Anyone here ever got bugged with i?

[nonequilibrium]

One silly scientific experiment is all it takes to wipe out a lifetime's worth of work by some theoretical physicist.

We like living on the edge ;)

Whether it is or isn't is irrelevant to mathematics.

I think micromass was simply trying to argue that a real number like pi does not have a physical realization, and of course whether something has a physical realization or not is dependent on, well, physical reality, and I'm fairly sure--for as far as a person can speak for somebody else--micromass didn't mean anything more by that.

The quote by fbs that elicited micromass' response (and whether space is discrete or not, I do not know; I'm sure that I've heard it is discrete, but there are only a few things I haven't heard from bad pop sci lit):

How about my rocket ship? If I send it out on a line, and I have some sort of mathematical definition for √2 and \pi, then I know at some point in time my rocket ship will have reached √2 and \pi meters from me, even if my measurements are precise only to a number of digits.

And I would like to say to fbs that it's a interesting remark, and I wonder what some would say to it, but as I see it, your remark is not justified. You're trying to prove that pi has a realization by assuming pi has a realization. I suppose this delves into the "continuity" issue: what do we mean by continuity? As I see it, it seems like a very theory-laden remark: you first have to assume space can be identified with \mathbb R^3, or more generally a manifold upon which you can measure the length of a certain path. This is however part of the model of reality, and not necessarily reality itself. But I feel you: it is a slippery sloap: what then is a part of reality itself? Let's not go into that question here, but it just goes to show that your picture is a bit too naive. As an aside, I wonder if you take out all the irational numbers out of, say, quantum mechanics and relativity, whether that would change the physics in any way... It sure would make the calculations trickier.

But a more important aside to this issue: to you it seems evident that pi is a part of physical reality because it is used in a model of reality, but what about i? It's an important part of quantum mechanics... Good you might say: "but in quantum mechanics it's a computational tool", but then you're being rather arbitrary: why are you so convinced pi is not a computation tool? I for one, am not (rather, I refrain from claiming anything about the physical existence of both pi and i)! That being said, there are lot of physicists who genuinely believe that quantum mechanics implies that the imaginary number i is a crucial part of physical reality! You see how deep the rabbit hole goes...

 

[fbs7]

But I'm arguing that this is not the case. How does \pi show up in the real world?? How do large numbers such as Graham's number show up in the real world?? They don't.

How about this: I draw a circle and measure the perimeter. My real-world circle will of course not be perfectly circular, and my real-measurement of the real-world perimeter will only have a few digits. But that's a first approximation. Then, as I refine my real-world capability, I do it a second time, get a better approximation. If I repeat many times, the limit of what I'm measuring approaches 2\pi. Therefore

lim _{error -> 0} measurement with error = 2\pi

It's an actual limit; I can see it, although I cannot measure it exactly. So I look at the mathematical definition of \pi and confirm it matches pretty nicely - hooray. Similarly, I can measure e and √2 through some experiment.

 

[micromass]

How about this: I draw a circle and measure the perimeter. My real-world circle will of course not be perfectly circular, and my real-measurement of the real-world perimeter will only have a few digits. But that's a first approximation. Then, as I refine my real-world capability, I do it a second time, get a better approximation. If I repeat many times, the limit of what I'm measuring approaches 2\pi. Therefore

lim _{error -> 0} measurement with error = 2\pi

It's an actual limit; I can see it, although I cannot measure it exactly. So I look at the mathematical definition of \pi and confirm it matches pretty nicely - hooray. Similarly, I can measure e and √2 through some experiment.

Aaah, and what makes you think you can actually take limits in reality?? What makes you think the real world is continuous and complete??
Evidence suggests that this is NOT the case. So the real numbers don't exist.

 

[fbs7]

One silly scientific experiment is all it takes to wipe out a lifetime's worth of work by some theoretical physicist.

Correct, and it only takes a change in one axiom to start a completely different branch of Mathematics.

I once reached this web site, Quantonics, quite by accident, and didn't understand anything. I kept reading sporadically for one year, and still couldn't understand even a single concept. The first phrase in "Quantonics for students" starts with "Truth is an agent of its own change", and I have absolutely no idea what that means -- much less the second phrase, "Quantum~learning is holographic image fledging of holographic~flux mind", which for me is beyond absolutely incomprehensible.

But I always found that site fascinating, because I imagine that if we got hold of a Math paper from 500 years into the future, it might look as perplexing to everybody as that website is to me.

 

[D H]

What makes you think the real world is continuous and complete??

Most physicists think it is. A small minority think otherwise.

Evidence suggests that this is NOT the case.

There is no such evidence. That space and time are quantized is, as far as I know, viewed as fringy. The standard model of physics has space and time as continuous, as does general relativity.

