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Zero Infinity

Zero and Infinity are perhaps two of the most unique concepts in math.  One representing nothingness itself, while the other representing a quantity greater than can be imagined.  Naturally, when these two interact, the results can be both confusing and intriguing.  I will present a comprehensive view on solutions to such problems, namely that of ∞ - ∞, ∞ / ∞, ∞ * 0, ∞ / 0, 0 / ∞, ∞0, 1∞, 00, ∞∞, 0∞, 1 / ∞, 1 / 0, and 0 / 0.

Note that Absolute Infinity, the types of zero, and Null Infinity are newly introduced here as an exercise in mathematical theory, and are not officially defined or proven to work as advertised.  These concepts have been suggested previously in other publications, but this article demonstrates a cohesive view on their interactions and properties.  Additionally, the terminology of the different types is newly introduced as well, and equivalent existing terminology is given in the type declaration table.  However, aside from these new additions, every effort has been made to describe all remaining concepts and solutions as accurately as possible.
 

Quantization of Types of 0 and ∞, and inverses: 1 / ∞, 1 / 0

Despite their looks, both 0 and infinity have many types, and to make things more confusing, infinity even has many infinite subtypes, too!  The primary types of zero and infinity are as follows:
 

Symbol	Name	Description
	Absolute Infinity	"True" Infinity. No number is larger.
	Real Infinity	The size of the entire real number system.  Also known as an "uncountable" set infinity.
	Limit Infinity	A number approaching infinity.  This is the infinity used in calculus' limits.  Also known as a "countable" set infinity.
	General Infinity	Any sufficiently large number.  Also called "Engineer's Infinity".  This is the infinity used in computer programming.
--- Standard Real Numbers ---
	General Zero	Any sufficiently small number.  Also called "Engineer's Zero".  This is the zero used in computer programming.
	Limit Zero	A number approaching zero.  This is the zero used in calculus' limits.
	Real Zero	One unit of the real number system.
	Absolute Zero	"True" Zero. No number is smaller. (By the absolute value, for example -1 has a larger absolute value)  This is the zero used when only 0 is stated.

For negative and imaginary (i = √-1) numbers the same types still apply, but notice that = - = i  and = - = i  however, that is not true for any of the other zero or infinity types. (i.e. ≠ - ≠ i )  Also, x + = as it is absolute.

Each type is also the inverse of the other type, for example: 1 / = , and 1 / = .

Therefore, to solve the first two problems, 1 / 0 and 1 / ∞: 1 / 0_x = ∞_x and 1 / ∞_x = 0_x, where x is the type of the zero or infinity.

Additionally, each type is immutable - that is, no finite amount of it can "jump" types.  That is x * ∞_y ≠ ∞_z, if y any z are different types, and x is any non-zero and non-infinite number.  Also, for the absolute versions, x *  =  and x * = , where x is any non-absolute-type number, as they are absolute.

Finally, one more special number is important, Null Infinity: .  It is a "set number", meaning it has the potential to be any non-absolute number, even real, limit, or general zeros or infinities.  It is effectively a definition of "Indeterminate", and is not any one number, but a set of possibilities.
 

Type-Dependent Answers: ∞ - ∞, ∞ / 0, 0 / ∞

For ∞ - ∞, the solution is entirely dependent on the relation between the two.  If they are exactly equal, the answer is .  If not, then the answer could be anything in between the largest infinity present, and .  Also note that for - = it is the set of all possible answers, null infinity.

Generally speaking, if they are of two different types, there is no obvious simplification.  However, the relative value of the infinities determine the result if they are the same type, for example.  if we say 1 = (1 + 2 + 3 + 4 + ...), and 2 = (1 + 3 + 5 + 7 + ...).  Then 1 - 1 = , and 1 - 2 = (2 + 4 + 6 + 8 + ...) which is another limit infinity.  Sometimes the expression can also be transformed into either ∞ / ∞ or 0 / 0, which can then be solved using the next section.

For ∞ / 0 or 0 / ∞, the result is simply the multiplication of the two types, as 1 / 0_x = ∞_x and 1 / ∞_x = 0_x.  Therefore: ∞_x / 0_y = ∞_x * ∞_y and 0_x / ∞_y = 0_x * 0_y.
 

