What is the difference between the complex numbers i and -i?

Question

First, I begin by making some possibly dubious assumptions with unclear definitions.

1. An object is defined by its properties. (Two objects with the same properties are the same.)
2. Numbers are objects.

So I can begin to believe that mathematical statements are true and that they express relationships between objects. For example, 1 + 1 = 2 describes a relationship between the objects 1 and 2.

I continue by being pedantic and asserting that when I say 1 + 1 = 2, the symbols I write are possibly names for the objects I'm referring to, but I do believe the statement is true. I resolve this partly by asserting that everyone has a "universal reference" to 0. That is, I hope that mathematical objects like numbers are not just relational but there really is a 0 that we are all talking about.

Then I am happy to believe in the ontology of the integers, because they are in relation to the fundamental 0.

I ask briefly if 1 could equal -1, and I note this is not true since 1 satisfies x² = x, but -1 does not. That is, there is a property satisfied by 1 that isn't satisfied by -1.

Finally, I start thinking about the complex numbers. I know that i is not equal to -i, since i satisfies x ≠ -x, where if we evaluate this expression with x = i, the -x then refers to -i.

However, because the complex numbers have conjugation as an automorphism, I can't think of an actual property that distinguishes them. For example, they both satisfy x² = -1. I can say -i has the property x ≠ i, which distinguishes them, but I am unhappy with this, because it assumes I have a "universal reference" to i, as opposed to just a name. Since I can conjugate the names of i and -i, I remain confused.

So how can we "witness" a property distinguishing i and -i? and if we cannot, what does that imply?

Answer 1

The key is that the field of complex numbers C has a nontrivial automorphism group over the field of real numbers R: the nontrivial automorphism is the complex conjugate. So whatever construction you use to create C, from the perspective of R there is no way to distinguish i from -i. Mathematically, there is no problem: just choose one of them as i and the other as -i. If someone else has a different choice, then there is a unique translation from your perspective to theirs.

Answer 2

I would probably say that there is really no essential difference between the number i, considered in isolation, and the number -i, considered in isolation. If somebody hands us a system, and we examine the system and realize that it behaves like the complex numbers, then we will be able to identify two different things in the system that correspond to i and -i, but we will never have any way to figure out which one is which.

It's easy to find examples of this indistinguishability. For example, 2x2 matrices, where the top-left and bottom-right entries are the same and the top-right and bottom-left entries are opposites, form a model of the complex numbers, and the matrices

[  0 -1 ]           [  0  1 ]
[  1  0 ]    and    [ -1  0 ]

represent i and -i. But which one is which? There's no experiment we can perform which will tell us which matrix corresponds to which label; we just have to choose a labeling arbitrarily.

However, this "problem" (if you consider it a problem) only happens when you look at a single number in isolation. As soon as we consider two or more complex numbers together, some essential differences start to show up. For example, the pair (i, i) has the following properties:

• The sum is a number with magnitude 2.
• The difference is 0.
• The product is -1.

On the other hand, the pair (i, -i) has the following properties:

• The sum is 0.
• The difference is a number with magnitude 2.
• The product is 1.

So we can see that, even if you don't consider the number i to be essentially different from the number -i, it is certainly true that the pair (i, i) is essentially different from the pair (i, -i).

Answer 3

If I understand Stewart Shapiro in his Identity, Indiscernibility, and ante rem Structuralism: The Tale of i and −i correctly, his, "there is no requirement that mathematical objects be individuated in a non-trivial way" means we can discern the two as trivially as we want. We can just appeal to the need for mathematical practice to be efficient and not bogged down in philosophy, and simply discern them as trivially as we want.

Clearly are there are two square roots to x^2 = −1, i and -i, just like x^2 = 1 has two. Are i and -i equal and thus identical and indiscernible? Well no. Just like in an unlabeled 2-element graph with no edges say, we don't require a non-trivial mathematical fact to tell them (i and -i or the two unlabeled, unconnected vertices) apart. In practice we have reasons to tell them apart (square roots of -1 or vertices), even if the reasons seem trivial, so we do so, say for cataloguing different automorphisms or as a step in a proof. Those are reasons enough. As long as we keep consistent in how we label them, such as in describing two automorphisms, that's enough for mathematical work.