 

[micromass]

Most physicists think it is. A small minority think otherwise.


There is no such evidence. That space and time are quantized is, as far as I know, viewed as fringy. The standard model of physics has space and time as continuous, as does general relativity.

Oops

 

[jgens]

How about this: I draw a circle and measure the perimeter. My real-world circle will of course not be perfectly circular, and my real-measurement of the real-world perimeter will only have a few digits. But that's a first approximation. Then, as I refine my real-world capability, I do it a second time, get a better approximation. If I repeat many times, the limit of what I'm measuring approaches 2\pi. Therefore

lim _{error -> 0} measurement with error = 2\pi

It's an actual limit; I can see it, although I cannot measure it exactly. So I look at the mathematical definition of \pi and confirm it matches pretty nicely - hooray. Similarly, I can measure e and √2 through some experiment.

Nope. The best you can ever get are approximations. There are limits to how accurately we can measure something. In order to recover the quantity 2π you would need to measure something to arbitrary accuracy. With the technology of today, we certainly cannot measure something on the order of magnitude of 10^-100 m. So what about distances on the order of 10^-G m where G is Graham's number? My guess is that we will never be able to measure that accurately. If this is the case, then taking the limit as "error → 0" is simply meaningless.

 

[disregardthat]

Continuity is a mathematical notion. It doesn't make sense to say that space and time is continuous. Or what would that mean?

 

[nonequilibrium]

Am I the only one with the feeling that this thread is derailing into something meaningless? If the OP still has a problem, can he state concretely what is troubling him?

 

[D H]

It doesn't make sense to say that space and time is continuous. Or what would that mean?

That it isn't quantized. That calculus is a valid technique for describing reality.

The standard model is written in terms of differential equations, not difference equations. The same is true of general relativity. I'll not an expert (far, far from it!) in quantum gravity or string theory, but even there I don't think quantizing space and time is essential.

 

[fbs7]

Aaah, and what makes you think you can actually take limits in reality?? What makes you think the real world is continuous and complete?? Evidence suggests that this is NOT the case. So the real numbers don't exist.

Well, some things are discrete, so if I measure the weight of a round glass of water, that's a function of \pi, but if I measure it with great precision, I'm going to see it's a multiple of tiny values. So at some point my measurement will diverge from the model, but that's really down the line - for an everyday glass of water, it gets it right with some 20 digits or so.

So I give it, \pi will eventually break down, but up to 20 or 25 digits it's a mighty good model for physical quantities. And then some subatomic thingie may be related to pi, and then it's back at work again, for perhaps 20 digits more.

i, on the other hand, does not translate to a physical quantity at even one digit. There is nothing I can measure (that I know of) that when it is squared it gives me -1...

 

[fbs7]

Am I the only one with the feeling that this thread is derailing into something meaningless? If the OP still has a problem, can he state concretely what is troubling him?

That's a good point. My original question was if anyone was bothered that we use i all the time when it has no counterpart in reality, and looks like nobody is, for the several reasons listed through the thread.

So I'm pretty satisfied with that answer. Thanks everybody

 

[nonequilibrium]

Indeed, the main reason being that math is the queen of science and physics is its jester.

 

[Number Nine]

i, on the other hand, does not translate to a physical quantity at even one digit. There is nothing I can measure (that I know of) that when it is squared it gives me -1...

So what? Why should we care that i can't be represented as a physical quantity? The problem here is that you can't seem to get over your notion of a number as a unit of measurement. People made the same complaints when the notion of irrational numbers were introduced, and, before that, negative numbers.

I would strongly suggest studying mathematics further before attempting reason about things like complex numbers; you don't appear to have the necessary background and you don't seem to understand what mathematicians mean when the use the word "number", or how they go about constructing and describing things like the field of complex numbers.

That's a good point. My original question was if anyone was bothered that we use i all the time when it has no counterpart in reality

...what? How would you do physics or engineering without complex numbers? Hell, many properties of the integers can't be described without reference to complex numbers. The distribution of primes within the natural numbers is directly influenced by a complex function. Do the primes not exist?

 

[Jamma]

Q: What is a number?

A: Who cares? We can construct a definition if we want, but then the mystery of what a number is simply vanishes with the creation of that definition.

 

[fbs7]

I would strongly suggest studying mathematics further before attempting reason about things like complex numbers; you don't appear to have the necessary background and you don't seem to understand what mathematicians mean when the use the word "number"

Wow.

Pardon me, I'm just an old engineer asking questions. I'm not here to reason, just to improve my understanding - and I learned quite many interesting things from the answers around. Sorry if I have your disapproval - I didn't think I had to study a subject a priori to asking here.

