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For those who don't know what a complex number is, in simple terms, a complex number is the square root of a negative number! For example, the square root of -1 is called a complex number. Those numbers appear in laws describing real-world phenomena. Hence my question, how can we rationalize the fact that complex numbers exist?

I have some intuition about how to work with rational or real numbers, but complex numbers appear to be indistinguishable from magic! Why do we think relying on such numbers for physical theories is fine? Isn't this a huge problem, that most Physics is based on this?

** End of question by NoChance **

Comment: This is somewhat inaccurate. ... "Complex number"-s are the 'superset' of real numbers: they 'expand' the set of real numbers to 'include' the set of 'imaginary' numbers. .. Any positive number in the real number system has two square roots - positive and negative square root (I'm neglecting zero here because it is neither positive nor negative, but a number which is itself regardless of addition (by itself) or multiplication). But what about negative numbers? -- To help 'complete' this system of having square roots for ALL (up-until-then, known) numbers the concept of the imaginary number "&i$" was introduced [Reference: http://jeff560.tripod.com/i.html]: defined as the "number", when multiplied to itself, gives the number -1. Afterwards, the rest is - as they say - history! ... So, by definition a complex number is defined as a 'number' of the form $a + bi$ where $a$, $b$ $\in$ $\mathbb{R}$ [the set of all real numbers]. ... (Such numbers, considered by many, are their own mathematical objects - not the same as real numbers, but subtly related to them (via some very deep properties of mathematics which I don't know)). They are no less "numbers" than the "reals" or "rationals". ... Plus, from philosophy: what we call 'reality' can only be known from our experience(s) of it through our five senses: without them there is NO way for us to know anything of "reality" (you would have to ask a professor of philosophy why.. I don't really know why). Therefore, what we call 'intuitive' may / may not readily reflect the "ontological" nature of 'physical "reality"': hence, there are many [mathematical] concepts and constructs used in physics that are not readily humanly 'intuitive' [something to do with "positivism" 'n such.. again, I couldn't tell you what] - but, they are nonetheless central to investigating the measurable and experimentally-amenable features of our 'world'.]

NoChance
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    You are begging the question, when you say that a number cannot exist in this world whose square is -1. That is precisely what the imaginary unit *i* is! And if you acknowledge that the complex numbers have a basis in electromagnetism, why does it seem to you that this number is not meaningful as an instrument for talking about reality, or that reality also somehow disallows the algebra which complex numbers involve? – Niel de Beaudrap Sep 29 '11 at 20:03
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    Given that [Euler had problems with imaginary numbers](http://en.wikipedia.org/wiki/History_of_complex_numbers#History), this seems like a valid question. Furthermore, it isn't always the case that positing a new element/axiom to a formal system will keep it consistent, so asking whether adding _i_ will screw with algebra is a legitimate question. – labreuer Oct 28 '13 at 08:05
  • It reminded me of a [comment in a similar question on mathematics.se.](http://math.stackexchange.com/questions/154/do-complex-numbers-really-exist#comment252_154) – Red Banana Nov 02 '13 at 08:47
  • @Close/Reopen voters. I've an edit pending to substantially changed this question such that I think it is a real question that the answers address. Feel free to mention any problems in my edit! – Discrete lizard Mar 23 '18 at 11:23
  • @Discretelizard, thank you for your edit. I recommend you appended the modifications in your edit as a further clarification to the original question instead of changing the original question itself. – NoChance Mar 23 '18 at 16:34
  • @NoChance I thought is was needless chatter, but very well. – Discrete lizard Mar 23 '18 at 23:27
  • @NoChance Although I'm afraid I cannot make sense of it, it remains just too unclear to me. (are you answering commenters? Try make that a bit more clear) Feel free to improve. – Discrete lizard Mar 23 '18 at 23:29
  • What is unclear about the original question? – NoChance Mar 24 '18 at 23:51
  • You're willing to accept the real numbers, which require a second order construction, but not the complex numbers,which are an algebraic extension of the reals? You have problems with a square root of $-1$ (which, once you have the real numbers, is easy to adjoin) but not with a square root of $2$ (which takes a whole lot of hard work to construct)? What an odd set of priorities. – WillO Mar 25 '18 at 00:34
  • @WillO, I am not alone! Complex Numbers are also known as Imaginary Numbers. The question simply says, if something contradicts algebra as we know it, how can it be in harmony with our logic system? No odd set of priorities at all here. – NoChance Mar 25 '18 at 21:37
  • In what possible sense is the existence of a square root of 2 less contradictory to "algebra as we know it" than the existence of a square root of -1? – WillO Mar 25 '18 at 22:55
  • @WillO The square root of 2 as a numerical value obeys all rules of the "real" word. For example, its the "very exact" length of a side of a square of which the square's area is 2. To me that is real. The fact that we can't measure it to the last decimal is because our measuring device is inefficient. – NoChance Mar 26 '18 at 19:30

