2^?

[Griwes]

Although I rarely rely on wikipedia when talking about math....

A very sensible policy that I wholeheartedly agree with. According to that "definition" the tuple (0, -1) is i. It isn't.

it is consistent with what I was taught at uni

and I am going to trust own university teachers more than random people on the internet (no offense).

[Gigasoft]

I'm sorry, but you are still incorrect. You are confusing a property of a number with something that uniquely defines it. You might as well define 3 by saying it is a prime factor of 12. That is certainly a property of 3, but it doesn't define it.

Mathematicians define i as being the tuple (0, 1) in the complex plane - at least that's the way it was when I was doing my thesis in Complex Analysis.

Your concern about the nonuniqueness of i comes from starting out with complex numbers and trying to define "another" i, let's call it j. We run into trouble if we try to define something in multiple ways at once. Obviously, we have to start with the real numbers, or we will have the problem of not being able to prove nor disprove statements such as i = j. If we define i in this way, and complex numbers as numbers of the form (a + ib) = (a,b), it is equivalent to defining complex numbers first and then defining i as (0,1).

[trinopoty]

I'm sorry, but you are still incorrect. You are confusing a property of a number with something that uniquely defines it. You might as well define 3 by saying it is a prime factor of 12. That is certainly a property of 3, but it doesn't define it.

Mathematicians define i as being the tuple (0, 1) in the complex plane - at least that's the way it was when I was doing my thesis in Complex Analysis.

I'm not an expert in complex numbers. I will just write what I have been taught at college:
A complex number is a number that does not have a complete real world existence. 'i' is an imaginary number whose square is -1.
In simple math, we know that when we raise any number to an even power, the result is always positive; so, 'i' does not exist in real world.

Usually, we first learn about imaginary number 'i' and then move to complex number; not the other way.
In that context: i^2=-1 is the definition of 'i'.

[Combuster]

Technically, the math originally started with "assume we have a variable i, for which holds that i² = -1", after which follows all sorts of other conclusions. In terms of real numbers "i" can't exist, so a new set of numbers was needed, created, and then called the complex numbers. Then someone needed to draw something and mapped the real part to the x axis and the imaginary part to the y axis. Taking the simplest way, i as 0+1i would map to (0,1). And of course you can take the re^iφπ representation and map that to x and y axes instead as it's all just a representation. Or you can turn a sheet of paper upside down. None of it is ever going to magically change the actual math.

Now comes the good part: i² = j² = k² = -1; ij = k; jk = i; ki = j; ji = -k; kj = -i; ik = -j. If you have ever seen this one the answer will be obvious, but find yourself a nice way of representation to deal with this, and you'll see it's only about attaching a specific meaning to a concept. After all, we can't decently visualize a purely 4-dimensional system, let alone on paper.

EDIT: messed up my math on a late evening

[Griwes]

Now comes the good part: i² = j² = k² = -1; ij = k; jk = i; ki = j; ji = -1; kj = -1; ik = -1. If you have ever seen this one the answer will be obvious, but find yourself a nice way of representation to deal with this, and you'll see it's only about attaching a specific meaning to a concept. After all, we can't decently visualize a purely 4-dimensional system, let alone on paper.

You forgot to explicitly highlight the best part, i.e. ijk = -1 Also, ji = -k, kj = -i, ik = -j, not -1.

[Mikemk]

how do you all know all this?

When your in high school, you'll understand

Thanks, I'm a senior in high school

[Love4Boobies]

Others have already pointed out that i² = -1 is the correct definition of i. However, it is incomplete without specifying that i ∈ ℂ. The reason for which i² = j² = k² holds is that j, k ∉ ℂ; there's no magic behind it.

-i is conceptually the same as (0, -1) in R2, which is *completely different thing* than 1, which is, again conceptually, (1, 0) in R2.

Mathematicians define i as being the tuple (0, 1) in the complex plane - at least that's the way it was when I was doing my thesis in Complex Analysis.

This is all wrong, of course. While you can indeed use geometric interpretations for complex numbers, they are by no means mathematical truths. It is you who should specify any conventions in a way that makes sense for the job at hand. For example, in computer graphics, we generally use vectors and matrices in frames that are incompatible with the one you've both described. We might represent the Cartesian coordinates (x, y) as the column vector [x; y], meaning x * X + y * Y, where X and Y are the unitary vectors [1; 0] and [0; 1], respectively. If the imaginary unit, i, truly were equal to Y, then Y² = -1... which is false. That said, there are multiple n × n matrices that can represent i.

[iansjack]

The complex numbers are as "real", no more imaginary, as the negative integers. As Kroenecker said, "god made the positive integers, everything else is the work of man".

I've no wish to contradict what anyone was taught, or think they were taught, at University. I've no idea what subject you were studying (engineers might be given a different definition of i than mathematicians). I took a 3 year degree in Mathematics and then a 3 year PhD course specializing in Complex Analysis.

Mathematicians do not define complex numbers as the square root of anything; they are simply a system of numbers defined as tuples of real numbers (a, b) with two operations defined upon them:

(a, b) + (c, d) = (a + c, b + d)
(a, b) * (c, d) = (a * c - b * d, a * d + b * c)

There is (if you work it out) a one-to-one correspondence between the real numbers and those complex numbers of the form (a, 0). And, again, you can see that:

(0, 1) * (0, 1) = (-1, 0)

It is convenient to represent these numbers in the form a + ib, where i is the tuple (0, 1). In terms of the above, it is clear that i * i = -1; but that is a consequence of the way we define i, not a definition of i. It is, obviously, also true that -i * -i = -1.

