Minor nitpick, what you are describing here is algebraic closure not completeness. I know you were using ‘complete’ as shorthand, but complete has a mathematical meaning. The real numbers are complete, but not algebraically closed. The complex numbers are both complete and algebraically closed.
Not so dumb. e^{i\pi/2} is a square root of e^{i\pi} which is i, as it should be. (I am ignoring the possibility of -i).
I think it is worth pointing out that though there are different notions of completeness. Originally, when the real numbers were described as ‘complete’ what was meant is that it is complete as an ordered field. Famously, a complete Archimidean ordered field is isomorphic to the real numbers. The complex numbers are not an ordered field so this notion of completeness doesn’t apply to them. Whilst this is still what is meant often when the reals are described as complete, it can be seen as a special case of Cauchy completeness and the complex numbers are also Cauchy complete.
Fair enough.
Careful here. The non-zero complex numbers are a Lie group, but the whole ring of complex numbers is not a Lie algebra. It fails to satisfy the identities ab = -ba and (ab)c + (ca)b + (bc)a = 0. I imagine (it has been over 60 years since I studied this) that the Lie algebra of the non-zero complex is the commutative (meaning product of any two elements is 0) additive group of complex numbers.
It is impossible to include division by 0 without losing most of the basic axioms of arithmetic.
While the algebraic numbers allow for roots of all polynomials, they are not complete with respect to things like trig functions and logs. They are also not Cauchy complete: Cauchy sequences do not converge within the algebraic numbers. It would take much of a course in algebra to explain all this.
I did (perhaps not carefully) say “complex numbers under addition”, or more precisely in this case regarded as a 1-d complex vector space. As for the Lie algebra structure, to avoid confusion with the product, we can (and many do unless I am mistaken) call the operation the “Lie bracket” and write [a,b], in this case [a,b] = 0 for any two elements (since the related group is commutative)
Perhaps worth noting that since [a,b] =0 identically the exponential really “converts addition to multiplication” in this case: exey = ex+y. More generally exey = ez where z = x + y + [x,y]/2 + ([x,[x,y] + [y,[y,x])/12 - [y,[x,[x,y]]]/24 + … which is relevant for non-commutative Lie groups, such as arise in physics and elsewhere.
Personally, I like the name “imaginary”. Reasons:
- It’s poetic. It gives the numbers a sense of mystery and allure. I’m completely serious here; I think the name played no small part in 10-year-old me being interested in them, and then other related concepts. The complex numbers are one of the most beautiful objects in mathematics and the imaginary component deserves an appealing name.
- There is an element of truth to the real/imaginary distinction. In many cases, say the solutions to a parabolic trajectory, real solutions really do represent real-world solutions, while imaginary/complex ones do not. That does not make the numbers less legitimate, but does hint that they represent things that are somehow outside our world.
- Math is filled with names that bear little relation to their normal meanings. Surreal numbers are not particularly dreamlike. A field is not a flat expanse of grass. Sets and groups are near synonyms in English but mean very different things in math. And so on. It may actually be better to have names that don’t bear much relation to the concept.
“Sideways”, “perpendicular”, etc. are ugly words and carry baggage of their own (Why is the imaginary component the sideways one? I say that the real numbers are the sideways numbers!)
“Perpendicular” is a relative term. The reals and imaginary numbers are perpendicular to each other. It’s not like the x-axis is the one true axis and the y-axis is a weird sideways one. They are the two co-equal components of a two-dimensional space and are perpendicular to each other.
It’s still an ugly, boring word. Imaginary sounds so interesting in comparison.
Right, but the whole problem is that the word “imaginary” doesn’t match the real-world usefulness and relevance of the numbers. Lasers are really cool, but if we called them laser numbers that would probably confuse people too.
“Perpendicular” is even worse in that regard. And that word only makes sense once you already know what an imaginary number is, by which time the name doesn’t matter too much. The connection between i^2=-1 and i being at a right angle to 1 is not that obvious, or even that relevant for the initial applications of imaginary numbers. Also, “perpendicular” is limiting in its specificity; i is just as much a rotation as it is a direction.
If we’re giving up the name “imaginary”, are we giving up “real” as well? Descartes came up with both names as a contrast. “Real” is just as unhelpful by the same standard.
Rotational numbers and radial numbers?
For certain applications, sure. Those have a nice ring to them in any case. But they are still a bit overspecific.
There are lots of ways to view the complex numbers. You can see them as what you get when you adjoin i to the real numbers, where i^2=-1. You can view them as pairs of reals such that (a,b)+(c,d)=(a+c,b+d) and (a,b)(c,d)=(ac-bd,ad+bc). You can view them as pairs of reals such that (θ,r)(φ,s)=(θ+φ,rs) (and addition is more difficult). You can view them as matrices, where a+ib=[[a -b][b a]] and they follow the normal rules for matrix math. You can view them as a helix with a diameter, a spacing between loops, and a handedness.
I’m sure there are many other objects they map to. Some of these have an obvious rotational aspect, but others don’t. As is the case with many mathematical objects, thinking of them in several different ways leads to greater understanding of their nature.
That’s only because they’re taught poorly. If you start with the idea of them being a right turn, the whole body of knowledge becomes much easier to understand.
What’s something that, if you do it twice, results in you facing the opposite direction? A 90º turn. That’s it. That’s all there is to i.
Agreed.
No, @Dr.Strangelove is right. Most people first encounter imaginary numbers in an algebraic context (solutions to algebraic equations, which is also the context in which they first arose historically) where “the idea of them being a right turn” is not particularly relevant. It’s not necessarily poor teaching to focus on only those properties that are relevant to the task at hand.
Absolutely, yes!
It’s a terrible name that hinders the understanding of mathematics for many people.
How about, “Impressive Numbers”?
vs “Regular (boring) Numbers”
What @Dr.Strangelove and I were saying about groups, more explicitly, is that the complex numbers of unit magnitude are well regarded as the Lie group U(1). Now this group is isomorphic to SO(2), and what are those? Rotations in the plane. This reasoning can help one to understand/visualize why Hamilton’s quaternions represent 3-d rotations; the former are isomorphic to SU(2) which maps 2:1 onto SO(3).
The typical contrast, as noted, is of “complex” numbers versus “real” numbers, “p-adic” numbers, “finite fields”, etc.
This does sometimes lead to terminology like “simple complex Lie groups”, but if you are already studying that material you should be able to keep things straight.
Note, also, that people always talk about “Gaussian integers”, not imaginary integers…