# Where is the world of Imaginary numbers?

Complex numbers have two parts, Real and Imaginary parts. Real world is base of Real numbers. but where is (or what is) the world of Imaginary numbers?

• They're in a perpendicular universe. Jan 9, 2014 at 23:58
• @GitGud, many people think that, but the complex numbers are actually fairly close to Toledo, Ohio. Jan 10, 2014 at 0:01
• @WillJagy A laugh as always $\ddot \smile$ Jan 10, 2014 at 0:02
• most of us are complex as children, but, then we grow up and get real. Except the privileged few who keep a nontrivial imagination. Jan 10, 2014 at 0:21
• If you show me a real number, I'll show you an imaginary number. Imaginary numbers are just as real as a real number. Don't get confused by their names. They are named like this because of historical reasons. Jan 10, 2014 at 8:04

How can you say the real world is where real numbers are found? A real number requires acquiring an infinite amount of information to distinguish it from all other real numbers, something that never happens. Instead, we acquire the results of a finite number of tests and assign a real number with error bars.

That's not a facetious or rhetorical question either. There are real mathematical philosophers who believe that we should not assume a continuous reality, that we should look at discrete models for our mathematics. Ultrafinitists, for example.

But that kind of discussion also misses an important point: mathematics is not reality - it models it. Or rather, the relationship is even more contrived. Mathematics models our perception of reality, and we also assume that the perception<-->ontology relationship is also one of theory<-->model. So that we want ontology and our language to be in bisimulation over our perception.

Tarski wrote much about this, and it is a common enough discussion in philosophy (though I personally think philosophers qua philosophers rarely have the understanding of the mathematical relationship between syntax and semantics that is discussed here, and only real standouts like Putnam have taken the discussion to heart).

And if you take this as your foundation of science, then imaginary numbers bisimulate ontology in perception in all the models they are used - electromagnetism, quantum mechanics, etc. They are not needed - you can always translate to 2-vectors with the same rules, along with a variety of other constructs. But you never have a forced model, even for natural numbers and counting. There are many representations we can use.

• Great answer --- as it turns out, real numbers aren't that 'real' at all :-)
– Newb
Jan 10, 2014 at 0:28
• The name “real numbers” comes from how Descartes distinguished between quantités réelles and quantités imaginaires when talking about solutions of equations; he also divided the quantités réelles into fausses and vraies (false and true), the latter being the positive numbers. He considered “real” those solutions that he could represent on a drawing. Unfortunately, the Argand-Gauss plane was invented much time later. May 15, 2014 at 7:47

Complex numbers can be thought of as 'rotation numbers'. Real matrices with complex eigenvalues always involve some sort of rotation. Multiplying two complex numbers composes their rotations and multiplies their lengths. They find frequent use with alternating current which fluctuates periodically. Also, $e^{ix}=\cos x+i \sin x$, parametrizing a circle.

I've given a lengthier answer here, but for now suffice it is to say that both electrical engineering, as well as all physical study of wave-forms, would be dead without them, as would also mechanics or computer-graphics.

I would say that complex numbers are as real as physical measurements in the real world, because of at least one peculiar thing called quantum mechanics, in which the wave-function for any particle is a differentiable complex-valued function. The square of the absolute value of that function is essentially the density function, and the argument is quite like a phase, which can be measured somewhat by getting two particle with different phase to interact and produce an interference pattern.

Also, to partially answer the related question of whether there is a reason for using complex numbers instead of vectors or some other equivalent objects, there is at least one compelling reasons. Firstly $\mathbb{C}$ is the essentially unique algebraic closure of $\mathbb{R}$, which is the completion of $\mathbb{Q}$, which is the natural extension of $\mathbb{Z}$, which is indisputably real because we can conceive of doing something once, twice, ..., or undoing it once (-1), twice (-2), ... or doing nothing at all (0). The algebraic closure of $\mathbb{R}$ must contain a square-root of $-1$, and it is so beautiful that when you add just that one element it becomes the algebraic closure!