Maybe there is somewhere a less trivial case discerns the two as well, but all we need is the above. Again, with the caveat I'm not 100% sure I understand his paper.

^ Philosophia Mathematica (III) 16 (2008), 285–309. doi:10.1093/philmat/nkm042

Answer 4

Since i is defined as a solution to the equation x² + 1 = 0, and −i also satisfies that definition, there is no algebraic fact about i which ceases to be true when all instances of i are replaced with -i. That is, there is no distinguishing property which can tell these two numbers apart, so long as you are careful to replace all occurrences of i including those hidden in the definitions used by whatever property you are trying to describe.

Nonetheless there is no real problem with having two things which are equivalent in all properties and distinguished only by name. This is the difference between equivalence and identity. Even in cases where there are technically some distinguishing features, the fact that they have different names is often a better way to tell the two things apart. Consider for example newborn identical twins; theoretically we could find some genetic difference due to mutations after the initial cell separation, but in practice the twins are simply given different names and then told apart by their names. It is arbitrary which twin receives which name, yes.

Answer 5

The complex numbers aren't simply a 2d vector space. They also come naturally with a basis: 1 & i. Thus they are examples of framed vector spaces. Hence i & -i are distinguishable by use of this basis.

Answer 6

One possibility is "Arg(i) is positive, and Arg(-i) isn't", assuming that you are comfortable with the distinction between positive and negative numbers. And if the Arg() function displeases you, you could probably rig up something using exp() or ln() that boils down to the same thing.

If you want to reject this, you will need to start getting very rigorous about what language/properties you are willing to accept. There are certainly some formal languages in which every statement that i satisfies is also satisfied by -i.

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Edit to add: Well, since comments have objected to Arg(), how about

Im(exp(iπ/2)) > 0, while Im(exp(-iπ/2)) < 0

I'm assuming that the Im() function is now going to be objected to, but are people going to say we're going to study the complex numbers, but never use the Im() function?

Look, the OP has a valid point: C has an incredibly large set of automorphisms, no matter how you choose to axiomatize it. This is in strong contrast to R, for which the "complete ordered field" axioms characterize it uniquely (up to isomorphism, of course), and also only has the identity map as a structure-preserving self map.

If you do math, you may notice that R is often presented as a structure that obeys certain rules, and we lay down the rules and show what R must be like, based on those rules. In contrast, C is usually "constructed" (out of R) - there's no common axiomatic presentation. But as one constructs C, things like the Im() function pop-out.

As previously said, if you restrict what operations you are allowed to use with C, you can find yourself in systems where you can't distinguish i and -i. And it's definitely interesting to explore this. But to claim that you can never distinguish i and -i when studying C seems silly.

Answer 7

You say:

So how can we "witness" a property distinguishing i and -i?

The answers present here all essentially and convincingly acknowledge that taken in isolation, i and -i are indistinguishable. But as part of a broader system function as additive inverses. Some answers note that using that system to model physical phenomena, there are real world consequences.

You also say:

"knowing what i is already", so it seems to be circular?

Yes, and this sort of circularity might be seen as impredicativity. Yet, impredicativity can be useful as it is in intuitionistic type theory (SEP). That's because they are additive inverses. On the complex plane, for instance, to define -i is to start with i and rotate it pi radians, and as one answer notes in the language of category theory, this impredicativity therefore indicates a transformation.