Sets and Ratios: ∞ / ∞, 0 / 0,  ∞ * 0

First, notice that ∞ * 0 is really ∞ / ∞ in disguise, as ∞ * 0 = ∞ / (1 / 0), and 1 / 0_x = ∞_x.  Therefore ∞_x * 0_y = ∞_x / ∞_y.

Additionally, for the three non-absolute types (real, limit, general), the solutions to both ∞ / ∞ and 0 / 0 are simply the ratio of the two to each other.  For example, if you have 1 / 2 , then it is equal to the Rate of approach for 1 / Rate of approach for 2.  This is also known as L'Hôpital's rule in Calculus, as the rates are merely the derivatives of the functions of those infinities.  If the infinities or zeros are equivalent, then the answer is 1: / = 1.  Note that for reals, the ratio is based on their relative cardinality, for limits it is based on their rates (or derivatives, multiple if necessary), and for the generals it is based on their actual values, as approximate numbers will give incorrect results.

If one is absolute, and the other is not, then the result is simply the absolute type evaluated, for example: / = .

If both are absolute, the answer is in fact a set of all possible answers, including all non-absolute numbers, from all possible zeros to real numbers to all possible infinities!  This is called Null Infinity: ∞_ø. This is because that any non-absolute multiplied by an absolute equals that absolute:
/ = x
0₀ = * x [ By the definition of multiplication and division (a / b = c is the same as a = b * c) ]
This is true for any non-absolute x (in ), by the definition of absolutes. ( * anything = )
 

Powers: ∞⁰, 1^∞, 0⁰, ∞^∞, 0^∞
                                         
For all except ∞^∞ and 0^∞:

If the zeros and infinities are non-absolute, simply transforming the expression into either ∞ / ∞ or 0 / 0, then evaluating the ratio, is the way to evaluate the result.

For the absolutes, the same principles apply as with ∞ / ∞ or 0 / 0, whereby if one is absolute, only that is evaluated, treating the other as finite.  Additionally, both ∞⁰ and 1^∞ are both .  0⁰ when absolute is also technically indeterminate, and also equals , strictly speaking, however, it is "less" indeterminate, for reasons outside the scope of this article, and is often defined as equal to 1.

For ∞^∞, the answer is always some type of infinity, and can only be evaluated based on the types involved, similarly to ∞ - ∞.  For absolutes, it is simply .

For 0^∞, the result is always some multiple of a zero of the "largest" type present.  For example: ∞R = C * .  What C equals depends on the other type and the values of the zero and infinity.
 

Connections

Size of the Real Numbers: The size of the set of the real numbers is and is equal to the inverse of this, indicating if one were to draw a line 1 unit long, then each real number would be  units wide, and inversely, if you were to draw each real number 1 unit wide, the entire real number line would be long.

Slope of a vertical line: By the slope equation, the slope = Δy / Δx.  This simplifies to Δy / 0, where Δy is an arbitrary real number, which means the slope equals .

Equation of a vertical line: The equation of a vertical line is x = a, where a is the x-intercept.  However, it can also be written as y = (x - a) / 0, as when x - a, it simplifies to 0 / 0, which equals , which includes all possible points - a vertical line!  At all other times, it equals , and doesn't appear visible.
 

Reference of Properties

1 / 0_x = ∞_x
1 / ∞_x = 0_x

= - = i 
= - = i 
≠ - ≠ i  (Any non-absolute type)
x * =
x * 0₀ =
x + =
x + = x

= All possible numbers ≠ and ≠

- =
∞_x - ∞_x = (If their values are exactly equal, not just their types)

∞_x / 0_y = ∞_x * ∞_y
0_x / ∞_y = 0_x * 0_y

∞_x * 0_y = ∞_x / ∞_y
/ = / =

Absolutes only: ∞⁰ = 1^∞ = (Technically, 0⁰ = , however, it is often defined as 0⁰ = 1.)

Note that Absolute Infinity, the types of zero, and Null Infinity are newly introduced here as an exercise in mathematical theory, and are not officially defined or proven to work as advertised.  These concepts have been suggested previously in other publications, but this article demonstrates a cohesive view on their interactions and properties.  Additionally, the terminology of the different types is newly introduced as well, and equivalent existing terminology is given in the type declaration table.  However, aside from these new additions, every effort has been made to describe all remaining concepts and solutions as accurately as possible.