Defining +0 and -0 in the Integer Number Set: A Question of Limit Approaches

[ivan]

Somebody asked me how absolute value of a real number can be defined. I said |a| is defined as +a if a>=0 and -a if a<0 (instead of, |a| is defined as a if a>=0 and -a if a<0). Then came an objection that with such a definition if a=0 its absolute value should be +0 and there's no such thing.

Can't one define -0 and +0 in integer number set such that +0=-0=0?

 

[Hurkyl]

Somebody asked me how absolute value of a real number can be defined. I said |a| is defined as +a if a>=0 and -a if a<0 (instead of, |a| is defined as a if a>=0 and -a if a<0). Then came an objection that with such a definition if a=0 its absolute value should be +0 and there's no such thing.

Can't one define -0 and +0 in integer number set such that +0=-0=0?

Well, yes. The unary operators '+' and '-' are defined for all real numbers. '+x' is always 'x', and '-x' is always the additive inverse of 'x'.

 

[ivan]

Well, yes. The unary operators '+' and '-' are defined for all real numbers. '+x' is always 'x', and '-x' is always the additive inverse of 'x'.

Thanks.

So, as I understood since 0 is a real number unary operations both + and - is defined for 0 such that +0=0 and +0+(-0)=0 (per definition of additive inverse). Since +0+(-0)=0 and +0=0 => 0+(-0)=0 => -0=0 too. Am I right?

 

[HallsofIvy]

+0= -0 since 0 is its own additive inverse. You might want to ask your friend whether "4" is plus or minus. There is no "+" in front of it! The standard convention is that no sign in front of a number implies "+" so "4= +4" (or "0= +0") is understood. You certainly do not need to say "|0|= +0".

 

[CRGreathouse]

And here I was thinking the question was about the IEEE floating point +0 vs -0 distinction (4 / -0 = -infinity in IEEE arithmetic, for example -- signs propagate as usual).

 

[ivan]

Thank you all. Your explanations were very clear.

 

[Diffy]

Somebody asked me how absolute value of a real number can be defined. I said |a| is defined as +a if a>=0 and -a if a<0 (instead of, |a| is defined as a if a>=0 and -a if a<0). Then came an objection that with such a definition if a=0 its absolute value should be +0 and there's no such thing.

Can't one define -0 and +0 in integer number set such that +0=-0=0?

I think you avoid this complication if you explain that the definition of the absolute value of a number is its distance away from Zero.

My feeling is that thinking about absolute value as a distance helps in a lot of ways.

|x| = 2
What numbers distance away from 0 is 2? 2, and -2.

It helps more when there is other stuff inside the absolute value.
|x - 4| = 2

Here you can think about it as a numbers distance away from 4 is 2 thus the answers is 6 and 2.

|x + 4| = 2

Here same thing, a numbers distance away from -4 is 2, therefore the answers are -6 and -2.

|x - 4| = -2
This statement clearly doesn't make sense, how can a numbers distance away from 4 be -2, distances must be positive.

 

[Ephratah7]

...0 is either positive or negative... it's the same.

 

[jim mcnamara]

I'm with CRGreathouse - +0 and -0 in IEEE format floating point operations.
That's what I though this thread was about.

 

[K.J.Healey]

Is it an implied operation that:
-5 == (-1)*(5) ?
because then there would be a difference in the infinity case:

1/(-0) == 1/((-1)*(0)) = - (1/0) = -infinity

(Mathematica agrees with me)

 

[HallsofIvy]

Well, first "infinity" is not a real number so you can't expect formulas for real numbers to apply.

But I'm puzzled as to what difference you see!

 

[Hurkyl]

...0 is either positive or negative... it's the same.

Wrong: 0 is neither positive nor negative.

 

[Ephratah7]

Wrong: 0 is neither positive nor negative.

oppsss... ^^... sorry..^^ wrong grammar... hehe

 

[wildman]

This is maybe a dumb question but I am going to ask it and maybe Hurkyl or Hallsofivy can answer it:

Could - or + 0 be dependent on how one approaches 0 at the limit (positive side or negative side)?