Complex numbers can puzzle me a bit, and I think I have some gaps in my understanding that makes it confusing for me to wrap my head around.
The way I try to explain complex numbers to myself is this:
- If we do not include complex numbers, we can find numbers that cannot exist on the Cartesian plane, such as with complex solutions to certain quadratics. We're missing numbers on our Cartesian plane, and we're trying to represent all numbers on this plane.
- If we instead express all numbers as $a + bi$ instead of $a$ we can now identify all numbers and be able to express them on the Cartesian plane.
If my interpretation is wrong in some way please do explain why.
Regardless, my main question is why, this follows, that we now have the Argand diagram derivative of the reals-only Cartesian plane, with the imaginary axis on the $y$ axis and reals on the $x$ axis? Does this have to do with a logical progression of when I said "If we instead express all numbers as $a + bi$ instead of $a$ we can now identify all numbers and be able to express them on the Cartesian plane."? Let me elaborate:
- Since numbers are now expressed as in the form $a + bi$, if the imaginary axis is on the $y$ axis and the real number line is on the $x$ axis, all numbers can be accounted for elegantly.
- This is due to the fact that now, any number can be expressed by its projection in the imaginary and real axis.
But due to someone establishing this rule, we now have the following properties:
- We now use vectors to typify numbers. $3 + 2i$ is now a projection in this plane, and we now use things like vector addition. And then that can snowball into Euler's identity?!
- Multiplying by $i$ now causes projections to "rotate" in the plane.
This just seems bizarre to me, and it's because we've elected to define our $y$ axis as a number multiplied by the square root of $-1$, which sounds totally arbitrary.
Why couldn't I make my own diagram and instead of multiplying numbers in the y-axis by $i$, I multiply it by $k$ which I define as equal to $13$? And all these new properties I listed, like the extended Cartesian plane now being sort of like a vector space and have rotational properties seem crazy to me.
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So,\ to\ recap,\ my\ main\ question\ is:\ why\ has\ all\ this\ been\ implemented*,\ keeping\ in\ mind\ the\ new\ odd\ properties\ our\ extended\ Cartesian\ plane\ now\ has?\
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What's the deal about all of this? How and why did we go from our number line of reals and end up with all this?
I hope I've made my question stated clearly enough. Please let me know if something needs to be made more clear.
*And by "all this been implemented", I mean creating a $y$ axis, the imaginary axis, for the real number line and allowing all these odd properties to form, hence the title.
Best Answer
In principle we could do what you suggest -- take $\mathbb R^2$ and associate every point $(x,y)$ to the number $x+13k$. Though the trouble with that particular plan is that each number now represents many different points -- for example, $(13,0)$ and $(0,1)$ and $(26,-1)$ are now all associated to the number $13$. This means that we can't use the scheme for anything where we calculate a number and that number points to exactly one point in the plane.
We could, however, do something more general. Take some field that extends $\mathbb R$, pick some element $\alpha$ in that field, and then represent $(x,y)\in\mathbb R^2$ by $x+\alpha y$.
As it went for $13$, if we pick $\alpha\in\mathbb R$, then we get something where a number doesn't represent a unique point. Suppose, however, that we steer clear of that case, and furthermore that we end up in the lucky situation that every element of the field represents some $(x,y)$ in the plane.
Something wonderful happens then -- namely, we can then prove (though not in the space left for me in this margin) that the field we're using must be isomorphic to $\mathbb C$ -- in other words the field is essentially the complex numbers, just called something different. In particular, somewhere in the plane there is an $(x,y)$ whose corresponding number behaves exactly like $i$.
So we could actually have said: Pick some complex number $\alpha$, and let $(x,y)$ correspond to $x+\alpha y$. As long as $\alpha$ is not real, this will give us a perfectly good one-to-one correspondence between points and complex numbers.
Now, among all the possible choices of $\alpha$ it turns on that exactly when $\alpha=i$ or $\alpha=-i$ we get the additional nice property that multiplication by any fixed nonzero complex number will correspond to a transformation of the plane that takes geometric figures to similar geometric figures.
Having multiplication correspond to similarity transforms is a pretty nifty property, which is a reason to prefer the representation $x+iy$ over other possible $x+\alpha y$.