4 Answers4

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First of all, complex numbers (and imaginary numbers) do appear in real-world phenomena; they have lots of practical applications.

But now, on to the philosophical portion of the problem.

Numbers are abstractions. They don't exist in the same way that, say, physical objects exist. You can give me two apples, but you can't just give me a two.

As abstractions, they follow certain conceptual patterns. For positive integers, these abstractions are fairly intuitive; for other types of numbers (such as negative numbers, or rational numbers, or irrational numbers) they are less so.

Your question mentions the square root of -1, but let's take the square root of 2. Does that number "exist in the world" in any meaningful way? Can you give me the square root of 2 apples?

Fortunately, this doesn't hamper our ability to use the square root of 2; from a philosophical perspective, we can do this by adopting a position known as Fictionalism-- in short, we can treat numbers a fictional objects, and substituted them into formulas without making any ontological commitments as to their existence. As long as the substitution satisfies the constraints (which is to say, in a broader context, is adequate to the phenomena) we're golden.

So, to answer your question: we don't have to rationalize that complex numbers exist. It doesn't matter if they exist or not.

EDIT: I found an SEP article that specifically treats of Fictionalism in Mathematics; it is a nice reference for the more specific case.

Michael Dorfman
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    Well, the square root of 2 can be visualized as the hypotenuse of a right angled triangle with legs of sizes 1 and 1. But that is a distraction. As is well known by Galois theory, most real numbers can't be constructed in this way. – Rom Sep 14 '11 at 14:23
  • But I must object somewhat. I think, inspired by Godel's completeness theorem (although, unlike that theorem, applied to non-first-order-logic), we should define "existing" as consistent. It is surely important that the real numbers be consistent. In fact, that's their whole point. Complex numbers are also consistent. The proof is not obvious, but I outlined in the comments to the question. – Rom Sep 14 '11 at 14:32
  • I can certainly give you *approximately* sqrt(2) apples -- giving you a complex number of apples would definitely be an interesting feat – Joseph Weissman Sep 14 '11 at 14:43
  • It is true that the rational numbers are dense in the real numbers. In fact, the real numbers are the completion of the rational numbers. This has no bearing on anything, though. Just as I can't give you sqrt(2) apples, I can't give you i apples. Just as I can give you approximate sqrt(2) (=|sqrt(2)|) apples, I can give you approximately 1 (=|sqrt(i)|) apples. This has no bearing on whether the actual number "exists" or not. – Rom Sep 14 '11 at 14:46
  • @Rom: Why should we define "existing" as consistent? I agree that it is important that the real numbers be consistent; it is of no importance whatsoever whether or not they actually exist. Rejecting Platonism does not in any way hamper one's ability to use mathematics effectively. – Michael Dorfman Sep 14 '11 at 15:25
  • @Rom: Also, just to clarify: my argument in my answer is not "since you cannot give me the square root of 2 apples, the square root of 2 doesn't exist"; my point was rather that although positive integers are easy to represent with apples, the fact that some other numbers are not so easy to represent with apples has no bearing on whether or not the numbers exist. And, in fact, if one is a fictionalist with regard to numbers, it doesn't matter one whit. – Michael Dorfman Sep 14 '11 at 15:32
