- #51
Deveno
Science Advisor
Gold Member
MHB
- 2,726
- 6
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.
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.
