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I am trying to understand imaginary numbers as something as natural and common as negative numbers or zero. The description that they are a rotation, $i^4$ bringing you back to one, and that this creates a complex plane, where numbers of just $i$ components exist, hasn't totally settled.

Does that mean there is a second complex plane analogous to a z-axis? Or multiple number planes that are totally separate from each other? This is a little harder to grasp.

But if not, why is there just one plane? Why can numbers go in another direction?

I think the key part that confuses me is the jump from $i$ being a two step process between -1 and 1, to it forming a whole new set of numbers.

Thanks

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    $\begingroup$ The existence of the field of quaternions is an answer. Have a look at (math.stackexchange.com/q/1916870). $\endgroup$ – Jean Marie Oct 22 '16 at 20:54
  • $\begingroup$ If you develop the complex numbers rigorously, one of the ways of doing so is as $(\Bbb R^2,+,*)$ endowed with addition $(a,b)+(c,d)=(a+c,b+d)$ and multiplication $(a,b)*(c,d)=(ac-bd,ad+bc)$. When talking about it as such, it is a two-dimensional space and so can be described as a single plane. That is not to say that there do not exist higher dimensional number systems, take the quaternions for example which live in a four-dimensional setting, but once we start talking about those, we are no longer talking about complex numbers only. $\endgroup$ – JMoravitz Oct 22 '16 at 20:55
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One way to think of the complex plane $\mathbb{C}$ is to think of it as the real coordinate (or $xy$-) plane $\mathbb{R}^2$. In $\mathbb{R}^2$, we have two real-numbered axes (in particular, the $x$ and $y$ axes). An element of $\mathbb{R}^2$ is an ordered pair $(x, y)$, where $x$ and $y$ range over all the real numbers $\mathbb{R}$.

What I am about to say is not exactly rigorous, but I purposefully omit rigor in favor of your understanding.

Now, what happens if we decide to multiply $y$ by $i$, for all $y$ in $\mathbb{R}$? We obtain $i \mathbb{R}$, where each element is of the form of $i y$, where $i$ is of course defined as usual and $y$ is a real number. Consequently our vertical $y$-axis now becomes imaginary, and we obtain the complex plane $\mathbb{C}$. Note that just like the real plane $\mathbb{R}^2$, $\mathbb{C}$ is 2-dimensional. To recap, $\mathbb{C}$ has two axes, one real axis and one imaginary axis.

What does an element of $\mathbb{C}$ look like? An element $z$ of $\mathbb{C}$ has the form $a + bi$, where $a$ and $b$ are real numbers; we call $z$ a complex number. If we were to graph a complex number $z = a + bi$, we would graph it just like we would an ordered pair $(a, b)$ in $\mathbb{R}^2$. E.g., the number $z = 1 + i$ would correspond to the point $(1, 1)$; the number $z = 1$ would correspond to the point $(1, 0)$. This second example demonstrates that a real number is in fact a complex number as well.

I am not sure if you have seen linear algebra or not, but your statement in the second sentence of your first paragraph can be explained with such ideas. In the real plane $\mathbb{R}^2$, we have two directions; namely, the $x$ and $y$ directions. It turns out that we only need the number $x = 1$ to generate the entire $x$-axis. Similarly we may do the same with the $y$-axis. Together, we obtain the whole plane $\mathbb{R}^2$. $\mathbb{C}$ carries the same idea, except your vertical axis is generated by $i = 1\cdot i$ instead of just $1$.

To further address your question, recall $i$ is defined as $i = \sqrt{-1}$. Thus, $i^2 = -1$, $i^3 = -i$ and $i^4 = 1$. What have we accomplished here? We have obtained directions for the vertical and horizontal axes. From here we may generate all of $\mathbb{C}$.

I am not entirely sure how to answer your next few questions, but recall from above that $\mathbb{C}$ is 2-dimensional. Thus, $\mathbb{C}^2 = \mathbb{C} \times \mathbb{C}$ would be 4-dimensional. Generalizing this notion, $\mathbb{C}^n$ is $2n$-dimensional. Hence I do not think there is really an analogy to the $z$ axis if we are referring to the third axis in $\mathbb{R}^3$. The best analogy I can come up with are several-dimensional complex number systems such as $\mathbb{C}^2$.

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