# Do complex numbers really exist?

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.

• 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
• Richard Dedekind asked a question which is relevant (although he asked it in a different context): "Was sind und sollen die Zahlen?" which translates as "What are the numbers... and just what SHOULD they be?" (English needs the extra length to encapsulate the same meaning.) Essentially, Dedekind asserts in his question that humans are free to define just what numbers "are". What we decide numbers to be is usually driven by pragmatic needs and ends. Of course, once one realizes that complex numbers encapsulate rotation, applications abound and it becomes useful for $i$ to be a "number". – Robert Wolfe Jan 30 '18 at 22:01

## 36 Answers

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.

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

A number of answerers have answered by asking the question of whether the real numbers exist.

There are lots of physical examples that a person can give to a lay-person of real numbers, including fractions, zero and negative numbers. Vector addition all by itself provides plenty of examples. For instance, most anyone can develop an intuition for the idea that if a vector pointing north has a positive component, then it can be combined with a vector pointing south, and that the south-pointing vector intuitively and usefully can be assigned a negative component.

So the question is, are there intuitive physical examples of imaginary numbers. I've never seen one! For example, I'd like anyone to show me something even resembling or approximating a stick of length 2i, that when lined up perpendicularly with another stick of length 2i outlines a region with area -4. Given this, lay people have a point when they deny complex numbers exist.

Two advanced comments on this:

(1) A lovely and non-obvious fact about sqrt(-1) is that once you have introduced it and called it "i" you don't have to introduce a pile of other numbers. For example, the sqrt(i) = 1 / sqrt(2) + i / sqrt(2). Imaginary numbers would be a lot less compelling to mathematicians if introducing them immediately led to a bunch of other quantities having to be hypothesized and named.

(2) Quantum mechanics makes heavy use of imaginary numbers. Unlike in electrical engineering, it isn't just a convenience. At the end of the day, in electrical engineering, after having solved a difficult problem, you'll never actually go measure an imaginary voltage or current. The solution to electrical engineering problems always involves "throwing away" the imaginary part of the answer, even though it makes the legwork a lot simpler when coming to an answer. In quantum mechanics, you don't throw the complex part of the wave function away. You work with it at every step of the way, until at the end, you use it's norm-squared. This turns a complex "probability amplitude" into a "probability density."

So you might say that quantum mechanics provides an example of the "existence" of complex numbers. However, the problem with this example is that quantum mechanics remains difficult to intuit even for the most experienced physicists. "I think I can safely say that nobody understands quantum mechanics." http://en.wikiquote.org/wiki/Richard_Feynman

• no, "we" (I'm EE bkg) don't throw away the imaginary component. We sometimes want to know the reactive power, we study ccts that have "purely reactive load" (inductor/capacitor) etc – bobobobo Apr 6 '12 at 0:39
• Penrose has made the point that the fundamental nature of reality, as measured by quantum wave functions take values in complex Hilbert spaces, is essentially ruled by the complex numbers. In a way the complex numbers are more fundamental than the real numbers. – Cheerful Parsnip Mar 20 '13 at 1:33

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.

Erase from your mind that 'complex' numbers have anything to do with the square root of $-1$. Instead ask yourself if this is feasible:

Intelligent life in a distant part of the universe developed in a completely different way than we did. They often say

God made the polygons, all else is the work of ___.

This species made great strides in algebra at a very early stage. They built up considerable 'mathematical potential energy' in their ancient history.

In a short span of time, they went from the polygons to the circle, and then realized that there was a natural covering map of the 1-dim number line onto the circle group. For them, the circle group of rotations was just as useful as the number line.

As they worked with 2-dim space, one of their greatest minds realized that the most natural way to extend multiplication from the circle and number lines to all the points in the plane was to multiply number lengths and add their angles (dilation/rotation). It was a flash of intuition and insight, but the theory details gave then a tremendous technological advancement; many math problems they worked on had a new feel to them (for example, they found that every polynomial could now be completely factored).

As they advanced into quantum mechanics, they developed, what we humans call, the Schrödinger equation,

$i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)={\hat {H}}\Psi (\mathbf {r} ,t)$

Of course they did not express it as we do, and they were really surprised when they learned that $i$ meant "imaginary number" to so many people on earth. They were even more surprised to learn what percentage of people even knew what it stood for.

• Nice idea. Could you flesh it out a bit? The connection between complex numbers and circle groups isn't obvious. – Neil Mayhew Jul 8 '17 at 17:43

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

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

## protected by Qiaochu YuanJun 17 '11 at 10:02

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