So, when you ask:

So how can we "witness" a property distinguishing i and -i?

we now can put the answer of why we can't distinguish them in context. In isolation you can't, but pairwise they can, that's because i and -i have a special relationship: they are symmetrical on the complex plane. In abstract algebra, symmetry is understood to be a property of groups. For a group, we simply need a set, and an operation on the set that obeys three axioms. Consider the group of ({1,-1,i,-i,0},+) as representing the complex plane:

1. Existence of the identity: 1+0 = 0+1, etc. That is, we can compare vectors and determine if one vector is the same as another. If two vectors violate an operation with the identity, then they are not the same.
2. Existence of the inverse: 1+-1 = -1+1 = 0, etc. That is, we can return from the origin of a vector from its terminal end if we use the inverse operation.
3. Existence of associativity: 1+(-1+0) = (1+-1)+0, 1+(-i+0) = (1+-i)+0, etc. That is, the sum of a vector is path independent. No matter how many additions we perform, given the same vectors, the route is immaterial to the destination.

Now imagine that we see that not only are i and -i indistinguishable when taken in isolation, they are actually also indistinguishable from 1 and -1! That's because all four elements of the group are rotations of a vector about an origin lying at the center of the complex plane. I can rotate the plane 3 times from the original position, and still have the same vector representation, that of perpendicular vectors extending from the origin that establish the linear basis of the complex plane. So, yes, they all have the same properties taken in isolation (a magnitude, a direction), but the moment they are laid on a plane, they are all defined in relation to each other (actually, impredicatively as as a sequence rotations of the first) by pointing in four distinct directions on a Euclidean plane.

Thus, the fact that they aren't differentiated when each is considered in isolation isn't a problem, it's a property and regularity we rely on when defining the complex plane in terms of a single unit vector! This is a lot like how PA uses the first natural number and the successor function to define all numbers. And to get even more philosophical, Lakoff and Núñez have an appendix devoted to the claim that we metaphysically ground the complex plane as a series of rotations because our brain has repurposed it's visual system to do the mathematics in their work Where Mathematics Comes From, a claim sure to annoy neo-Platonic thinkers.;D

Answer 8

==
I'll give this a shot.

You could be looking for the mathematical difference or the physical difference. I'll tell you about the physical difference.

On the imaginary plane, things below the real axis are the Fourier transforms of their mirror images above the axis. Ultimately, this usually involves space and time, because those two kinds of distance have opposite signs in Einstein's spacetime interval metric.

Pairs of variables that obey the uncertainty principle are Fourier transforms of each other. (Technically, one is the inverse transform). These are called complex conjugate variables. Energy and frequency are an example.

• The imaginary value above the real line represents energy stored in space at a particular location. Mass is an example. The time of day doesn't matter. The mass just sits there with a constant positive value.

• The imaginary value below the real line represents energy stored in time. The way you store energy in time is with frequency.

I don't want to go deeper than that because I don't know if this helps, but maybe it does a little.

Answer 9

If i and -i were identical, then you could substitute one for the other in any mathematical identity. For instance, we have the identity i × -1 = -i. If we substitute, we get i × -1 = i, which is false as can be proven by squaring both sides, with a clearly contradictory result of 1 = -1.

Therefore, the two objects i and -i are distinct. As other answers have explained, the choice of which is which is arbitrary, but that does not mean that they are one and the same.

Answer 10

i + -i = 0 whereas i + i = 2i.

The name for i may be arbitrary, but the name for -i is not. You can't swap them over because the name -i means that it is the additive inverse of i.

Answer 11

So how can we "witness" a property distinguishing i and -i?

Simple. The condition x * i = -1 is fulfilled only for i, but not for -i.

However, I suspect that you won't like this condition because the condition itself involves i – and it must do so because conjugation (switching i and -i) is an automorphism, so every algebraic condition you express without using i will hold for both.

I suspect you'd rather have a property that can be "checked" before "choosing" i or -i. This is as unnecessary (the mathematics is well defined either way and lead to equivalent results) as impossible (due to the mentioned automorphism).

And I challenge your assertion that this is somehow exceptional. This is rule. Before you construct something you cannot have properties.

You assert that this is different for integers because 0 is a "universal reference":

I resolve this partly by asserting that everyone has a "universal reference" to 0. That is, I hope that mathematical objects like numbers are not just relational but there really is a 0 that we are all talking about.

