Is zero even or odd?

[Nicholas O. Lindan]

Sorry to nit pick, but in most any proper clac it should be written as
(-1)^(1/2), less you get -1 from -1^1/2, due to the '-' being evaluated
last. At least thats what happens in my TI86.

That's the problem here. Folks are using the wrong calculator.

On an hp49 (physically a horrible peice of junk, now made by
Casio(?), don't buy one) there is at least an 'oo' key.

It complains about oo when 1^0/ is _entered_, but it happily
uses 10^500 when it comes time to numerically evaluate oo.

Those for who Reversed not their Polish is 1^0/ is an
asciigram for '1 [enter] 0 [divide]'

 

[Nicholas O. Lindan]

Am I missing something here?

No, not really - nobody has said anything profound yet.

you're still DIVIDING [by 0],

The nub of the matter for sure. If we stopped this thread
would cease to be.

so when x = 0, the result [1/x] is undefined.

Well, we sure can't find a definition.

It is, in truth, 'arguably undefined'.

 

[John Larkin]

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.

I got tired of worrying about math exceptions in embedded systems so
wrote a fixed-point math package that always produces legal answers
and never crashes. The basic variable format is a 64-bit thing, 32.32
bits (32 bit signed integer plus 32 bits of fraction) which handles
most practical realtime values pretty well. The biggest possible
positive and negative values are defined to be "infinity", and
anything that computes bigger is just clamped there.


0 / anything = 0

nonzero/0 = infinity (sign maintained, of course)

small / big = 0

big * big = infinity

sqrt(negative) = 0


and stuff like that.


John

 

[John Fields]

Try it on a calc for starters. You just can't divide by zero.

 

[David Kastrup]

Er..., it can also be divided by every other number (rational,
irrational, and imaginary) without a remainder,

Which means that it is also a multiple of 3, 4, 5, 6... against which
there is no law. It does make 0 the center of the additive universe.

although some of us are amused by the strange concept of dividing
nothing and the absurd idea that there may be a `remainder'. Then
comes the wild assertion that when a number is divided by nothing,
it becomes infinite.

Numbers don't become, they are. 4 does not "become" 2 if I divide it
by 2. Half of 4 _is_ 2.

And if I divide 1 by 0, the related question is "if I have one piece
of candy, and I hand out equal amounts of candy to nobody until there
is no candy left, ..." Bzzzzt. No need to look further.

 

[David Kastrup]

---
Depends on the calculator.

I have an _old_ Commodore C8, and if you divide by zero (0.0,
actually) the display will count up.

I had an even older calculator, and its motor would just keep spinning
without the total ever changing.

 

[k wallace]

-0 often/usually signifies a limit approaching from the negative
direction.

But that is an indication of direction to approach from, NOT a sign on
the zero. When approaching f(x) from -0, we are not somehow computing
with "negative zero".
So while "-0" may have a defined meaning, it is certainly not
"negative zero".
This is getting too silly.

kwallace

 

[Nicholas O. Lindan]

If n/0 for n not 0 had a value then it would be equal to 0 and not equal
to 0 at the same time.

???

The common contrarian take is:

1 / 0 = oo
n / 0 = n * oo
0 / 0 = 0 * oo = 1

I take the stance that 0 and oo are imaginary and sticky. Once
you have an equation with a 0 in it you are stuck with the zero.

2 * 0 == 0 + 0 and it doesn't simplify

As doesn't

2 * sqrt(-1) = sqrt(-1) + sqrt(-1)

Contradictions are not permitted.

Who made that up?

Anyway, I am of contradictions, they were taken out and
slapped across my brow several days ago.

 

[John Fields]

Exactly: graph y = 1 / x

You get a graph that loos like this:
(Both the ---- line and the horizontal segments of ... are y = 0, drawn
as such since to show what the graph line looks like without being
overlapped by the origin (zero point) line.)


y = 1 / x:

| (doesn't quite reach 0,
|. <-- since y = undefiend for x = 0)
|.
| ..
(<- etc) ....... | ....... (etc ->)
-------..-+----------
.|
--> .|
(doesn't quite |
reach 0, since y = undefiend for x = 0)

---
Unde_fiend_? I like that!-)

How about if we redraw the graph to look something like this:


y = 1 / x:

OXO
|
-

-
|.
| .
| .
| .
--.------0,0--------
. |
. |
. |
.|
-

-
|
OXO

with the discontinuities in plus and minus Y being used to allow us to
ignore the unimportant values (to us) of Y so that we can get to
infinity (ASCII OXO)?

That way we could (by sliding the discontinuity up and down) also plot
y = tan phi when phi was at, and also close to, 90°.

 

[Kevin Aylward]

???

The common contrarian take is:

1 / 0 = oo
n / 0 = n * oo
0 / 0 = 0 * oo = 1

I take the stance that 0 and oo are imaginary and sticky. Once
you have an equation with a 0 in it you are stuck with the zero.

Oh?

2^0 = 1

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.

 

[Alfred Z. Newmane]

If n/0 for n not 0 had a value then it would be equal to 0 and not
equal to 0 at the same time. Contradictions are not permitted.

I know, that was my silent point (that ultimately, you cannot div by
zero.)

 

[Alfred Z. Newmane]

???

The common contrarian take is:

1 / 0 = oo
n / 0 = n * oo
0 / 0 = 0 * oo = 1

oo (infinity isn't a number) so you cannot use it this way.

 

[Alfred Z. Newmane]

That's the problem here. Folks are using the wrong calculator.
Exactly.

