Is zero even or odd?

[Alfred Z. Newmane]

0 has no flat edges, so it can't be even.

It does on my digital clock (as does my digital watch)

 

[Alfred Z. Newmane]

No, you *do* have +0 and -0, and they are both equal to 0.

Well I basically said that in the part you didn't quote

 

[Clifford Heath]

Lim{x->0}[(-x^)2] = 0

Either you mangled your parentheses, or this is a insidious way of
sneaking in a smilie.

Nothing is mangled if you view plain text.

Perhaps you could explain what (-x^)2 means then?

 

[Gideon]

I remember this as a homework problem in 9th grade algebra class many years
ago. More recently, my son encounted the same question in 7th grade algebra
class as an extra credit homework question. He did well but failed to realized
that one cannot just assume that zero must be even or odd but not both. Most
other students make the same mistake. The basic proof provided by many of us
in my algebra class is listed below:

--------

Homework question:
Is zero and even number or an odd number?

Math definitions:
Even numbers are numbers that can be written in the form 2n,
where n is an integer.
Odd numbers are numbers that can be written in the form 2n + 1,
where n is an integer.

First question and answer:
Is zero an even number?
That is: Is there an integer n, such that 0 = 2n ?
Yes. 0=2n => 0/2=n => 0=n (an obvious integer)
Therefore, 0 is an even number because if can satisfy the definition.

Second question and answer:
Is zero an odd number?
That is: Is there an integer n, such that 0 = 2n +1 ?
No. 0=2n+1 => -1=2n => -1/2=n which shows that n can not
be an integer in this case.
Therefore, 0 is not an odd number because it fails to satisfy the
definition.

--------

Any other math fact about even and odd numbers aren't needed to answer
the simple question "Is zero even, odd or both?" Of course, other facts about
even & odd numbers can be used to answer the question if those facts have
been rigorously proven. For example: A number is even if and only if it is the
sum of 2 even numbers.


Note that we cannot automatically assume that the set of even numbers and the
set of odd numbers are mutually exclusive just from the definitions above. Of
course, the proof of this fact is rather trivial:
x = 2n plus x=2m+1 =>
2n=2m+1 =>
2n/2 = 2m/2 +1/2 =>
n=m+1/2, which is impossible since n & m must both be integers

Based just upon the definitions, we also cannot make the assumption that every
integer must satisfy at least one of the definitions above and must therefore
be even or odd.

I point these out because the even/odd situation is an early introduction to
number theory and primative proofs for young students. And most students fall
into the trap of making assumptions which are not supported by the definitions
alone:
1) All even or odd numbers must be integers
2) All integers must be even or odd numbers
3) A number cannot be both even and odd

All three statements above are true but must be substantiated via proofs.


The debate over whether zero is positive or negative is also solved rather
quickly by reverting to the math definitions:
Positive numbers are numbers that are greater than zero.
Negative numbers are numbers that are less than zero.

 

[Don Klipstein]

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

Zero is definitely even. Dividing zero by 2 leaves no "remainder" or
"fraction".

However, as a bit of a digression, there are "odd functions" and "even
functions" - odd ones must have output zero when input is zero. Even
functions are permitted to have output zero or nonzero when input is
zero.
Functions can be odd, even or neither. With an even function, F(x) =
F(-x). With an odd function, F(-x) = -F(x).

- Don Klipstein ([email protected])

 

[Kevin Aylward]

No it cant.

Anything (or nothing) divided by itself = 1

O is specifically excluded from this result as dividing by zero causes
contradictions.

Division by zero is *defined* to be undefined. And my usual, "this is
not debatable" applies to this one. Read it up in any text book.

The limit:

L = f(x)/g(x) x->xo, where f(xo)=g(xo)=0

May or may not exist. If it dose, the limit may be any specific value
depending on the way the limit is approached.

In many case the *limit* represents physical reality. The notation 0/0
is a limit, and as such, is meaningless in mathematics.

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.

 

[Kevin Aylward]

0/0={ SET OF ALL INTEGERS }
No.


n/0= NULL SET for n<>0

It is very well-defined.