Before you construct the integers you cannot agree on a "fundamental 0". You think the integer 0 is universal because of the connection to the real world (counting etc.). And there is a relation to the real world. But how do you establish it? Via the properties of the integers after constructing them.

For example by counting: You count objects, starting at 0 and adding 1 for every object you see. But to do so, you must have already chosen some mathematical object as 0 (and defined the addition accordingly). The real-world meaning might be universal, but its translation into mathematics makes sense only after you define one object as 0.

Or by using additive neutrality: If I join a group of 0 objects to another one, the number of objects doesn't change. Again, to translate this into mathematics you need to have the integers already defined, and whatever you choose as 0, it will work.

I could choose what you call number 1 as my number 0, maintain your definition of addition and would get the same mathematical properties and the same connection to the real world (starting my counting by my own 0, thus your 1, of course).

Answer 12

Taken just so far, the sequence of iⁿ does not reveal the fuller significance of its terms. That is, if we have thought our way to just i and -i, as such, we would not have a way to attribute to them the specificity we would expect of "real objects."

But so that is where the vision of the complex plane, and the unit sphere of the quaternions, comes into play. True, there remains something "arbitrary" about how we orient the complex grid, so that we "could've just as well have" located -i opposite i in the grid except also on the opposite initial side (of the grid, or our consideration of it rather). This arbitrariness escalates to uncountable infinity once we pass to the quaternions, and proceeds apace into even higher realms (e.g. sedenions).

"Algebraically," we have that the properties of the terms of iⁿ are distinguishable: per se, i¹ doesn't equal i³, etc. Also, i and -i have the meta-property, "If you multiply them by themselves, you get -1; but if you multiply them by each other, you get 1," which becomes the commutativity of multiplying i by i³. But are any of these the kind of "real properties" you're asking about? They actually seem like they might well not be...

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Two caveats:

1. Per set theory: it is possible to have these things called "ur-elements" which are qualitatively indiscernible (from the vantage of the witnesses of the higher infinite) but which are numerically distinct nevertheless. So either we analogize the difference between the imaginary and quaternion/et. al. units to the difference between ur-elements, or we analogize it to a difference in the order of possible consideration (AKA the normal order, as when we attend to one thing first, another thing next, and so on). In either case, we have that i and -i are differentiable apart from any nontrivial identity conditions (they are either numerically distinct or such that our contemplation of one depends, in actual epistemic time, on our contemplation of the other).

2. There is a similar matter in physics, however, where it is "just a matter of convention" that we interpret electrons as negatively charged and positrons as positively charged. Having a certain pair of particle types, we are required to assign one the plus sign, the other the minus sign, in a certain connection; but otherwise, it is perhaps somewhat "arbitrary" which one gets which sign. But so aren't these particles "real objects"?

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EDIT: although familiar with automorphism-talk, I was not deeply familiar enough to recognize that what I said about "you could've swapped their initial positions as long as you reflected the whole environment in the right way, too" was therefore in reference to a fact involving automorphisms in this context. So I should add that what I said about ur-elements also pertains to automorphism-talk because I was indicating this particular subtopic. But so then I ended up having a very different thought about all this, based on my previous emphasis upon the representation iⁿ.

For we can go to i^x, and consider then iⁱ = 2.078... and -iⁱ = -2.078... Then i^-i = 4.8104... and -i^-i = -4.8104... Or i^1/2 equals a positive number reflected by a counterpart negative number for -i^1/2, then there are a bunch more symmetries and diagonal rotations and whatnot when we think through raising i and -i to the power of -1/2, etc.

And so "everyone" keeps bringing up the thing about automorphisms, which should call to mind the fact that there is already an automorphism for ℤ, per negation. Which could call to mind the question as to whether 1 and -1 are fully discernible on a certain level, or 2 and -2, or for that matter are the surreals ω and -ω so much to be told apart aside from by "fiat" (by our decision to orient ourselves through the surreals in whichever way)?

