Oh goody, one of my favourite subjects; zero! All or nothing:
How many numbers in mathematics symbolize an everything in the number world? Is there some place on the real number plane which symbolizes the sum or the whole of all numbers? Interestingly, the answer to this question is
no. As everyone knows, there is always a next greater number when counting and it isn't possible to count to a final largest number. There is just something different about the nature of the system of mathematics which makes it impossible for it to represent itself as a whole.
We could use the term positive infinity to refer to all the positive numbers combined together, but such a term would not actually represent a completed sum or combined whole. Since there is always a next greater number in this group there cannot be a single definite value. This positive infinity is more a representation of a never ending process; a series of numbers, and not a number itself. Of course the same is true of the infinity of negative numbers. Like the positive side, there isn't a unified sum of all the negative numbers.
But what if we combine together all the positive numbers with all the negative numbers? We can write this as an equation. At first it seems like if we try to sum all numbers into a single ultimate number; if we sum all the positive numbers with all negative numbers, then the total combination of all in question would sum up to zero, i.e.
(1 + (-1)) + (2 + (-2)) + (3 + (-3)) +... = 0 + 0 + 0 + ... = 0
Wouldn't that be strange if the sum of all numbers somehow equaled zero. We could then say that zero represents the everything of math, couldn't we. And that really wouldn't make sense, because the meaning of zero is very related to the word nothing.
The equation above makes it seem like zero is the sum total of all real numbers. There is always a negative value for every positive value, as shown above with integers. However, there is a problem with the consistency of this approach. It is possible to sum all numbers several different ways, and the sum does not always have the same answer. Several equations sum all real numbers yet each yields a different product. The two equations below add up all integers but as you can see, they have different sums:
(1 + 0) + (2 + (-1)) + (3 + (-2)) + (4 + (-3)) + ... = 1+ 1 + 1 + ...
next:
((- 1) + 0) + ((-2) + 1) + ((-3) + 2) + ((-4) + 3) + ... = (-1) + (-1) + (-1) + ...
These two equations, and the first equation that equals zero, each include all integers in the equation, yet we find three different solutions to the same equation. It is the same problem. In these equations we are summing definite things or values, which holds us in the realm of the finite, where a definite quantity of things is greater than zero things. The equations above sum a definite series of values, they don't sum the whole, and consequently it is said in mathematics that the sum of all real numbers is undefined. Which really kind of makes sense. Otherwise, zero would be a mathematical nothing and an everything simultaneously. So to be consistent, in ordinary math zero represents nothing and there is no ultimate number that represents all numbers, because math is the counting of definite things.
Zero cannot represent both nothing and everything in the same mathematical system of values. QED half of nothing is a quarter of nothing.