The following thoughts on the idea of zero are not new and have undoubtedly been entertained by every mathematician at some point in their lives. The introduction of negative numbers hundreds of years ago probably created the idea of zero as a result of an operation such as : (+1) + (-1) = 0
The number zero has created endless hours of discussion, and indeed
confusion.
Consider the non-intuitive ideas such as 0! = 1 and x0 = 1 and
1/0 = infinity and the undefined value of 0/0.
Even simpler operations such as x + 0 = x, x * 0 = 0 and x - x = 0
accomplish nothing when zero is involved.
Further examples are formulae such as (bn - 1)/(b - 1) which is
the formula for the sum of a geometric sequence.
This has the equivalent form of b(n-1) + ... + b2 + b + 1
If you substitute b=1 in the later expression, the sum is n.
Now consider what happens with the first form of the expression when b=1 is
substituted.
(bn - 1)/(b - 1) = (1n - 1)/(1 - 1) = n
This means that 0/0 = n
Similar proofs exist which show, for example, that 0 = 1.
These are totally illogical, and reflect the problems inherent in using
zero as a number.
Zero behaves differently from other numbers.
The idea of zero is synonymous with absence, so I choose to view the
set of all positive integers, and zero as opposite
ideas rather than considering zero to be a distinct number like 1, 2 or 3.
In set theory, the idea of a null (or empty) set, is a similar idea.
I emphacise that it is the number zero, not the idea, that
I choose to eliminate. Presence is the opposite idea of zero and in the
realm of numbers, zero reflects the absence of numbers.
In otherwords, something either exists, or it doesn't. If it exists,
then it has a quality that we call number associated with it, and if it
doesn't exist we call this absence, zero.
I will leave this discussion to the philosophers to ponder in more depth.
Perhaps someday I will be convinced that zero is a necessary number, however in the world of positive integers, I can presently see no need for it.
An Alternate Number System (described also in these pages) eliminates the need for the digit zero completely, so unless I use the Existing Number System, you will not see much use of the symbol 0 on these pages.
Enjoy