Re: The infintely small number b
- From: BuddhaThu <softspokenbuddha@xxxxxxxxx>
- Date: Sat, 17 Nov 2007 12:24:15 -0800 (PST)
On Nov 16, 10:21 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:
Following is the definition and properties of the infinitely small
number b which I called sookshma number in my recent posts. This may
also be called infinitesimal, but I'm not sure if the formal
definition and properties of infinitesimal are the same. I'm using b
as the symbol for this, simply because it was used in the recent
discussions.
I knew this definition is neither rigorous nor complete, but I think
it has the essential ingredients to make it more formal.
The infinitely small number b
-----------------------------
b is the number that represents smallest extent.
b is not equal to zero because zero represents non-existence of
extent, while b represents existence of extent.
b is not equal to b in arithmetic sense (addition, subtraction).
b is equal to b in geometric sense: b*1 = b; This simply means that b
exists
Operations with zero (non-existence of extent)
b + 0 = b
b - 0 = b
b * 0 = 0
(b / 0) is undefined and not to be taken as inf.
Operations with itself
b + b = b
b * b = b
b / b = 1
(b - b) is undefined.
Operations with n where b<n<inf:
n * b = b
n / b = inf. (indicates that n/0 is undefined and can not be taken as
inf.)
n + b = n + b
n - b = n - b
We can't precipitate the last two arithmetic operations (addition,
subtraction) because it requires comparison of b with b itself in
arithmetic sense which is undefined.
- venkat
Let me take a crack at this.
When dealing with the continuum, there is need to see that we are
dealing with two distinct structures. The first is the infinite real
number line and the second is the infinitesimal number line.
Their distinction is as follows.
The real number line no matter how small is will aways be finite.
But once you can generate a rule, you can have the infinitesimal.
The infinitesimal is a generalization of quantity. You cannot sum
infinity. You cannot subtract infinity. You cannot multiply or divide
infinity. It simply does not make sense. But you can generalized
infinity into a mathematical rule.
In this, I am with Gauss on relating to any infinite structure.
Infinity is not a number, but a quality of things. To say how infinity
adds or subtracts is to say how beauty adds or subtract. It does not
make sense, and in a way, is a wrong grammar rule. It is not something
that can be dicatated with quantity, but of quality.
However, once you can generalize it, as in distinct from summing it,
into a set, then this generalized set with its own set of rule
governing boundaries can exert a fair amount of addition, subtraction,
multiplication, and division. It is now a quantity. It is now
countable. This is because it is behaving as if it is finite. Finitude
here is within a boundary set by a rule.
In this instance, you can order them into ordinals and compare them.
You can split them, as long as they are reasonably defined within
their own finite set of boundaries.
When we say infinite + 1, we are not saying that 1 is absolutely
merged into that infinity. It cannt sum.
But we can say infinity + 1 within their own rigidly separated sets of
quanity. This makes them into two cardinals instead of one. With this,
you can set an ordering mechanism, much like an ordinal process. This
is if you keep them in their separate cardinal formulas. But once you
merge, then the whole deal is off.
infinity + 1 will not make sense.
B.T.
.
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