Algebraic Properties
With these definitions R is not a field and not even a ring. However, it still has several convenient properties:
- a + (b + c) and (a + b) + c are either equal or both undefined.
- a + b and b + a are either equal or both undefined.
- a × (b × c) and (a × b) × c are either equal or both undefined.
- a × b and b × a are either equal or both undefined
- a × (b + c) and (a × b) + (a × c) are equal if both are defined.
- if a ≤ b and if both a + c and b + c are defined, then a + c ≤ b + c.
- if a ≤ b and c > 0 and both a × c and b × c are defined, then a × c ≤ b × c.
In general, all laws of arithmetic are valid in R as long as all occurring expressions are defined.
Read more about this topic: Extended Real Number Line
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