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But "the number whose square is -1" is not the definition of i.

At last, somebody agrees! That's exactly what I have been saying all along, in response to:

i is defined as follows:

i^2 = -1

No other definition of i is correct

As outlined in the link I gave, i is defined merely as (0, 1); nothing more, nothing less.

This seems to be where I came in, so enough said.

You are again defining imaginary unit in terms of complex numbers, while you should define complex numbers in terms of imaginary unit, since that's where their properties come from...

I hear a politician talk rather than an academic here.

iansjack wrote:The formal definition of a complex number, as I outlined earlier, is given here.

It's not. Learn to read your sources properly:

The paper referenced by iansjack wrote:The construction of the system of complex numbers begins by appending
to the system of real numbers a number which we call
i
with the property that
i² = -1
(Note that there is no real number whose square is -1)
The system of complex numbers consists of all numbers of the form
a + bi
where a and b are real numbers. We deÖne addition and multiplication for
complex numbers in such a way that the rules of addition and multiplication
are consistent with the rules for real numbers

Now then, where does that definition say that

iansjack wrote:i is the complex number (0,1)

or

iansjack wrote:(a, b) + (c, d) = (a + c, b + d)
(a, b) * (c, d) = (a * c - b * d, a * d + b * c)

I don't think I can value your fallacious appeal to your university degree, it's an dishonour to mine.

Now then, where does that definition say that

Learn to read your sources properly (and fully). I'll leave you to find where the link I gave says exactly that.

Hint - don't just read the first sentence, or even paragraph, of a reference.

You are again defining imaginary unit in terms of complex numbers, while you should define complex numbers in terms of imaginary unit, since that's where their properties come from...

Read my link - from a mathematician - rather than relying upon Wikipedia. The properties come from the formal definition of a complex number. "Imaginary" is - well, it's just in your imagination.

I am relying on information given me by mathematicians at my uni, and, again, as they all agree, I will believe them, not some papers or people on the Internet.

Complex numbers are defined in terms of i, not the other way around, and, as stated in post you seem to have missed, that ambiguity does not matter more than whether, say, my currently used pen is blue or black or even yellow.

If you define complex numbers "formally", you need to use some weird formulas that seem to have fallen from the sky, while with sane definition as a+bi complex multiplication can be understood by ten year old child, since it works exactly the same way as for real numbers and you don't even have to define it!

iansjack wrote:Quite a good description, for those who are interested, here: http://math.kennesaw.edu/~sellerme/sfeh ... utline.pdf of the formal definition of complex numbers. Note that the author asserts i * i = -1 as a property of i, not a definition. The formal definition of a complex number, as I outlined earlier, is given here.

This is fairly standard stuff in any undergraduate course in algebra.

A property of a thing is just a statement involving that thing. Obviously, most statements involve more than one thing. For example, the statement i = (0,1) is a statement involving i and (0,1). Now, is this a property of i or a property of (0,1)? This is of course a meaningless question. Now, a definition is nothing more than a collection of axioms, which are also just statements involving things. In the first section, he actually defines complex numbers twice, and then he defines i in C to be equal to (0,1). It is no more reasonable to call this "the definition of i" than "the definition of (0,1)" or "the definition of C". The statement i = (0,1) doesn't really tell you anything you didn't already know about complex numbers. It just says that we will use the label (0,1) to denote what we also call i, and we will not use the label (0,-1). Of course, if we wanted, we could have written it {0@1#, or we could have used the letters A-J instead of digits, and so on. We could also define complex numbers as a subset of the quaternions, which can be done in an infinite number of ways, and we'd have an infinite number of possible "outcomes". Clearly, it doesn't matter "where" i lives, as it behaves exactly the same. The difference between i and all the other possible i's is simply that we call one of them i, and the others we don't. The fact that an associate professor I've never even heard of also decides to call it (0,1) hardly contributes anything to its definition. Perhaps there is an intelligent alien species on a remote planet that calls it A. Or -A. Who knows? Would the definition of i suddenly become ambiguous if we were ever visited by these extraterrestrials?

iansjack wrote:

I am a self-taught in computer science/engineering and mathematics. I don't see how this is relevant, unless you wish to appeal to accomplishment. Let's stick to the math, shall we?

If I were to discuss heart surgery with you, then it would be relevant information that you had studied heart surgery and that you were not just regurgitating "facts" gleaned from Wikipedia. This is just as true if we are discussing abstract Mathematics. I can believe that you are self-taught.

Your second fallacy so far is a false analogy since "self-taught" does not mean that I am, as you say, regurgitating "facts" gleaned from Wikipedia. I assure you I've done my share of study. I'm just skeptical of yours.

iansjack wrote:I haven't really got time to discuss all the intricacies of Complex Analysis with you; I can only say that if your understanding of the complex plane is that it is merely a geometric representation of complex numbers then you have had a poor teacher.

Luckly for you, I am not ignorant of the subject so you won't have to. Unfortunately for you, however, the definition of i cannot come from complex analysis since complex numbers are a prerequisite to it. You need a complex plane before you can look at functions whose domains and codomains are in it. Since you don't have time/ability to make your case using complex analysis, I suggest you try to muster one using abstract algebra instead.

iansjack wrote:Why do I think "the number whose square is -1 is ambiguous"? Because there are two such numbers. It is as ambiguous as "the number whose square is 4"; that is not a definition of the number 2 ("the positive integer whose square is 4" would be an acceptable definition; that is unambiguous). A definition that does not define is not a definition.

I would have expected a PhD to know that all reals are fixed by two field automorphisms: the identity and the conjugate. I've vaguely mentioned the ambiguity that results from representing i as a matrix: the automorphism group of the special orthogonal group SO(2, ℝ) has the identity and the automorphism which changes the chosen direction on the unit circle. Conceptually, you have the same problem with the complex field, ℝ[x]/(x² + 1) .

iansjack wrote:Your insistence on the "right one", the fact that you even introduce such a concept, demonstrates that you fail to grasp the basics of Mathematics. Hardly surprising; it is a highly formalized system that cannot easily be grasped by self-tuition alone. By all means stick with your self-taught beliefs; that you cannot distinguish between a properties of a formal system and the axioms that define that system is of no consequence to me.

My grasp on mathematics seems to be better than your ability to comprehend written English. I'm claiming the exact opposite: there is no standardized direction. You can represent i as you like so long as you do so in a consistent manner. Your ordered pair formalism is one way to be more precise, my matrix formalism is another. But the ambiguity is still hidden in there, even if your notation is consistent.

ITT: angry nerds \o/

gravaera wrote:ITT: angry nerds \o/