You might object: "Oh, but I do know that -1 has the property that raising some n to the power of -1 is the same as having the fraction 1/n. By contrast, raising n to the power of 1 never would result in such a thing unless we were already working with a fraction." Very well, but then with respect to taking the square root of -1, before any question of multiple solutions to this is in view, we can see that our whole apprehension of the eventual terms is caught up in their role as those solutions. Now the taking of roots succeeds division, in the order of the negative/inverse hyperoperation sequence (whyso addition and multiplication have only one inverse, but from exponentiation onward, they all have two inverses, is addressed here). And division is iterated subtraction with an "eye towards" zero, in this context. E.g. 4/2 = 2 means that subtracting 2 from 4 twice will yield 0. So the internal nature of the concept of i is a relatively deep evolution of the concept of mathematical negation.^!!!

But so then the automorphism stuff that shows up with respect to ℂ is preemptively mirrored by the automorphism of ℤ, which is itself a matter of negation theory. Negation theory itself is what questions of identity and discernibility have to do with, so in a sense, the inability to nontrivially differentiate between i and -i, or at least the inability to nontrivially differentiate between them other than modulo the abstract basis for ℤ, if that is what we are indeed left with, doesn't have to be too troubling, I would think. Or rather, consider then the trivial/nontrivial distinction, as an item of negation theory; is there then a distinction between a trivial and a nontrivial version of that very distinction?

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^!!!I checked what the "square super root of -1" would be, via Wolfram Alpha. It gives the result 1.6903... + 1.8699...i, which means that though we unearthed the whole new plane of complex numbers just by digging into the square root of -1, we delved not really any further down, in the sense of into an even newer world, by moving one step further along the list of negative hyperoperations. So if we take the disclosure of the complex plane as an elaborate revelation about the possibilities of negation theory, then we will take the lack of disclosure of an even greater plane as an important non-revelation about the possibilities of negation theory.

Answer 13

If you write non-zero complex numbers in polar coordinates then "i" has positive argument, namely pi/2, and "-i" has negative argument when restricting the argument to the open interval ]-pi,+pi[. Hence both complex numbers can be distinguished as soon as you can distinguish positive from negative real numbers.

Note: The argument of a non-zero complex number is determined by its logarithm up to an additive summand k*2pi with an integer k.

Answer 14

Well, things can be only different or same in relation to a sense. For example, in the sense of simply being human, all humans are the same. But, does that really mean, for any property of a human, that all humans would have that property virtue of being human? No.

In the case of $i$ and $-i$, they could be seen in the sense of the geometric meaning of multiplying a given number by them. $i$ rotates clockwise ninty, and $-i$ counterclockwise ninty degrees.

But, in another sense, say being solution to the equation $x^2+1$, there is really no difference between them. But, then again, even if we had a polynomial $(x-1)(x-2)$, is there any difference between $1$ and $2$ in regards of being simply solutions?

Answer 15

From a set theoretic perspective the complex numbers are 2-tuples of real numbers with multiplication defined by

(a, b) * (c, d) = (ac - bd, ad + bc)

Of course the other field axioms are defined straightforwardly as well. In this sense we have

i = (0, 1)

-i = (0, -1)

It can be seen from the multiplication rule that i^2 = (-i)^2 = (-1, 0).

We see that i and -i are explicitly/constructively defined. So the difference between i and -i is that i = (0, 1), but -i = (0, -1).

There is of course a major symmetry of the complex numbers that we could define j = -i and work with j instead of i and you wouldn't be able to tell the difference from an "application" point of view. But the same symmetry exists in the reals. If one day the whole world decided to swap the definition of negative and positive everything would continue to plug along exactly as it had done, we would just have some say the opposite word in some scenarios.

Answer 16

A visitor from outer space asks me, "Where is the country called Australia?" I tell them, "Oh, it is below the equator". It is easy for them to understand "equator" but I do not think they can understand "below" without a lot of explanation. Isn't this question simply about chirality, or left- and right- handedness? To explain the difference, you only have to explain one example, but you cannot even do that to some creature lacking any part of your own experience. As Bill Clinton once said, "It all depends what the meaning of the word "is" is!"