No it isnt.

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.

 

[Bondo]

you can say that again! Hilarious!

 

[Bondo]

Nice statement. However, this 'proves' once again to me that I
was never any good at math because I'm nogood at thinking, just
too lazy! What a shame.
Bondo

 

[Ben Bradley]

In sci.math,
comp.soft-sys.matlab,
sci.physics,
alt.math.undergrad,
rec.puzzles,
sci.astro,
sci.electronics.design and
comp.lang.perl.misc, on Mon, 20 Dec 2004 21:02:32 +0000, John Woodgate

0/0 can take ANY value.

Furthermore, 0/0 can GIVE any value. What a versatile expression!

 

[Josef Moellers]

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.

Not only usenet: someone once postulated that the only three "values" to
be used in providing computer resources should be 0, 1 and infinity
(which he meant to mean "unlimited" (for all practical purposes) (*)).

(*) My math teacher always said "A sphere's surface is unlimited but not
infinite", just to highlight the difference between the two.

 

[John Woodgate]

I read in sci.electronics.design that Clifford Heath <[email protected]>

Lim{x->0}[(-x^)2] = 0
Either you mangled your parentheses, or this is a insidious way of
sneaking in a smilie.

Nothing is mangled if you view plain text.

Perhaps you could explain what (-x^)2 means then?

How did you manage to move the ^? (;-)

 

[[email protected]]

It's a placeholder you twit.

It is a valid number. And it is even.

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

 

[[email protected]]

Except for the fact that: 0 / 0 = undefined

Or actually more correct: n / 0 = undefined

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"

 

[David Kastrup]

0/0={ SET OF ALL INTEGERS }

n/0= NULL SET for n<>0

It is very well-defined.

So { SET OF ALL INTEGERS } = 0/0 = (0+0)/0 = (2*0)/0 = 2*(0/0)
= 2* {SET OF ALL INTEGERS } = {SET OF ALL EVEN INTEGERS}?

Odd.

 

[David Kastrup]

The limit:

L = f(x)/g(x) x->xo, where f(xo)=g(xo)=0

May or may not exist. If it dose, the limit may be any specific value
depending on the way the limit is approached.

In many case the *limit* represents physical reality. The notation
0/0 is a limit, and as such, is meaningless in mathematics.

Hogwash. The notation 0/0 is most certainly not a limit, like 4/2 is
not a limit. And how could you define a limit if there were no
function values to start with?

0/0 is clearly, if anything, a constant expression. And it turns out
that its value is undefined. And limits have nothing to do with that.

There are "limits of the form 0/0", but this is a shorthand for
something completely different, and such limits in general _have_ a
value (depending on just what is taken to the limit here).

 

[Fred Bloggs]

No it isnt.

Kevin Aylward

You apparently have stumbled on something else you know damn little
about. In case you need help with this , you might note that "/" is NOT
an operator on the integers, it is the "inverse" of a multiplication
operator. 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. 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.

 

[Peter Wyzl]

: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

I think it's odd that you even need to ask...

 

[Fred Bloggs]

So { SET OF ALL INTEGERS } = 0/0 = (0+0)/0 = (2*0)/0 = 2*(0/0)
= 2* {SET OF ALL INTEGERS } = {SET OF ALL EVEN INTEGERS}?

Odd.

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

 

[Kevin Aylward]

I should clarify this. This is referring to the notion that different
f(x) and g(x) will lead to different limits. Usually for the limit to
have meaning, it must be the same independent of the way a specific f(x)
and g(x) approaches the limit.

Hogwash. The notation 0/0 is most certainly not a limit, like 4/2 is

This was a typo, for which I apologise. It should have been abundantly
clear from the context that I was saying "is not a limit". Unfortunately
when I spell checked I inadvertently deleted a word.

The above wouldn't make logical sense at all otherwise, as I already
defined "L" a limit, and distinguished it from 0/0. How can a limit be
physically meaningfull, yet meaningless?

not a limit. And how could you define a limit if there were no
function values to start with?

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

Which is what I said i.e. "0/0 is meaningless"

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.