Is zero even or odd?

[Kevin Aylward]

You apparently have stumbled on something else you know damn little
about.

There is a lot I don't know, but this isn't an example of such.

In case you need help with this , you might note that "/" is
NOT an operator on the integers,

No it isn't, it is an operator on all numbers, integer or otherwise.

it is the "inverse" of a
multiplication operator.

Sure, you can have *another* meaning to the / operator in a different
context, but this aint that context. This discussion is about a/b as
usually understood in arithmetic.

Inverse is a well-defined concept but not
necessarily a function, it is a set theoretic mapping. E.G. m/n={ q:
m=q*n} by definition, so that m/n which is actually a set which can
be empty, a singleton, or infinite.

My, my, aint you a clever dude...

In the case of m/n, it is then
m/n = F^-1(m) where F(x)= n*x. Your reasoning would lead one to
believe /: I x I -> I is a function, which it isn't.

Nope. I am using a well understood definition of division as applicable
to this argument.

Kevin Aylward
[email protected]
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

 

[Fred Bloggs]

There is a lot I don't know, but this isn't an example of such.




No it isn't, it is an operator on all numbers, integer or otherwise.

We were talking about integers, and therefore 0/0={all integers}. You
want to talk about reals then 0/0={ all reals }. Are you saying that 0*x

Sure, you can have *another* meaning to the / operator in a different
context, but this aint that context. This discussion is about a/b as
usually understood in arithmetic.

I just told you how it is understood.

My, my, aint you a clever dude...




Nope. I am using a well understood definition of division as applicable
to this argument.

Really? You never have told us what your "well understood" definition
is- so what exactly are you "using" here?

 

[John Woodgate]

I read in sci.electronics.design that Kevin Aylward

How can a limit be
physically meaningfull, yet meaningless?

I don't know, but the word 'meaningless' is meaningful. I hope that is
of no help whatsoever. (;-)

 

[John Fields]

Ah, so 1 = 0/0 = (0+0)/0 = (0/0) + (0/0) = 2 ?

 

[John Fields]

And two isn't prime because it's even?

 

[John Fields]

a = b
a^2 = ab
a^2 - b^2 = ab - b^2
(a+b)(a-b) = b(a-b)
a+b = b
but a = b
a+a = a
2a = a
2 = 1

What could be clearer?

---

b = 1
a = b
a^2 = ab
a^2 - b^2 = ab - b^2
(a+b)(a-b) = b(a-b)
a+b = b
1+1 = 1

 

[BB]

Huh? 0/2 is somehow undefined because 2 > 0? Interesting.

Well sure. 0 /+N is illogical. It's like asking:
"How many universes are in a black hole ?"
0/-N makes more sense. Therefore, black holes
have lots of useless anti-matter inside of them. ;-)

 

[Nicholas O. Lindan]

Except for the fact that: 0 / 0 = undefined
Or actually more correct: n / 0 = undefined

Man, like, we don' need no steenkin' facts ...

 

[Nicholas O. Lindan]

0/0 is clearly, if anything, a constant expression. And it turns out
[to some] that its value is undefined.

Better minds than can be found here have argued this and not reached
any conclusion. 'Undefined' is the answer given by the teacher in the
7th grade, and will serve for all practical purposes.

Maybe what is needed is a New Number = '*' (or something) = Any Number You Want.

 

[Nicholas O. Lindan]

"How many universes are in a black hole ?"

Oh, this sounds like even more fun. Something we know even less
about ...

I would say about a black-hole's-worth.

But I don't believe in black holes.

Makes me an expert on the subject ...

 

[BB]

Oh, this sounds like even more fun. Something we know even less
about ...

How about some easier questions:

How many black holes are there in the universe ?

Is it meaningful to ask infinity/0 ?

Are we going to need some kind of mathematics
where the second question is somewhat meaningful
in order to answer the first question ?

Is the last (sic) question meaningful ?

I would say about a black-hole's-worth.

But I don't believe in black holes.

Makes me an expert on the subject ...

There are black holes stealing odd socks out of
my laundry.

 

[Richards Noah \(IFR LIT MET\)]

Really? You never have told us what your "well understood" definition
is- so what exactly are you "using" here?

You guys are arguing two different things. The argument that 0/0 is the set
of all integers/reals/whatever you are using is the set theory response to
the question. However, the more commonly used form is the algebraicly
accepted argument that states that division is a function of the forms: Z /
Z -> Q, R / R -> R, etc. In this definition, division by 0 is undefined for
all Z or R, including 0. So, you are both correct, but arguing different
things.

 

[Steven Lord]

0/0 is clearly, if anything, a constant expression. And it turns out
[to some] that its value is undefined.

