebraic terms that would really enrich my understanding of this topic? I don't want to leave anything about this topic in ambiguity, at least within my scope of learning at the moment.

4. Originally Posted by Skinner
Is there anything else I should know in algebraic terms that would really enrich my understanding of this topic? I don't want to leave anything about this topic in ambiguity, at least within my scope of learning at the moment.
I agree with Dan that you seem to have a basic grasp of the concept.
I do wonder how you are dealing with the multiplication of complex numbers?

Just as we have the algebraic structure of a field for the real numbers, we must have that same structure on the complex numbers. So we must have additive and multiplicative identities. All numbers must have additive inverses all nonzero numbers must have multiplicative inverses. On that last point, it is important for you to under stand that the expression $\frac{1}{{3 + 4i}}$ is not considered a proper complex number. Its correct form is $\frac{1}{5} - \frac{4}{5}i$. You may want to find out why that is the case.

One finial point: I would encourage you to consider the number i as what it is.
The number i is simply a number that solves the equation $
z^2 + 1 = 1$
.