# Why do we call complex numbers “numbers” but we don’t consider 2-vectors numbers?

We refer to complex numbers as numbers. However we refer to vectors as arrays of numbers. There doesn’t seem to be anything that makes one more numeric than the other. Is this just a quirk of history and naming or is there something more fundamental?

• Usually, the difference would be whether you're in a context where complex multiplication would be interesting. It's kind of like the question: is $\mathbb{Z}$ a group or a ring? It depends on which operations are of interest for the question you're studying. – Nate Eldredge Mar 7 '19 at 23:20
• What's a "number" anyway? – Asaf Karagila Mar 7 '19 at 23:33
• @QthePlatypus: No, that's exactly the point. The question of whether the set $\mathbb{R}^2$ of pairs of real numbers should be considered as the set of complex numbers, or as a set of vectors, depends on which operations you wish to equip that set with. – Nate Eldredge Mar 7 '19 at 23:48
• I think the correct question were: is there a rigorous definition of the object "number" (without further characteristics such as "integer", "real", "complex" etc.)? – user Mar 7 '19 at 23:57
• Some considerations can be found here: math.stackexchange.com/q/865409 – user Mar 8 '19 at 0:01

They're called "numbers" for historical reasons, since the motivation in the development of the complex numbers was solving polynomial equations. They were viewed as natural extensions of the real numbers. It's somehow quite natural and satisfying to say "every polynomial equation can be solved by some (complex) number". Is it more natural to regard $$i$$ as being a number which, when squared, is equal to $$-1$$, or is it more natural to regard $$i$$ as being some non-number thingamajig which when squared is equal to $$-1$$? Clearly the former.

"Higher" number systems, like quaternions, aren't really called numbers very often, for the simple fact that they are not as intimately connected with number theory and analysis in the same way that complex numbers are.

Beyond these social conventions, I can't see any other reason. The word "number" doesn't have a strict or absolute definition in pure mathematics. $$\mathbb{N}$$, $$\mathbb{Z}$$, $$\mathbb{Q}$$, $$\mathbb{R}$$ and $$\mathbb{C}$$ are technically just sets with a certain algebraic structure.

I partially disagree with the other answers which claim that complex numbers are numbers simply by virtue of the fact that you can add and multiply them. Well, if that's the rationale, is every ring also a set of numbers?

• Wikipedia calls the quaternions a number system. – JJJ Mar 8 '19 at 4:52
• You can also tell what opinion people had of a number system based on its name, generally (it is not a coincidence that the word rational means both a number system and a way of thinking). At least the old ones. – PyRulez Mar 8 '19 at 6:26
• Agreed; the use of the word number is definitely a matter of convention. As for opinions; the use of the word 'real' numbers is regarded as an incredibly triggering piece of propaganda by physical finitists... – Eelco Hoogendoorn Mar 8 '19 at 12:40
• @PyRulez I would have expected that "rational" comes from "ratio", not from "ration". – Thern Mar 8 '19 at 16:31
• @Thern The English words "ratio", "rational" and "reason" all come from the same Latin root "ratio", meaning "reckoning" or "reasoning" – David Richerby Mar 8 '19 at 20:12

The two fundamental operations for numbers are "addition" and "multiplication" which obey very nice "laws" of arithmetic. Taking powers is also important. You can do all of those things with complex numbers. You can add two vectors but the "dot" product of two vectors is not a vector and the "cross" product of two vectors does not satisfy the "nice laws". Neither the dot product nor the cross product of vectors can be used to define powers.