On an hp49 (physically a horrible peice of junk, now made by
Casio(?), don't buy one) there is at least an 'oo' key.

Good grief... I thought all of those calculators were ordered destroyed
for fear of national security?

It complains about oo when 1^0/ is _entered_, but it happily
uses 10^500 when it comes time to numerically evaluate oo.

My TI simply gives 1e500

Those for who Reversed not their Polish is 1^0/ is an
asciigram for '1 [enter] 0 [divide]'

Ah, the old HP Reversed Polish notation

 

[Alfred Z. Newmane]

But that is an indication of direction to approach from, NOT a sign on
the zero. When approaching f(x) from -0, we are not somehow computing
with "negative zero".
So while "-0" may have a defined meaning, it is certainly not
"negative zero".

Exactly. I was gonan pot the same thing jsut before I saw your post.

This is getting too silly.

Well it seems a lot of debates I've seen on usenet become over inflated
by people who know nothing of what they are talking about, and the OP is
no wiser than before he made the initial post. Even with all the
knowledgeable posts abroad, they get lost in the mix of utter ignorance,
or so it seems.

It's really sad if you really think about it...

 

[Alfred Z. Newmane]

]

y = 1 / x:

| (doesn't quite reach 0,
|. <-- since y = undefiend for x = 0)
|.
| ..
(<- etc) ....... | ....... (etc ->)
-------..-+----------
.|
--> .|
(doesn't quite |
reach 0, since y = undefiend for x = 0)

Oops... nicks picks, nit picks... ;-P

How about if we redraw the graph to look something like this:


y = 1 / x:

OXO
|
-

-
|.
| .
| .
| .
--.------0,0--------
. |
. |
. |
.|
-

-
|
OXO

with the discontinuities in plus and minus Y being used to allow us to
ignore the unimportant values (to us) of Y so that we can get to
infinity (ASCII OXO)?

I like your graph better than mine, well done.

That way we could (by sliding the discontinuity up and down) also plot
y = tan phi when phi was at, and also close to, 90°.

Good point

 

[Nicholas O. Lindan]

oo (infinity isn't a number) so you cannot use it this way.

Yes, that's my point. Keep track of oo, don't merge it with
numbers.

j [sqrt(-1)] isn't a number but we still mix it up with numbers.
For infinity I can look in the sky. For 0 I can examine my
bank balance. But for j I can't look anywhere, but still
it has use. With that perspective oo + 1 may be worth
manipulating.

I will admit, I find no use for 1/0 and 1 + oo > oo.
It's a mental itch. And here's this scratching post.

As justification, j was pretty useless/undefined/don't
talk about it for till (someone famous) came up with e^jx,
which on the face of it makes even less sense.

So, as hobby, I am exploring the idea that if you can
keep track of oo and 1/0 and 4*0 it might have some
advantage.

 

[Nicholas O. Lindan]

Well it seems a lot of debates I've seen on usenet become over inflated
by people who know nothing of what they are talking about, and the OP is
no wiser than before he made the initial post. Even with all the
knowledgeable posts abroad, they get lost in the mix of utter ignorance,
or so it seems. It's really sad if you really think about it...

"A tale, told by and idiot, full of sound and fury, signifying nothing."
Willy the Shake

For me, my choice is to endlessly converse about 1/0 or go back to
making cold sales calls.

Sort of like: "What will the dog eat before it consents to eating dog food."

 

[Nicholas O. Lindan]

OXO
-
|.
| .
| .
| .
--.------0,0--------
. |
. |
. |
.|
-

-
|
OXO

One infinity, one zero. +oo == -oo; +0 == -0. Neither
actually exist and you can approach from the direction of
your choice.

From the graph I would say 1/0 is oo.

Somebody wrote a whole book on '0', I have (had?) a copy
but darned if I can find it.

 

[John Fields]

One infinity, one zero. +oo == -oo; +0 == -0. Neither
actually exist and you can approach from the direction of
your choice.

From the graph I would say 1/0 is oo.

 

[John Savard]

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.

It is possible to make a set of consistent rules for dividing by zero.

In general for ordinary multiplication and division, if a/b = c, then a
= b*c, and if a = b*c, then a/b = c.

We also know that 0 times any of the old-fashioned numbers we know about
makes 0.

So, if 5/0 = ?, then 5 = 0*?. ? cannot possibly be any number we know
about. Could ? possibly be positive infinity?

What about 0/0 = ?. 0 = 0 * ? is true if ? is any number, positive or
negative; ? can also be zero. Can we say that ? is any finite number?

It turns out those answers are not quite true.

0 = 0 * 0.

Also, 0 = -1 * 0.

So, if 5 = 0 * ?, it's also true that 5 = 0 * -1 * ?, and it's also true
that 5 = 0 * 0 * ?. So ? has to be either plus or minus infinity, or
infinity squared, or infinity cubed, and so on. And even that isn't
*quite* right, but it comes close.

If 5 = 0 * ?, then 0 * 0 * ? can be 0 * 5, or it can be 0 * ?, depending
on which two items you multiply by first.

This means that 0/0 has to be allowed to be plus or minus infinity as
well as any finite number, including zero.

Because the rules break down so badly for dividing by zero, including
the fact that multiplication now stops being associative, mathematicians
have chosen to concentrate on studying only the "real numbers", which
are all finite quantities. This way, they can deduce new theorems from
the properties that multiplication and division have on those numbers;
generalizing to division by zero is not normally done because it appears
that it would just create awkward exceptions in every mathematical
proof, without being fruitful, without producing new, useful results.

John Savard
http://home.ecn.ab.ca/~jsavard/index.html