Infinity, Negative Zero, and Not a Number

Mathematics always go beyond its own current dimension. Natural numbers was created, but it does not satisfy existence of zero and negative numbers. Integer was then created, but what happened if we scale integer into smaller numbers. Rational was then create, but it cannot represents number liked root of 2. Real numbers was then created, but some create Complex numbers to satisfy root of -1, and some create Extended real, to satisfy division of zero.

Extended real is very interesting, it doesn't comply with number system as following:

Infinity + 1 = Infinity
1 = Infinity - Infinity
1 = 0 ???

Therefore, infinity is not existed and division by zero is undefined. But extended real try to create its own rules by introducing non-number liked Infinity, Negative Infinity, Negative Zero, and Not a number.

Infinity can be created by 1/0. Some basic properties of infinity are:

1.0 / 0.0 = Inf
1.0 / Inf = 0.0
1.0 + Inf = Inf
Inf - Inf = NaN
Inf * 0.0 = NaN
Inf * -1.0 = -Inf

Infinity in extended real has its own sign. It can be positive or negative. Negative infinity can be created by -1/0. Some basic properties of negative infinity are:

-1.0 / 0.0 = -Inf
1.0 / -Inf = -0.0
1.0 / -0.0 = -Inf
1.0 + -Inf = -Inf
-Inf * -1.0 = Inf

Zero in extended real also has its own sign. It can be negative. Negative zero can be create by -0. Some basic properties of negative zero are:

-(0.0) = -0.0
-0.0 * -1.0 = 0.0
-0.0 * -0.0 = 0.0
-0.0 * 0.0 = -0.0
-0.0 + -0.0 = -0.0
-0.0 + 0.0 = 0.0
-1.0 % 1.0 = -0.0

All undefined numbers are Not a Number (NaN). All arithmatic operation with NaN result to NaN, therefore extended real has no arithmatic error. NaN can be created from:

0.0 / 0.0 = NaN
Inf - Inf = NaN
Inf * 0.0 = NaN
1.0 % 0.0 = NaN
Inf % 1.0 = NaN
Sqrt(-1.0) = NaN

Arithmatic comparisons for extended real are as follows:

Inf > 1.0
1.0 > 0.0
0.0 == -0.0
0.0 > -1.0
-1.0 > -Inf
-Inf > NaN
NaN != NaN

NaN doesn't equal to itself, to check NaN use double.Equals(x, NaN) instead. Extended real is IEEE standard and it is called "floating point". In C#, 2 types for floating point are provided, float and double. Float is single precision, its maximum precision is around 1.4E-45. Double is much precise, its maximum precision is up to around 4.94E-324.

Next post, I will introduce type with unbounded precision.