Why is the extended real number line not considered a field?

Question

Here is a reminder of what a FIELD is: A field is a set F with two operations; addition and multiplication which satisfy these axioms:

A1) If x,y is an element of F, then x + y is in F
A2) x + y = y + x (commutative)
A3) (x + y) + z = x + (y + z) (Associative)
A4) F contains an element 0 such that 0 + x = x for every x in F
A5) For every x in F, there is a corresponding -x in F such that x + (-x) = 0

M1) if x and y are in F, there product is also in F
M2) Multiplication is commutative: xy = yx
M3) Multiplication is associative: (xy)z = x(yz)
M4) F contains an element 1 not equal to 0 such that 1x = x for every x in F
M5) If x is in F and x is not equal to 0 then there exists an element 1/x such that x (1/x) = 1.

Finally, there is the distributive law: x(y+z) = xy + xz

Now here is the definition for the EXTENDED REALS: The set obtained by adding the symbols -00 and +00 (infinity symbols) to R (the real set).

Thanks...

Answer 1

The easiest way to see why the extended real numbers are not a field is that ∞ and -∞ don't have multiplicative inverses. That is, there is no extended real number a such that a * ∞ = 1, and there is no extended real number b such that b * -∞ = 1. So condition M5) is certainly violated.



The way the extended real numbers are usually defined, a number of the other properties also fail. Most of the time, ∞ + (-∞) is left undefined in the extended real numbers, to reflect the fact that ∞ - ∞ is an indeterminate form.



I'd guess that this is what the other answerer above had in mind--if ∞ + (-∞) is undefined, then ∞ and -∞ don't have additive inverses, which violates condition A5).



Of course, if ∞ + (-∞) is undefined, then this means condition A1) is also violated.



One could even argue that A2) and A3) are violated, because they don't make sense when x = ∞ and y = -∞.



M1) might fail as well, depending on how you define 0 * ∞. (Some sources leave it undefined, because 0 * ∞ is an indeterminate form. Some other sources define 0 * ∞ = 0, which makes sense in measure theory and probability, but not in some other contexts.)



If 0 * ∞ is undefined, then one could argue that M2) and M3) also fail to be satisfied, because they don't make sense when x = 0 and y = ∞. The same is true about the distributive property.



So, really, the only ones of these which the extended real numbers satisfy are A4) and M4).



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It turns out that there are, in fact, ordered fields which are larger than the real numbers and contain the real numbers. Such fields have "infinitely large" elements and "infinitely small" elements, but their structure is more complicated than that of the extended real numbers, so I won't describe such fields here.

Answer 2

Extended Real Numbers

Answer 3

It looks like they violate what you have called A5 and M5. The infinite elements have neither additive nor multiplicative inverses.

Answer 4

What is the multiplicative inverse either of the infinite elements? There is none, so it isn't a field.

Answer 5

It is impossible for the human mind to calculate infinity. It is impossible for anything to contain such knowledge and understanding that they understand what infinity it. You can't write infinity in numerical form, for it is never ending; therefore, it is not a number. Hope that made any sense...