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Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that complex numbers are necessary and meaningful, in the same way that real numbers are?

This is not a Platonic question about the reality of mathematics, or whether abstractions are as real as physical entities, but an attempt to bridge a comprehension gap that many people experience when encountering complex numbers for the first time. The wording, although provocative, is deliberately designed to match the way that many people do in fact ask this question.

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The question isn't whether or not complex numbers exist, but whether or not there exists a complex number system (intuitively, a number system isomorphic to $\mathbb{C}$). According to ZFC, it does. This is unsurprising, given the usual geometric interpretation of $\mathbb{C}$ as a complex plane. For a treatment of these kinds of issues (but not $\mathbb{C}$ in particular), check out Goldrei's exceptional book. – goblin Nov 20 '13 at 4:44
I don't know if this was mentioned before, but one way I like to tackle this is to explain that irrational numbers aren't any more "real" than imaginary numbers. They are simply abstract answers to abstract questions. – Shai Deshe Mar 31 '15 at 14:50

35 Answers 35

I'm of the idea that complex numbres exist in the same way real number exists, we can be ware of real number every time because they're involved in our daily operations. Lets talk about phasors, they aren't more that convenient ways to express senoidal equations, based on the Eulers Identity:

$$e^{jt} = \cos(t) + j \sin(t) $$ Because this is easir to express and solve differential equations that involves senoidal functions in terms of exponential notation, then they become the prefered manner for analysis of electrical circuits of alternating current (CA). Some quantities sucha as impedance (Z=V/I) where V,I are phasors, have no sense without the complex numbers. In this application, physically the complex part is associated with reactive elements (inductors and capacitors) while the real part is associated with resistive elements (resistors). Also other kinds of transformation such as Laplace transformation (for solving linear differential equations) are based on complex numbers. I have had some work about these kind of notations and transformations and their application in Electrical Enginnering.

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Really, I could say that numbers themselves don't exist. Imagine I am an alien from the planet Krypton who just came here knowing nothing about numbers. All I know that are real are trees and apples and popcorn.

"Numbers?!", I might say. "Well if I can't see them, how do I know they are real?" Then you might get three apples and say to me, "Look here, see these apples?" I would nod my head, and you might say, "Well, numbers basically show us how much of something there is. I will explain the words we use to describe numbers." Then you may put away two of the apples, leaving one in your hand. Then you would say, "Here is one apple." Then you would get another apple and say, "Here are two apples." You would get the last one and say, "Three apples". Then you would start teaching me more numbers.

*Fast forward 15 years later*

Now I come to you again, asking you, "What are complex numbers?" How would you explain that to me? You might say, "Remember when I taught you what numbers were? It was like learning a totally new language! From there you learnt many things about numbers, which we humans call Mathematics. You learnt about algebra, which is just another part of the language of mathematics. You learnt many, many things about math. Now I am going to teach you about complex numbers. Think of it as yet another part of the language of mathematics."

"Remember when I taught you about the natural numbers ($\mathbb N$), then integers ($\mathbb Z$), then rational numbers ($\mathbb Q$), them real numbers ($\mathbb R$)? $\mathbb Z$ closed off $\mathbb N$ with negative numbers, $\mathbb Q$ closed off $\mathbb Z$ with fractions and decimals, and $\mathbb R$ closed off $\mathbb Q$ with irrational numbers like $\pi$. What then, closes off $\mathbb R$? Complex numbers ($\mathbb C$) do." After explaining what complex numbers are, you would say, "See how complex numbers close off the real numbers and fill in all the remaining gaps? If only real numbers exist, we would never be able to explore negative square roots. But with complex numbers, we can!" Now I might ask, "What closes off $\mathbb C$ then?" You would reply, "No one knows. Maybe one day, someone will invent a new number system that closes off $\mathbb C$, but as of now, there is nothing that closes off $\mathbb C$."

That is what I have to offer on the subject of complex numbers.

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I'm not a mathematician and for a long time I've struggled with the question if numbers exist.

Mathematics is a language. Numbers are part of its vocabulary, like "apple" is part of the English vocabulary. Just like you can build a sentence with the word "apple" you can build equations with numbers, and just like you have to follow certain rules to make a correct sentence you have to follow certain rules for your equation to hold true.
Now, and I'm trying not go into semiotics (of which I know nothing), the word "apple" is not an apple: you can't eat the word "apple" for one thing. But it exists in the language. You can invent things in a language to represent things in the real world. Does the verb "to walk" exist? It represents an action in the real world, but in itself it is nothing.
I think the same is true with numbers: numbers exist because we defined them in the language of mathematics, but they only make sense if you can connect them to something real. Natural numbers often represent the cardinality of sets: 10 may the number of real world apples in a real world basket. Rational numbers come in handy when comparing things: one apple mat be 1.2 times as large as another one. To non-mathematicians irrational numbers make less sense, for everyday non-mathematical use even pi is rational: 22/7.
Now for complex numbers same thing: we define them in the language of mathematics, but just like irrational numbers it's much harder to make them represent something real, although especially physicists are very good at describing the real world with them. What about other numbers like quaternions? They're part of the language, and are often required to make the grammar fit.

So, are complex numbers real? (mind the pun! :-))
They obviously exist in the language of mathematics, and may be used to describe real world things and events, but in themselves I don't think they exist.

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Your line of argumentation also shows that real numbers do not exist, not natural numbers, nor groups, nor anything else! – Mariano Suárez-Alvarez Sep 11 '10 at 12:37
-1; this doesn't answer the OP's question. – Qiaochu Yuan Nov 7 '10 at 11:11

In the en they exists as a consistent definition, you cannot be agnostic about it.

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Some more elaboration would be nice. – J. M. Apr 14 '13 at 2:34

I posted a similar question recently about how complex analysis describes reality better than real analysis and I got very interesting answers.

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protected by Qiaochu Yuan Jun 17 '11 at 10:02

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