  • Existing implies consistency. In the case of 1st order logic, which is not the case of the axioms of the real numbers, then the reverse is also true (Godel's completeness theorem). In the case of non 1st order logic, in effect every proof I've ever seen for the consistency of some object is that it exists (constructing it, using the assumption of consistency of ZF), so in practice there's almost no difference. But people have issues with the word "existing", and we need some working definition. So why not consistency? – Rom Sep 14 '11 at 15:40
  • @Rom: I guess I don't see a connection between consistency and ontology. I don't see any reason why inconsistent things can't exist, and why consistent things can be non-existent. But, more to the point of this question, I don't see any reason to distinguish between different classes of mathematical objects when it comes to existence-- there's no reason to impute a different ontological status to complex numbers than, say, integers or reals. – Michael Dorfman Sep 14 '11 at 18:42
  • "I don't see any reason why inconsistent things can't exist" -- well, then, clearly your definition of existence is way too permissive. – Rom Sep 14 '11 at 19:03
  • Not all of us buy into fictionalism. The profound utility of complex numbers in describing certain physical phenomenon is actually one of many good arguments against it. (From your article, I fall into the platonist camp.) – Rune Aamodt Sep 17 '11 at 21:45
  • @codebolt: if you are a Platonist about complex numbers, all the better--then you have no problem with the question at all. The questioner was starting from the premise that the square root of negative one *does not exist*, so I tried to provide an answer on those terms. – Michael Dorfman Sep 18 '11 at 07:01
  • @Michael Dorfman Thank you for adding the link, I will read it. – NoChance Sep 22 '11 at 09:06
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Forget for a moment that you ever learned anything about so-called "complex" or "imaginary" numbers. Let's start with the numbers known as "real" numbers. We know how to do arithmetic on those. What about pairs of real numbers though? Suppose I had two elements,(a, b) and (c, d), where a, b, c, and d were all real numbers. Surely if real numbers exist then ordered pairs of real numbers exist. Can I meaningfully perform arithmetic on (a, b) and (c, d)?

I will attempt to define arithmetic in this way:

  • (a, b) + (c, d) = (a + c, b + d)
  • (a, b) * (c, d) = (ac - bd, bc + ad)

My means of defining multiplication might look a little strange, but I don't think that should be a problem in understanding.

So so far we've only dealt with real numbers and pairs of real numbers...certainly those exist. Let's make a strange observation about these pairs of numbers and their arithmetic:

(0, 1) * (0, 1) = (-1, 0)

So we can call the element (0, 1) the square root of the element (-1, 0). Surely this guy, (0, 1), exists in the same sense that the real numbers exist...he's just an ordered pair of real numbers in a system with a funky way of doing arithmetic.

Now those elements (a, b) seem like they should exist in the same way as regular real numbers, but they are a bit more complicated. They need some kind of name...since they are more complicated, how about we call them "complex" numbers. Let's also mandate that complex arithmetic works in the way that I described.

So we've just set up a system of numbers that we called "complex numbers" that doesn't reference things that don't exist. We just had to multiplication in a funny way. Now let's remember our old friend (0, 1)...let's give him a name. I'm lazy and don't want to give him a long name, so let's just call him i. I think you can see that any complex number (a, b) can now instead be rewritten as a + bi. So the complex numbers are numbers of the form a + bi where a and b are real numbers and i is our old friend (0, 1). There's nothing imaginary about him...he exists in the same way that his cousin (1, 0) exists.