Answer 17

Your premise #1 is incorrect in mathematics, since unequal isomorphic objects have the same mathematical properties within the setting that makes them isomorphic.

Example. In Z as an additive group, 1 and -1 can't be distinguished since they are both generators of this group and the map f(n) = -n on Z is an automorphism of Z as a group that exchanges 1 and -1. But when you view Z as a ring, 1 and -1 are distinguishable since 1 is the multiplicative identity in Z but -1 is not the multiplicative identity (or 1 is a square in Z but -1 is not). So in the setting of group theory you can't tell apart 1 and -1 in Z, and if you change your setting to ring theory then you can tell apart 1 and -1 in Z.

Example. In C as a field extension of R there is no mathematical property that distinguishes i and -i: these are simply the two solutions to x² = -1 in C. When you label one solution as i, the other solution is -i since the two solutions are additive inverses. Operations like "Im" (imaginary part) and "Arg" (angle relative to the positive real axis) do not distinguish between the two solutions of x² = -1 since each solution has its own imaginary part and angle functions: when j is a square root of -1 in C, each complex number is a+bj for unique real numbers a and b and each nonzero complex number is re^jt for unique r > 0 and real t; the choice of j gives us the functions Im_j(a+bj) = b and Arg_j(re^jt) = t. The two imaginary part functions Im_i and Im_-i get interchanged when we replace i and -i, and likewise for the two angle functions Arg_i and Arg_-i. That multiplication by one solution of x² = -1 is a clockwise rotation by 90 degrees and multiplication by the other solution is a counterclockwise rotation by 90 degrees is not an essential difference since swapping the two solutions of x² = -1 interchanges clockwise and counterclockwise rotations: the 90 degree rotation that is called clockwise is an effect of your choice of which square root of -1 you choose to call i and this applies to both solutions of x² = -1.

Answer 18

If you are an algebraist or algebraic number theorist, there is indeed no difference between i and -i: they have the same properties. The difference appears only in analysis. For example, multiplication by i defines a counterclockwise rotation in the complex plane, whereas multiplication by -i defines a clockwise rotation.

Answer 19

Your premise 1. is simply incorrect, as far as mathematicians are concerned. Objects are what they are, they have no particular reason to be distinguishable by their properties.

Of course, whether two objects can be distinguished by (a certain family of) properties is often an interesting question with a non-trivial answer.

In some cases, we can simply group together indistinguishable objects and say that they are "the same object". This grouping will of course depend on what properties are allowed to be taken into consideration. For instance, a circle and a square are the same from the point of view of topology, but not from the point of view of differential geometry. The group of isometries of an equilateral triangle and the group of permutations of numbers 1,2,3 look the same and most mathematicians would say that they are the same group, although strictly speaking there is some identification happening here. Incidentally, the idea of grouping together indistinguishable objects corresponds roughly to the operation of taking a quotient.

In other cases, we cannot quite group indistinguishable objects together, mostly because we construe them as elements of a larger whole. This is the case for i and -i - they have the same properties, but if we tried to group them together we'd not be able to do arithmetic since e.g. i+i = 2i is different than i + (-i) = 0. A simpler example is given by the integers +1 and -1, which have the same properties with respect to addition: in other words, they cannot be distinguished by any sentence involving only addition, variables, and the constant 0. In this situation, interesting mathematics arises because we can study automorphisms, that is, maps from a structure into itself which preserve all the relevant properties. For instance, the conjugation map on the complex plane C (i.e., the map that takes x+iy with x,y real to x-iy) is a ring automorphism, meaning that it preserves addition and multiplication. Likewise, the map on R that takes x to -x is an (additive) group automorphism, meaning that that it preserves addition. As a matter of fact, the study of automorphisms of number fields (i.e. what we obtain by adding to the rational numbers roots of polynomials, such as i, which is the root of X^2+1) has grown into a significant branch of mathematics, known as Galois theory.