It would be ridiculous to define 2 by saying "2 is the number whose square is 4” - fine in a universe of positive integers, but a useless definition when we have to consider negative integers too. I don't want to go to my bank, thinking I am one hundred pounds in credit, to find that I am one hundred pound oeverdrawn. It is no comfort to me that the manager says "in this bank we define one hundred as the number whose square is ten thousand". There is no "the number" who's square is ten thousand any more than there is "the number" whose square is -1. In each case there is "a number..." ( and there is "another number...").

A definition must, by definition, identify something uniquely - two possible outcomes is of little use when defining something. "A" is no use for a definition - it has to be "the". In the case of i it is "the" tuple (0, 1); that is unique and thus can serve as a definition.

[iansjack]

This is all wrong, of course. While you can indeed use geometric interpretations for complex numbers

Nowhere have I made a geometric interpretation of complex numbers. A tuple is just that - a collection of two real numbers; and complex numbers are just the universe of such tuples with two operations defined upon them. Everything else follows from that with no need for an interpretation in physical terms. It's purely a logical system. (But you can form a one-to-one mapping of this to a plane just as you can form a one-to-one mapping of the reals to a line; these are convenient representations - but not definitions - of the two number systems.)

Mathematicians like logic; and they don't like "definitions" that are ambiguous - that's caused all sorts of trouble through the ages. "The number whose square is -1” is ambiguous and thus of no use as a definition of anything.

[Combuster]

Mathematicians do not define complex numbers as the square root of anything; they are simply a system of numbers defined as tuples of real numbers (a, b) with two operations defined upon them:

(a, b) + (c, d) = (a + c, b + d)
(a, b) * (c, d) = (a * c - b * d, a * d + b * c)

Wrong: it is the other way around again. Those equations are again the mathematical consequence of the assumption that i²=-1 and that (a,b) is a representation of a + bi

(a + bi) + (c + di) = a + c + bi + di = (a+c) + (b+d)i (nothing really interesting here)
(a + bi) * (c + di) = ac + bdi² + adi + cbi = ac - bd + adi + cbi = (ac-bd) + (ad+cb)i (here, one substitution of the assumption is used)

Mathematicians like logic; and they don't like "definitions" that are ambiguous

Go and define "right" or "left" without using the other - if you get away with it, you'll be doing so using a geometric interpretation. Now, is i or -i the right one?

[Love4Boobies]

Beat me to it.

I've no wish to contradict what anyone was taught, or think they were taught, at University. I've no idea what subject you were studying (engineers might be given a different definition of i than mathematicians). I took a 3 year degree in Mathematics and then a 3 year PhD course specializing in Complex Analysis.

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?

Nowhere have I made a geometric interpretation of complex numbers.

Mathematicians define i as being the tuple (0, 1) in the complex plane

This is precisely what the complex plane is: a geometric interpretation of complex numbers. Again, not a mathematical truth but a type of representation.

A tuple is just that - a collection of two real numbers;

No, a tuple is a sequence of zero or more elements that can be part of any set you like. They can even be other tuples. At any rate, from the rest of the paragraph and the first sentence I've quoted in this reply, I take it you're willing to drop the complex plane business.

Mathematicians like logic; and they don't like "definitions" that are ambiguous - that's caused all sorts of trouble through the ages. "The number whose square is -1” is ambiguous and thus of no use as a definition of anything.

You mean formalism, not logic. But why do you think it's ambiguous?

[iansjack]

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.

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.

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.

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.

[iansjack]

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.

[Griwes]

We're talking about definition of imaginary unit, not complex numbers.

Anyway, it really doesn't matter if i^2 = -1 is ambiguous, since i is just a symbol. Let's say that sqrt(-1) = { I, -I }, where I and -I are unambiguous. Now, whether you say "i = I" or "i = -I" doesn't matter, since, in geometric interpretation, it just switches the Im axis from having increasing values up to having increasing values down.

Anyway, *as you cannot compare complex numbers*, you cannot say whether i > -i or the other way around, so orientation of Im axis is irrelevant, and the statements I wrote in previous paragraph are *meaningless*, because you cannot "increase" or "decrease" value when moving along Im axis. Therefore, whether i is I or -I doesn't really matter, and since there is an agreement that *i* is an unambiguous symbol, that has property such that i^2 = -1, there is no ambiguity, since we do not care whether the i we use is I or -I.

Hence, the whole argument is pointless.

(This whole post is not based on some high level academic knowledge, just on what I already know and what logic tells me; if you disagree, feel free to prove any error here.)

[Gigasoft]

A definition must, by definition, identify something uniquely - two possible outcomes is of little use when defining something. "A" is no use for a definition - it has to be "the". In the case of i it is "the" tuple (0, 1); that is unique and thus can serve as a definition.

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.

But "the number whose square is -1" is not the definition of i. There is no "the" in there. Again, you are trying to "pick" a number from the complex numbers you already know. In other words, you are reading i^2 = -1 as an equation and trying to solve it and "find" i. We define i by starting with real numbers and saying that i is to be another number, and we define i^2 as being equal to -1. Once you do away with your "other" complex numbers, there is no ambiguity, and only one outcome. Sure, once we have defined i, we find that there is another number whose square also happens to equal -1, but that doesn't bother us since we never defined i as the square root of -1. Every statement that can be made using my complex numbers is equally true when written in your complex numbers, and vice versa, even though we can't say for sure whether my i equals your (0,1) or your (0,-1).