This question is based off of the observation that the notion of zero depends on the presence of absence. I can say there is 1 banana right over there or there are 2 bananas right over there and when I do you can look over there and see the 1 or 2 bananas. But when I say there are zero banana's over there, when you look, you will just wonder why I pointed it out. Is there not then, some sense in which zero is not a "number" at all, but the noticing of the presence of absence? Which is a lot more like there being -1 banana(or maybe just 1-1 bananas), than it is quite like there being "zero" bananas.
This sort of thing (claiming 1-1 does not equal zero) happens all the time in elementary algebra classes, where it is referred to as an arithmetic error or "mistake".
Regarding whether or not zero qualifies as a "real" number, it occupies a "real" position on the number line and in decimal notation indicates the absence of specific powers of ten in the number containing it. In this sense it is as real as any other digit.
Greek and Roman number systems did not have zero. Roman numerals were still preferred in accounting in some places until the 1800s. Zeno's paradox depends in large part on uncertainty on interpreting zero.
The crucial function of zero is as a placeholder in a place-value system, in our case normally base ten.
Egyptians have the earliest recorded use of zero, using the same hieroglyph for financial records showing neither debt nor credit, and to mark the base level of tombs with measurements taken above and below. They did not have a positional system however, which we got from India, along with our numerals. A case has been made that the spiritual ideas of India helped them be comfortable with zero, with the same base word 'sunya' used in the defining Buddhist term sunyata (emptiness, or dependent origination).
I find it hard it hard to understand your specific suggestion. There is
1 - (-1)
And there is 'one lot of minus one', which is 1 x - 1
When you 'point at zero', it only becomes zero bananas in context of banana calculations, and this is telling. Zero occurs meaningfully and naturally in accounting and geometry. It seems to be it occuring 'in the numberline itself' that people have a problem with, like with imaginary numbers. Numbers are a tool, we map them to things in ways that are useful, and recycle what we learn for other mappings.
Your question makes me think of:
Zero doesn't depend on the "presence of absence"; it is just the cardinal number of the empty set. Even in ancient times, people were well aware of emptiness, or empty sets, or classes that had no members:
- That jug is half full of beer, and that other jug is empty.
- Abraham has 1000 sheep and Lot has no sheep.
- Cyrus has 10,000 horses and Xenophon has no horses.
Zero is just a convenient notation, a sign for this well-understood concept that allows arithmetic to be more regular. Instead of saying "4 jugs - 4 jugs leaves you with no jugs", you can can say "4 jugs - 4 jugs = 0 jugs", and then simplify further to "4-4=0".
There is nothing mysterious about zero just as there is nothing mysterious about negative numbers or fractions, once you understand what they are intended to represent.
Substraction is defined as adding the ( additive) inverse of a number.
So , by definition : n- m = n+(-m)
By definition a number b is the additive inverse of a number a iff a+b = 0.
( In general, the " inverse" of an objet is the one that yields the identity or neutral element for a given operation; the identity element for addition is the number 0)
The additive inverse of 1 is -1.
Putting all this together we get :
(1) 1-1 = 1 + (-1)
(2) 1+(-1) = 0.