Michael McGowan
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  • This is a brilliant way to explain imaginary numbers. I hope authors on the subject adopt a similar approach. I have no problem on starting with a def. and building a theory that is consistent with it No mater how strange the outcomes may look. In fact, this is the case in many mathematical subjects. Whether those mathematical abstractions exist or not is not quite what my question is about. What is truly special about (i) is that it is used to represent real things in this universe. That is we use this (strange and imaginary) quantity to measure (real) things that do exist in this universe. – NoChance Oct 01 '11 at 14:09
  • The 'brilliance' here is known as 'abstract algebra'! Do try and read some books on it if you like this! I think it is pretty doable for philosophers with a bit of background (say, a single course) in basic logic. The main idea is this: when working with numbers, we only care about what 'operations' we allow. So, we ignore everything _but_ the operations and even ignore how the operations work in practice and just define them using a few 'logical' axioms! This is a very nice field that really shows the power of abstraction in Mathematics! – Discrete lizard Mar 23 '18 at 11:28
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Geometry was axiomatized 2500 years ago. It was only towards the end of the 19thc that arithmetic was. Numbers are just so 'obvious' that it is hard to think about them.

Usually when you are introduced to complex numbers in school, they are quite obscure. Negative numbers at one point in history (of mathematical thought in Europe) were not considered numbers. Before that, there was controversy about zero, and even about 'one' being a number.

One can question all these things (and it is a good thing to question them) but at some point the questioning stops because you realize what you can and can't do. No, I can't hold '2+3i' apples, but that's OK, the formal rules that apply to such numbers don't apply to the situations of holding apples. I can see 5 apples, but not -5 of them, but that's OK, '-5' is not something to be seen (well, actually, if you see them in someone else's hands you might be considered to be seeing '-5 apples'). But do you really see '-5' by itself or even '5' by itself. I don't think so. Existence of numbers is not like the existence of real word objects.

Anyway, 5, -5, 3+2i don't actually exist 'out there', but we can use them when talking about 'out there' objects.

Mitch
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  • You last sentence is important, how can we describe reality with non-real abstractions (complex numbers)? Whys are we not using real abstractions to describe the real world? – NoChance Sep 15 '11 at 04:06
  • Because every polynomial has a complex root, but not nec. a real one. – Rom Sep 17 '11 at 04:55
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    @Emmad: In your mind, what's the difference between a non-real and a real abstraction? In mathematics, the label is merely historical baggage, not an ontological qualifier. Would the symmetric group on *n* objects be a real abstraction? What about the group of integers modulo *n*? – anon Sep 21 '11 at 21:26
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As others have already mentioned, numbers are of course abstractions. However, at least you could claim that whole integers hold meaning since they are representations of quantity in the real world. You have five apples in front of you, for instance.

Now if I take one apple, slice it in half, and leave the rest, you don't have four apples and neither do you have five still. Hence, real numbers can represent the real world as well.

Now if I took all apples away, what are you left with? Zero too is abstract, however without it, you could not represent a quantity representative of no apples. Same could be said of negative numbers which otherwise couldn't represent bank debt or a fall in market prices.

If you continue this line of reasoning, you'd understand that complex numbers are simply another extension to representation in the real world, albeit a bit technical. Just because you can't see the application yourself doesn't mean there isn't one.

Neil
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    A quibble: if you take an apple and slice it in two portions, you have a demonstration of *rational* numbers, not real numbers. – Michael Dorfman Sep 14 '11 at 18:44
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    Rational numbers is a subset of real numbers, being only a number that can be written in fraction form. I was referring to the ability to be able to have a quantity which cannot be expressed in terms of integers. You don't count the powder granules of flour when you wish to express a quantity, for instance. It may be an *irrational* weight that you have to deal with. Besides, for all intents and purposes, it looks like half an apple, but is actually a value which cannot be expressed as a rational number. – Neil Sep 15 '11 at 12:36
  • Since an apple has a finite number of atoms, and the number of atoms in each portion of the divided number is an integer, it looks like half an apple but is in actuality a rational number. Similarly, the granules of flour will always form a rational weight. Irrational numbers don't appear often in real-world examples (if we take measurements to their ultimate conclusion.) But as I said, this is just a minor quibble, and your answer was quite good. – Michael Dorfman Sep 15 '11 at 14:08