The infintely small number b

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
.

Relevant Pages

  • Re: The infintely small number b
    ... b is not equal to zero because zero represents non-existence of ... b is not equal to b in arithmetic sense (addition, subtraction). ... is undefined and not to be taken as inf. ...
    (sci.math)
  • Re: The infintely small number b
    ... b is not equal to zero because zero represents non-existence of ... b is not equal to b in arithmetic sense (addition, subtraction). ... is undefined and not to be taken as inf. ...
    (sci.math)
  • Re: Simple subtraction formula returning strange results = Excel g
    ... subtraction differ only in some number of the least significant bits ... The 64-bit floating-point representation is shown in the stylized hex form &hEEEMMMMM,M...M, where "E" is the biased exponent and "M" is the mantissa. ... always results in a zero in A3. ... A1 displays as 1.79769313486232E+308, but the first 30 digits of its exact representation are 179769313486231,57081452742373. ...
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  • Re: Simple subtraction formula returning strange results = Excel g
    ... I was merely expanding on the explanation from http://support.microsoft.com/kb/78113, which states: "Should an addition or subtraction operation result in a value at or very ... Consider the following counter-example where the two operands differ in the first 15 sig digits, ... A3 is exactly zero. ... A1 displays as 1.79769313486232E+308, but the first 30 digits of its exact representation are 179769313486231,57081452742373. ...
    (microsoft.public.excel.worksheet.functions)
  • Re: quick question
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