 

[chiro]

Q: What is a number?

A: Who cares? We can construct a definition if we want, but then the mystery of what a number is simply vanishes with the creation of that definition.

Numbers are just used to represent a form of variation.

That is one of the points in mathematics: instead of stating useful things that talk about specific instances or cases, we generate statements about things that take on some level of variation. The more variability, the more general (and as a consequence powerful) the statement is.

Mathematics without variability would be pretty pointless.

 

[fbs7]

Indeed, the main reason being that math is the queen of science and physics is its jester.

I'm an engineer... I wonder where my profession would fall in this sequence...

 

[Number Nine]

Wow.

Pardon me, I'm just an old engineer asking questions. I'm not here to reason, just to improve my understanding - and I learned quite many interesting things from the answers around. Sorry if I have your disapproval - I didn't think I had to study a subject a priori to asking here.

Who said you couldn't ask a question? I said that your definition of "number" diverges from the mathematical definition, and that it fails even in the case of the familiar reals. The fact is that there is a certain amount of knowledge (particularly of analysis and algebra) required to fully understand the structure and construction of the complex numbers and number systems in general (even the reals), and intuition will not take you very far here (particularly if your definition of number is "a unit of measurement"). More, you've repeatedly made claims like "i has no basis in reality", which is demonstrably false.

 

[IsometricPion]

Many materials have an index of refraction that is (measured to be) a complex number. Of course, the caveat is that one could take the imaginary part of said complex number and obtain a real number that characterizes the fraction of incident light a given thickness of the material will absorb.

 

[Deveno]

my own thoughts on i are something like this:

we establish an intuition early on, that "numbers are something we can measure". first, by counting, and later by more sophisticated means of comparing ratios, and doing other fancy algebraic things (like taking roots, and subtracting and what-not).

of course, the word "measure" brings to mind some kind of yard-stick, and that's essentially what the real numbers are, the possible arbitrary markings on a blank (and perfectly straight! and infinitely long! whatever...) yardstick. in other words, they "idealize" our notion of measurement (limited, of course, by our finite capacity for accuracy).

but in the real world, we might notice an quantity that oscillates back and forth, like a sine wave. and it turns out that it is useful to think of it just being "something" in constant motion, around a circle. so there's perhaps some OTHER quantity (which we can measure, too) and a trade-off between the two (like a trade-off between potential and kinetic energy). so instead of having TWO equations:

x = cos t
y = sin t,

we just have ONE:

|z| = 1.

of course, now we need TWO measuring sticks, which introduces geometry into arithmetic. numbers have somehow become "spatial".

you might be comforted (or perhaps horrified), to learn that the number of "dimensions" numbers can have is severely limited to powers of 2. and for n = 4, we already lose commutativity of (multiplication of) numbers (making things more cumbersome to calculate), while at n = 8, we lose associativity (which is even worse). only very brave souls even consider n = 16, where things are very complicated, indeed.

as to your argument that "arrays are not numbers", perhaps you should think about the following set of 2x2 matrices, of the form:

[a 0]
[0 a]

where a is a real number. such matrices act so much like real numbers, a blind man might not be able to tell the difference (i suppose they are a bit chunkier, and don't drip off the chips so easily).

in general, most matrices of the form

[a b]
[c d]

behave quite poorly. they don't commute with respect to matrix multiplication, and a great many of them fail to have inverses. but we can do algebra (of a limited sort) with them, and equations involving matrices (as letters) occur in many places for "real-world problems". that is, the matrix equation:

Ax = b

is solved the same way we solve:

ax = b, by "dividing by a" (that is, finding A^-1 in the matrix case).

it turns out that matrices of the form:

[a -b]
[b a ]

not only have inverses (unless a = b = 0), but actually commute with each other, so it doesn't matter "which one we multiply by first". and, of course:

[0 -1][0 -1]...[1 0]
[1 0 ][1 0 ] = [0 1]

and there is good reason to associate the latter matrix with the number 1.

all of which is to say, there is some reason to consider "some" arrays, as being "numbers", because the algebra works out. if you want to distinguish these from "one-dimensional numbers", go right ahead, but there are some good reasons to consider any field as "a dimension unto itself" (it certainly reduces the storage space for doing linear algebra calculations with these fields).

most of the numbers we "enlarged" our original concept with, came from the desire to work with certain equations:

x + 1 = 0 ---> negative numbers
2x = 1 ---> fractions
x² - 2 = 0 ---> irrational numbers
x² + 1 = 0 ---> complex numbers

most of these constructions involve using "pairs" (or worse) of the previous set, to get going:

the number -2 is formally defined as the pair (0,2) (0 positive part, 2 negative part...or some other pair like (3,5), (1,3) and so forth), the number 3/4 is defined as the pair (3,4) (which to be perfectly honest, should be "the pair of pairs" {(3,0),(4,0)}). a number like √2, is even worse, it's the infinite set:

