De-zeroed (symbol: 0])([0) is one of the 14 original Bloxin Hyper-zeroes. It is said to be smaller in size than 0, yet larger in size than hegirondo. This makes it the 8th absolutely smallest Bloxin Number, first introduced on September 24th of 2021, in the video (SNEAK PEAK) Between -0 and 0.
The first 7 of these (the smallest) do not have any definitions other than a name and a "symbol" (possibly intended as a numeral akin to the numeral "0" for zero). These 7 name/symbol pairs are (in increasing size) branoro -=-=-, tutanoro [0-0], gihenoro [*--*], jiwanoro /.,.,.\, kodanoro " ' ^ ' ", arrunoro <><>, and hegirondo ([{}]).
Definition[]
De-zeroed is the first one (in ascending order) to have in addition to a name and symbol, what appears to be a "definition". The definition of De-zeroed is 0^12. This requires some interpretation.
Standard Interpretation[]
In standard mathematics, any positive real power of 0 is equal to 0. 0^x = 0 for all real x > 0. 0^0 is one of the seven classical indeterminate forms of calculus. This means treated as a limit of 0 to the power of a limit of 0, this expression can actually equal anything, much like 0/0. 0 to a negative power is undefined in a way similiar to an expression like 1/0. That being said it is strictly correct to say:
-0 <= 0^12 <= 0
However, these terms have no particular order relation in the conventional sense since we can easily rearrange these terms in the inequality and it would still be equally correct:
0^12 <= -0 <= 0
0^12 <= 0 <= -0
0 <= 0^12 <= -0
0 <= -0 <= 0^12
-0 <= 0 <= 0^12
This is because they are all in fact equal. This has lead to debates as to whether these numbers should be treated as distinct at all.
Non-Standard Interpretation[]
Putting the Standard interpretation aside for a moment, there is a sense in which one may become convinced that 0^12 is something "smaller than" 0. The standard interpretation is based on the idea of multiplication as repeated addition. That is a*b is by definition "b groups of a". Notice that order here has some meaning even though multiplication commonly turns out to be commutative in most number systems. So when we have a*0 the standard interpretation is that this is 0 groups of a. 0 groups of anything would be simply having "nothing", in otherwords, 0. So when we encounter 0*0 we naturally get 0 since we have 0 groups of 0. A good way to describe this is to imagine 0 boxes each of which contain 0 items. Even if one had boxes there would be no items, but no boxes with no items is obviously no items. Hence 0*0 = 0. For this reason we can say that 0^2 = 0, and in fact we can iterate this to conclude 0 = 0^2 = 0^3 = 0^4 = etc. since any new power can be interpreted as the previous power times 0.
0^3 = 0^2 * 0 = 0 * 0 = 0
0^4 = 0^3 * 0 = 0 * 0 = 0
0^5 = 0^4 * 0 = 0 * 0 = 0
etc.
Given this, how could we conclude that 0^12 < 0, instead of 0^12 = 0? We can do this in a couple of ways, but the simplest is to define what we mean by "small number" and then make a generalizing assumption. Firstly we define a "small" number the set of all reals in the interval (0,1). Note that if we chose any real number in this interval, integer powers form a strictly decreasing hierarchy:
0.1^1 > 0.1^2 > 0.1^3 > 0.1^4 > ...
This holds for any member of this set. We now make the following generalizations (1) 0 is a "small" number just like any of these small numbers (2) this property holds for 0 as well. So we obtain:
0^1 > 0^2 > 0^3 > 0^4 > ...
A possible interpretation of this is that 0^2 actually means treating 0 like a unit of sorts, and then having 0 of that unit, thus making it even smaller "at that scale". That is 0^2 is to 0 as 0 to 1. Put another way this implies that 0/0^2 = 1/0. Put another way, we may presume that 0 is infinitely smaller than 1 and 0^2 is infinitely smaller than 0.
This undertanding is clearly distinct from that of the standard interpretation, therefore it may well be fair to say that this is actually an abuse of notation. That is we are using 0^2 for two distinct meanings. To distinguish these meanings we may think of 0^12 as an operation resulting in 0, and 0^12 as singular symbol representing the concept of going multiple levels down below zero. In otherwords, 0^12 is not actually evaluated but is interpretted as a new size smaller than 0. What properties these expressions have other than 0^a < 0^b if a > b is not known, in the context of The Bloxin Hyper-zeroes, but presumably we can multiply them together to obtain yet smaller numbers. Thus we may say:
0^a * 0^b = 0^(a+b)
and that:
0^a * 0 = 0 * 0^a = 0^(a+1)
For simplicity we can probably also assume that, for any real non-zero x, we have:
0^a * x = x * 0^a = 0^a
This gives them a shared property with 0 since it is also true for real non-zero x, 0*x = x*0 = 0.
In any case, it is in this obtuse sense, and only in this obtuse sense, that we may say that De-zeroed is in fact a number "smaller than 0", that is a "hyper-zero". Some may reject this and claim that any hyper-zero would have to necessarily be smaller than any positive integer power of 0 (notational abuse not withstanding), since these are all in fact equal in standard mathematics.