Anyone here ever got bugged with i?

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  • #51
Deveno
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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.
 
  • #52
1,768
126
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.
 
  • #53
345
37
Thanks all for the comments about that. All input was quite interesting.
 
  • #54
105
0
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.
 
  • #55
disregardthat
Science Advisor
1,854
33
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.
 
  • #56
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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.
 
  • #57
345
37
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.
 
  • #58
345
37
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.
 
  • #59
D H
Staff Emeritus
Science Advisor
Insights Author
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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 [itex]\hat{\imath}[/itex], [itex]\hat{\jmath}[/itex], and [itex]\hat k[/itex] come from the quaternions.
 
  • #60
Curious3141
Homework Helper
2,843
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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".
 
  • #61
To this I have to add, not meaning to be facetious, negative chairs and negative money also don't exist but make for very useful tools for accounting. Negative numbers, of course, were once the imaginary numbers of now.
 
  • #62
345
37
To this I have to add, not meaning to be facetious, negative chairs and negative money also don't exist but make for very useful tools for accounting. Negative numbers, of course, were once the imaginary numbers of now.

Haha, true, and zero didn't exist until it was created too.

But negatives are easier to instinctively associate... like move 10m North then 10m South, and you're back to where you were.

Anyway, nevermind "i"... this whole tread got me deeply troubled with my beloved reals... ay-ay-ay, that's a much bigger problem.. :cry:
 

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