Do you believe in √(-1)?

[A18]

Discuss.

[So the Heroes Fall]

No.

[IDS Judicial Overlord John Dibble]

The letter i is merely a concept, much like the invisible pink unicorn, the flying spaghetti monster, zombie Jesus, or the Teenage Mutant Ninja Turtles. It also isn't nearly as cool as any of those things, so that makes me believe in it even less. Seriously, it's so damn boring.

[memphis]

It's just a place holder much like zero is. No need to get all fussy about it.

[dead0man]

It was at this point in my math learning career that I started to get frustrated.  The f of X sh**t was the end.

[Tik]

In the same way that I believe in the idea of ghosts - impossible, but good for scaring people.

[The Mikado]

It's true in the same way a corporation being a "person" is true.  Strictly speaking, it's a nonsense mathematical term, but in a broader sense, it's a necessary fiction to make everything else work.

[GMantis]

Imaginary numbers, despite their name, have actually real life application in modern physics, so yes.

[realisticidealist]

i is the most awesome number ever. Give it a break.

[muon2]

When I was young, I found Isaac Asimov's rebuttal to the unreality of imaginary numbers one of the most compelling. The story has stayed with me throughout my academic career, but I am afraid I might not get it all verbatim. So, I found this rendition of it from his later autobiography, In Memory Yet Green (1979):

    On September 27, 1938, I registered for my fourth year at Columbia. I was taking integral calculus now and Sidney was taking more sociology while I was doing that. After calculus, I would go around to the sociology class, where the professor held court after the lecture was over. When that was done, Sidney and I would have lunch.

    It made for dull listening, generally, for I never have been impressed by the soft sciences. On October 10, I found the sociology professor (his name was Casey) had made a table on the board in the course of his lecture in which he divided people into rationalists and mystics. Under mystics he had listed mathematicians.

    I studied that for a while and then, even though I was not a class member, I interrupted the postlecture session by saying, “Sir, why do you list mathematicians as mystics?”

    He said, “Because they believe in the reality of the square root of minus one.”

    I said, “The square root of minus one is perfectly real.”

    He said, “Then hand me the square root of minus one piece of chalk.”

    I said, “The cardinal numbers are used for counting. The so-called imaginary numbers, like the square root of minus one, have other functions. If you had me a one-half piece of chalk, however, I’ll hand you a square root of minus one piece of chalk.”

    Whereupon Casey promptly broke a piece of chalk in half and handed it to me with a smile. “Now your turn,” he said.

    “Not yet,” I said. “That is one piece of chalk you’re handing me.”

    “It is half a regulation length of chalk.”

    “Are you sure?” I said. “Will you swear it is not 0.52 times a regulation length or 0.48 times that?”

    By now Casey realized it was time for hard logic if he was to win the argument, so he decided that since I was not a member of the class, I would have to leave the room at once. I left, laughing rather derisively, and after that I waited for Sidney in the hall.

Mathematics deals in abstract quantities that include thing like zero, negative numbers, variables, functions, groups, vector spaces, algebras, and a host of other concepts. One of them happens to be complex numbers which includes the square roots of -1, and for historical reasons they were dubbed imaginary numbers. They are no more imaginary than any of the other abstract objects in mathematics.

[A18]

I'm shocked that you didn't mention infinitesimals.

[Stranger in a strange land]

Imaginary numbers, despite their name, have actually real life application in modern physics, so yes.

[Gustaf]

Yes. But I am religious.

Seriously though I think some of these concepts are mystic by nature. The human mind (or at least my mind) cannot comprehend many of the concepts used in modern matehmatics and physics. Sure, I can follow the proofs and I can use it (I owned imaginary numbers back when we had exams on it) but I can't really wrap my mind around it and get it. It is about as mystic to me as God.

[Citizen James]

I believe in i.

I also believe in pi, although it is irrational.

[Ghyl Tarvoke]

i is just an abstraction of an abstraction. If you accept the existence of 'two' then you should probably accept the existence of i.

