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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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You might want to say something about in what way you think real numbers are "real" and "valid." The real numbers (unlike say the rational numbers or the computable numbers) are a purely mathematical abstraction. –  Noah Snyder Jul 20 '10 at 22:20
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It would be deliciously ironic if this got closed for not being a real question. –  Larry Wang Jul 21 '10 at 1:30
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If by ironic you mean "fitting", then yes. –  user126 Jul 21 '10 at 2:23
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@harry-gindi,@kaestur-hakarl, this site is for mathematicians of all levels, including those in high school, to get help with understanding math. These kind of comments don't help with that at all. For example, by the time someone knows what a field structure on R^2 is, they will already be very familiar with complex numbers. –  Neil Mayhew Jul 21 '10 at 4:12
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Do fractions really exist? Show me 3/5 of a piece of chalk. Isn't that a piece of chalk? –  Peter Shor Nov 21 '10 at 16:13
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34 Answers

There is a lovely way of motivating the "existence" of the complex numbers just by using a little calculus on the real numbers. I found this in Visual Complex Analysis, and it tickled me, so I thought I'd share it here, despite the lateness of the answer.

If $r_1,...,r_n$ are real numbers, define:

$$f(x)= \frac{1}{(x-r_1)(x-r_2)...(x-r_n)}$$

When $a\notin \{r_1,...,,r_n\}$ we can find the Taylor series around $a$:

$$\sum_{k=0}^\infty \frac{f^{(k)}(a)}{k!}(x-a)^k$$

The question is, for what (real) $x$ does this series converge to $f(x)$?

As it turns out, if we let $R=\min_{k} |a-r_k|$, then if $|x-a|<R$ this series converges to $f(x)$ and if $|x-a|>R$, then it doesn't converge.

So, in a sense, the $r_i$ "block" the ability of the Taylor series to converge around them.

Now, what about the Taylor series for $g(x)=\frac{1}{x^2+1}$?

Given an $a$, this function has no "real" blockages - it is defined on all of $\mathbb R$ - but the Taylor series for $g(x)$ around $a$ has a similar $R$ value, and that $R$ value is $\sqrt{1+a^2}$, a value that can be computed entirely with real number calculations.

That then looks like there is some geometric obstruction to the Taylor series, an obstruction not on the real line, but a unit distance away from the real line in a perpendicular direction away from $0$. It "looks like" an "imaginary" root of $x^2+1=0$.

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

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