• So Quaternions which have Non commutativity multiplication would also be considered as non numbers? – Q the Platypus Mar 7 '19 at 23:34
• That's a good point- though I have never liked quaternions! Multiplication of quaternions is not commutative but it is associative which allows powers. That's what I was really thinking of. – user247327 Mar 7 '19 at 23:42
• You can also add and multiply matrices, but I don't think I've ever heard anyone call a matrix a "number". – Hong Ooi Mar 8 '19 at 2:48
• @user247327: It's worth pointing out that IEEE 754 floating point numbers are non-associative, closed under division, non-reflexive (i.e. they violate $\forall a\:a = a$), non-substitutable (i.e. numbers which are "equal" cannot always be substituted for one another in expressions or equations), and also the most widely used approximation of the reals in practical computing applications. – Kevin Mar 8 '19 at 2:48
• @Todd Sewell - I think this may relate to positive and negative zero. Positive and negative zero are equal, yet if you divide something by one or the other you may get positive or negative infinity. – Hammerite Mar 8 '19 at 13:27

Numbers appeared first when we started to count things. $$1$$ tree, $$2$$ trees, $$3$$ trees, and so forth. That make up the set of non-zero natural numbers: $$\mathbb{N}$$. Afterwards, people started to "count backwards" to get $$\mathbb{Z}$$ (I'm just kidding, you can do some research on how negative numbers appeared historically). With the urge to divide things without remainders, the set of rational numbers $$\mathbb{Q}$$ made its way to the world. A certain idea of geometric continuity gives us $$\mathbb{R}$$. Finally we want all equations to have a root, that's how $$\mathbb{C}$$ comes into play.

I guess what is considered "numbers" is rather a social question. The use of $$\mathbb{C}$$ in physics and its $$2$$-D representation must have given a good intuition for a large set of people to accept it's intuitive enough to be considered "numbers".

I think it's not that natural to think of $$\mathbb{C}$$ as a $$\mathbb{R}$$-vector space of dimension $$2$$, not more natural than to think of $$\mathbb{R}$$ as an infinite-dimensional $$\mathbb{Q}$$-vector space, nor of $$\mathbb{Q}$$ as a non-infinitely-generated $$\mathbb{Z}$$-module.

• People do think of complex numbers as points on the number plane. – Q the Platypus Mar 7 '19 at 23:48
• Thank you. I'm sorry, that final part is just a personal opinion. edited. – BQT Mar 7 '19 at 23:48

I think naming is a rather soft topic, so there may not be a hard answer. However, I think it is worth noting that complex numbers are one of the three associative real division algebras (real numbers, complex numbers, and quaternions). These are all linked by the idea that division is meaningful in those three systems.

In general, a 2 dimensional real vector cannot admit a concept of division which matches what we expect division to do. However, if said real vector defines addition and multiplication the way complex numbers do, division is a natural result of those definitions.

Both, complex numbers and vector spaces are mathematical spaces, i.e., sets equipped it with some structure, e.g., operations (such as addition or multiplication) and rules obeyed by them (such as commutativity or associativity). Without that structure, these sets are pretty useless: You can hardly make any interesting statements about them or connect them to reality.

For example, without any structure, the only difference between ℝ and ℝ² is how we tend to name the objects within: We can find bijective maps between them. Of course, once we impose the usual structure onto these sets, we will note that these maps do not preserve any of it (they are not even continuous) and thus are not very relevant.

Now, for ℝ² (the space of 2-vectors) and ℂ, some structure such as addition is the same, but some isn’t, e.g., there is no equivalent to complex multiplication in ℝ². This distinction in structure is the only relevant difference between ℝ² and ℂ. If you define a multiplication of 2-vectors that just mirrors complex multiplication, you would just be inventing complex numbers under a different name.

So much for why it is justified to apply different labels to complex numbers and 2-vectors. As for why we call complex numbers numbers in the first place, there’s certainly a bit of history involved (of which I know little) and number is not a mathematically defined concept anyway. That being said, calling complex numbers numbers is not completely deviod of consistency, given that complex numbers feature most of the structural properties of other sets that we call numbers (for even more historical reasons), e.g., the vast majority of things you can do with real numbers, you can also do with complex numbers. This does not hold for 2-vectors.