[Tonberry]

i do.

[Tender Branson]

No.  Imaginary numbers are, by definition, imaginary, i.e. products of the imagination.  Therefore, they are not real; if they were, they would be real numbers.  Since the square root of -1 is clearly not a real number but an imaginary number, then by its very definition it cannot exist.

[Gustaf]

No.  Imaginary numbers are, by definition, imaginary, i.e. products of the imagination.  Therefore, they are not real; if they were, they would be real numbers.  Since the square root of -1 is clearly not a real number but an imaginary number, then by its very definition it cannot exist.

All abstractions are products of the imagination.

[muon2]

No.  Imaginary numbers are, by definition, imaginary, i.e. products of the imagination.  Therefore, they are not real; if they were, they would be real numbers.  Since the square root of -1 is clearly not a real number but an imaginary number, then by its very definition it cannot exist.

The name has little to do with its meaning and more to do with its history. That would be like saying a strange quark has less reality than an electron because of its fanciful name.

But I would claim the imaginary numbers to be just as functional as any real number. If you believe that a line segment can be rotated in a plane then you have worked with imaginary numbers, just not in their algebraic form. For instance, consider the line segment in the 2-dimensional plane from (0,0) to (3,4). A 90 degree rotation around the origin takes the other end to (-4,3). But this is just multiplying the 2-dimensional point by the imaginary unit i. If I wrote it with algebra it would read (3+4i)i = (-4+3i), and I would have used imaginary numbers.

[Magic 8-Ball]

I am faithful.

[pbrower2a]

i is an answer; one can debate the relevance of the answer.

We have a hierarchy of genuineness of numbers. First we have the natural numbers that one counts with. Add the number "zero" for completeness through the inclusion of the concept of "nothingness" or a void.

The ancient Egyptians and Greeks hardly accepted zero as a number, perhaps suggesting that "nothingness" had no significance. Neither did they have the mirror image of the whole numbers, the negative integers, which with the whole numbers and zero comprise the integers, the "complete" numbers. 

The Egyptians found fractions -- "broken numbers" useful, as did the Greeks. Add to such numbers as 3/8  and 2/5  such inverses as 8/3 and 5/2 and one has all the broken, if sensible "rational" numbers -- if one allows 1 as a divisor. Zero is of course prohibited as a divisor.

Trouble arose with numbers that could not be seen as fractions -- like the square roots of 2 or 23, or the cube roots of 4 or 25. They could only be approximated with fractions, but they were solutions. The Egyptians knew well of the 3-4-5 right triangle and perhaps 5-12-13, 7-24-25, 8-15-17, 9-40-41, and 11-60-61 ... but the great Euclid could prove that the diagonal of a right triangle with rational sides could never be expressed precisely as a fraction. Such numbers didn't seam quite 'reasonable', so they were called 'irrational'. The weird constant π created its own problems.

Some cultures discovered zero -- our use of zero derives from Indian practice. They had the concept of the void and nothingness and had a number to express it. The Arabs found it useful, and we learned it from the Arabs.

In the Renaissance, people started trying to solve equations that supposedly had no answers -- like x^4 =1. "1" and "-1" are solutions, but so are "i" and "-i"; after all, i^4 =1. Numbers involving i were thus "imaginary" even if three-dimensional mathematics is  a sane proposition. But they were algebraic, establishing them as solutions of polynomial equations.  One could do about everything but establish order among them.

Numbers like π and the frequently-used e (base of the natural logarithms, and heavily in use) could later be proved impossible to find polynomial equations with π or e (among other numbers) from which one can find a root. Such numbers are called transcendental -- as they "go beyond" computation as they are un-algebraic.

[Communists For McCain]

√(-1) is dead.

[realisticidealist]

√(-1) is dead.

You killed him, you monster.

[Communists For McCain]

√(-1) is dead.

You killed him, you monster.

That wasn't a nice thing to say. A real gentleman of your persuasion would've said "you bastard."

[Storebought]

No. I believe in quaternions.