Better minds than can be found here have argued this and not reached
any conclusion. 'Undefined' is the answer given by the teacher in the
7th grade, and will serve for all practical purposes.

Maybe what is needed is a New Number = '*' (or something) = Any Number You

Want.

Just FYI, if you're performing arithmetic using the IEEE 754 standard, then
n/0 for n not equal to 0 is the infinity with the same sign as n (i.e. -1/0
is -Inf while 1/0 is +Inf). Under the standard, 0/0 is NaN (Not a Number).
If you scroll down to "Special Operations" on this page:

http://stevehollasch.com/cgindex/coding/ieeefloat.html

you'll see some of the operations on numbers that can be represented in the
form given by the standard that give "special" results.

Professor William Kahan also discusses some of these types of operations in
these lecture notes, starting around page 6:

http://www.cs.berkeley.edu/~wkahan/ieee754status/IEEE754.PDF

 

[Nicholas O. Lindan]

Just FYI, if you're performing arithmetic using the IEEE 754 standard, then
n/0 for n not equal to ...

Ever hear the one about the Grandmother and blowing eggs?

IEEE passed a standard. Well, heck then, the issue is settled.
Lets all go home.

 

[Vince Fiscus, KB7ADL]

I know 0 is neither negative or positive but what about odd/even? I think
it's even.

Odd numbers start at 1 and go every other number 1,3,5,7;1,-1,-3,-5,-7
Even starts at 2 and go every other number 2,4,6,8;2,0,-2,-4,-6,-8

An even number plus an even number equals an even number.

An odd number plus an even number equals an odd number.

An odd number plus an odd number equals an even number.

0 + 1 = odd number

0 + 2 = even number, 2 is not odd, so zero must be even.



KB7ADL

 

[Tam/WB2TT]

The two are not the same.

The definition of the ratio a/b is

a/b = r iff b*r = a

for the case of n/0 there is no r such that r*0 = n (follows from the
definition of zero. Therefore n/0 (for non zero n) *does not exist*.

On the other hand, for 0/0, every r qualifies since for every r, r*0 =
0 (the definition of zero, again). Therefore, 0/0 is truly undefined,
in the sense that it is impossible to *uniquely* assign a value to the
ratio r.

Mati Meron | "When you argue with a fool,
[email protected] | chances are he is doing just the same"

It depends on how you get there, [sin(x)]/x is certainly defined for all
values of x including 0 and infinity.

Tam

 

[Chris Mattern]

It depends on how you get there, [sin(x)]/x is certainly defined for all
values of x including 0 and infinity.

Tam

No, it most certainly is *not*. [sin(x)]/x for x=0 is
0/0 and is undefined. The *limit as x approaches 0* of
[sin(x)]/x is 1, but that's not even vaguely the same
thing. The difference is huge.
--
Christopher Mattern

"Which one you figure tracked us?"
"The ugly one, sir."
"...Could you be more specific?"

 

[John W. Kennedy]

Thats becuase, when translated to reality, that statement becomes (0)^2
= 0, because 0 has no sign. I really wish people would stop trying to
spread the false hood that0 actually has a sign.

In the days before IEEE format, at least one FORTRAN was designed to
read, write, and test equality on -0.0, so that it could be used as NaN
(usually for "datum missing"), but I grant that having a real NaN is
ever so much nicer.

 

[Richards Noah \(IFR LIT MET\)]

Wrong- where do you get off saying (2*0)/0= 2*(0/0) ?

How about the following:

(2 * 0) / 0 = (2 * 0) * (1 / 0 ) <- Definition of division as the
inverse of multiplication
(2 * 0) * (1 / 0) = 2 * (0 * (1 / 0)) <- Associative property of
multiplication
2 * (0 * (1 / 0)) = 2 * (0 / (0 / 1)) <- Definition of division
2 * (0 / (0 / 1)) = 2 * (0 / 0) <- 0 / 1 = 0

He was just leaving out some unnecessary steps, being as that they are
rather common and generally just understood.

Of course, this is following the same strange assumptions of the fact that 0
/ 0 is a defined operation, or that 0 has an inverse.

 

[Nick Atty]

[huge cross-posting continued remorselessly - fu to rec.puzzles 'cos
that's where I read it]

Sure it can: 0 / 0 = 0 * (1 / 0) = 0 * infinity = 1

It works if the only three numbers in the universe are
0, 1, and infinity -- A number system that seems very
suited to usenet.

Someone, and I can't remember who, once said something to the effect
that all computer programs should work like this. They should allow no
instances of something, one instance of it, or any number at all.

It's not a bad idea - think how many bugs are a result of programs
dealing with far more things than the programmer ever expected.

And let's not, please, have the endless(!) debate about whether infinity
means anything to computers, whether C is a turing complete language
etc.

Please.