The answer by @MathematicsStudent1122 is good but doesn't address the question about vectors. The reason for using a distinctive name for such objects is that, in general, they do not admit all the arithmetical operations that have we come to expect with numbers. In particular, there is generally no natural or canonical way of multiplying or dividing vectors. Exceptionally, for two, three, or four dimensions, operations that correspond to multiplication or division of vectors do exist, although they are no longer commutative for dimensions 3 or 4; moreover, in three dimensions, vector multiplication has un-numberlike properties (e.g. $$\pmb x\times\pmb x=\pmb0$$), and there is no natural way to define division. As you point out, real 2-D vectors can be identified with complex numbers. However, even in low-dimensional cases, typically vectors are not used in a context where any vector multiplication that could be defined has meaning.

In my observation, the term number is used almost exclusively in the following cases:

• The natural numbers, basically the origin of the term.

• Any system which extends a system already called “numbers” by adding numbers that are in some sense “missing” That is, there are some constructions which sometimes give a number, but sometimes don't, although is “looks like” such a number should exist.

The actual construction of the extension is then different from just adding those “missing” numbers just in order to make sure that what you do makes sense. However the goal is always to add the “missing” numbers, and any further numbers that are implied by their existence, but nothing else.

The classic sequence of number sets is exactly of this type:

• The natural numbers allow to solve equations like $$2+x=6$$, but not equations like $$5+x=2$$. Adding solutions to the latter gives the integers (which, of course, are considered to be numbers).

• The integers allow to solve equations like $$2x=6$$, but not equations like $$5x=2$$. Adding solutions to the latter gives the rational numbers.

• In the rational numbers, all convergent sequences are Cauchy, but not all Cauchy sequences converge. But they look like they should converge. Thus the missing limits are added. (Note that there are other constructions of the real numbers, based on other constructions where numbers are found "missing").

• In the real numbers, equations like $$x^2=2$$ can be solved, but equations like $$x^2=-2$$ cannot. Adding those numbers then gives the complex numbers.

But is also is true for most other systems which are generally called numbers. For example:

• Finite sets always have an integer number of elements. For infinite sets, there's no natural number describing the set's size. The cardinal numbers add those numbers describing the size of infinite sets.

• Similarly, when considering well-orderings, you need a generalization of the terms “first”, “second”, … to infinite size orderings. The ordinal numbers provide those missing numbers.

• When informally talking about differentiation and integration, one often uses the concept of “infinitesimal quantities”. Those don't exist in the real or complex numbers. Non-standard analysis adds those “missing” infinitesimals (and everything implied by them), arriving at the hyperreal numbers.

• When looking at the sequence $$r_n$$ of remainders modulo $$p^n$$, $$p$$ prime, every integer gives an unique sequence. But not every sequence gives a number. The $$p$$-adic numbers add those “missing” numbers.

Indeed, off the top of my head, I can only think of two constructions called “numbers” that fall outside this scheme:

• The quaternions are not constructed as extension, but explicitly as “numbers” that allow to describe three-dimensional geometry, after seeing that complex numbers describe two-dimensional geometry. It turned out that you need a fourth dimension to get something meaningful. However not everyone considers the quaternions to be numbers.

• The surreal numbers are not constructed as extension to anything, but arise independently from combinatoric game theory. However since not only behave very much like numbers do, but in addition contain subsets isomorphic to several other number sets or classes (you'll find subsets isomorphic to the real numbers, to all systems of hyperreal numbers, and to the ordinal numbers under natural arithmetic). So while the surreal numbers are not constructed as extension, they can be viewed in some sense as such, justifying the term “number”.

[Possibly not a direct answer, but]

Notice ordinal sum and product are not commutative, that full distributivity of multiplication over addition and of exponentiation over product fails, and they're still regarded as numbers. OTOH, the corresponding operations for vector spaces are well-behaved (probably because they do become cardinal-like once you identify the isomorphic ones) and yet we don't tend to think of or to refer to them as numbers.