{((1,0),(1,0)), ((14,0),(10,0)), ((141,0),(100,0)), ((1414,0),(1000,0)), ((14142,0),(10000,0))...}

so, an infinite set of pairs of pairs.

to reduce such an infinite set of pairs of pairs to, let's say, tally marks, could be done, but the margin of this book is too small...decimal notation has its drawbacks, but at least its concise.

so, regarding complex numbers as pairs of real numbers, isn't that much different than what we do with negative numbers (using -b as shorthand for (a,b-a)), or fractions (using a/b as shorthand for (ac,bc)), and the "usual rules for arithmetic" still apply (no need to learn new rules, like with integers modulo n, for example). we even have a nifty word for describing the relationship between 1 and i, we say they are orthogonal.

 

[homeomorphic]

Since I recently posted something about this subject in another thread, I'll be brief.

I used to think imaginary numbers were weird until I read Visual Complex Analysis. Nuff said.

 

[fbs7]

Thanks all for the comments about that. All input was quite interesting.

 

[Mensanator]

I know how to use i... and I know that complex numbers are mighty useful (I'm an electronic engineer, so I used them a lot)... and how they keep popping all over the place in physics. So I'm sure the algebra behind i works.

But, it never bugged anyone here that we use something that doesn't really exist?

What makes you think it doesn't exist? "Imaginary" means "not real", it has nothing
to do with existence.

 

[disregardthat]

That it isn't quantized. That calculus is a valid technique for describing reality.

The standard model is written in terms of differential equations, not difference equations. The same is true of general relativity. I'll not an expert (far, far from it!) in quantum gravity or string theory, but even there I don't think quantizing space and time is essential.

But what would quantized space mean? The discussion of space being continuous or quantized seems strange and metaphysical to me (and too much dependent of mathematical notions such as real numbers, continuity, etc..), but above all, uneccessary. Space isn't a topological space. After all, calculus being valid as a model of representation depends on its success, not the nature of space.

 

[Jamma]

you might be comforted (or perhaps horrified), to learn that the number of "dimensions" numbers can have is severely limited to powers of 2. and for n = 4, we already lose commutativity of (multiplication of) numbers (making things more cumbersome to calculate), while at n = 8, we lose associativity (which is even worse). only very brave souls even consider n = 16, where things are very complicated, indeed.

I assume that you know the result that the only real division algebras are of dimension 1,2,4 and 8? It's a beautiful result (can be proved using algebraic topology). Of all these division algebras, the complex numbers arguably have the "nicest" properties.

 

[fbs7]

I assume that you know the result that the only real division algebras are of dimension 1,2,4 and 8? It's a beautiful result (can be proved using algebraic topology). Of all these division algebras, the complex numbers arguably have the "nicest" properties.

Wow, that's pretty amazing.

 

[fbs7]

you might be comforted (or perhaps horrified), to learn that the number of "dimensions" numbers can have is severely limited to powers of 2. and for n = 4, we already lose commutativity of (multiplication of) numbers (making things more cumbersome to calculate), while at n = 8, we lose associativity (which is even worse). only very brave souls even consider n = 16, where things are very complicated, indeed.

That's amazing. So there's no such thing as 3 + 2.i₁ + 1.i₂, where i₁ and i₂ are types of imaginary numbers, right?

That's really impressive.

 

[D H]

That's amazing. So there's no such thing as 3 + 2.i₁ + 1.i₂, where i₁ and i₂ are types of imaginary numbers, right?

That's really impressive.

Extending the complex numbers from two dimensions to three dimensions doesn't work. It does however work for four dimensions. For example, 1+2i+3j+4k. Here each of i, j, and k are in a sense a type of imaginary number, where i²=j²=k²=ijk=-1. These are the quaternions.

Have you ever wondered why i, j, and k are used as the canonical unit vectors for ℝ³ (standard 3 dimensional Euclidean space)? The answer is that the development of the quaternions preceded the development of vectors, at least amongst physicists. Our use of \hat{\imath}, \hat{\jmath}, and \hat k come from the quaternions.

 

[Curious3141]

That's amazing. So there's no such thing as 3 + 2.i₁ + 1.i₂, where i₁ and i₂ are types of imaginary numbers, right?

That's really impressive.

Not for 3d, but for 4 and 8 dimensions, then yes, there are. Quaternions and Octonions (Cayley numbers), for example, which fall under the umbrella of "hypercomplex numbers".