What's 1 divided by 0
Mathoman
Virgil
>Prove that 1 divided by 0 yields infinity.
Division by zero doesn't yield anything except a lot of arguments.
For any division, a/b, if a/b = c, then you must also have a = b*c.
But if a/0 = c, then a = 0*c = 0, so the only division
by zero that is remotely possible would be 0/0.
Note that 0/0 causes problems, too, since 0/0 = x could be any number, x,
and still have 0 = 0*x. Thus if 0/0 could be defined, we would have 0/0 =
1 and
0/0 = 2 and 0/0 = 3, and so on, and all numbers would end up being equal.
We summarize this state of affairs by saying that division by zero is
undefined (and undefinable).
--
Virgil
vm...@frii.com
Jake Wildstrom
>Prove that 1 divided by 0 yields infinity.
By no means. It is true that lim (1/x)=oo, but to conclude from this that
x->0+
1/0=oo is failure to recognize both the definition of a limit and the
particular meaning of the somewhat misleading construct lim f(x)=oo.
x->a
Let us suppse, just for a moment, that 1/0 had a value in the reals, which
we shall call r. THen, 1 = 0*r. Note that
0*r = (0+0)*r
0*r = 0*r + 0*r
-(0*r)+0*r = -(0*r) + 0*r + 0*r
0 = 0 + 0*r
0 = 0*r
Thus, 0=1. Not exactly a healthy conclusion to draw about the reals, that.
1/0 is not efined. It's not even indeterminate (like for example 0/0)
+--First Church of Briantology--Order of the Holy Quaternion--+
| A mathematician is a device for turning coffee into |
| theorems. -Paul Erdos |
+-------------------------------------------------------------+
| Jake Wildstrom |
+-------------------------------------------------------------+
Virgil
>Why is one divided by zero undefined?
The usual definition of division requires that if a /b = x is possible,
then the division has to "check" as b*x = a.
If you want to have a number x such that 1/0 = x,
then x must also satisfy 0*x = 1. It can't happen.
If you want to define a/0 = x, you will have to have 0*x = a.
This only works if a is zero.
But you can't divide zero by zero because every number as result will check.
(If 0/0 = x and 0/0 = y then x = y, so if 0/0 is possible, all numbers are
equal.)
So you can't ever divide by zero.
(For all you nitpickers, I am assuming a number system containing at least
a 0 and a 1.)
--
Virgil
vm...@frii.com
cliff...@my-deja.com
Maybe 1/0 is “undefined” because we do not fully
comprehend the properties, operations, and
relations of such an implication. After all, the
real number “zero” denotes a void; and to
manipulate it as such is mind-boggling --
especially in our realm where there is
always “something”… :-)
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Jake Wildstrom
You posted this in response to my post which explained exactly WHY.
Listen before you speak. It'll do you some good.
>Maybe 1/0 is “undefined” because we do not fully
>comprehend the properties, operations, and
>relations of such an implication. After all, the
No, we understand fully the implications of allowing division by zero. And they
are not implications which are particularly conducive to a meaningful algebraic
structure. Essentially, if you allow division by zero, you'd have to decide
that one of the following properties doesn't hold in the integers:
1) For all a, b, and c, (a+b)c=ac+bc.
2) There is a number 0 such that for all a, 0 + a = a
3) For all a, there is a number -a such that (-a) + a = 0
4) 0 != 1
None of these properties are really dispensible.
>real number “zero” denotes a void; and to
>manipulate it as such is mind-boggling --
Bullshit. We are not talking about philosophy here. It follows logically that
division by zero leads directly to the conclusion (true in only one ring, and
not a very interesting one) that 0=1.
>Share what you know. Learn what you don't.
Share what you refuse to learn you don't know.
Richard Farrer
(Lim x->0+) 1/x is infinity
(Lim x->0-) 1/x is minus infinity
One definition of a limit existing is that the limit from above and the
limit from below give the same value. Since this is not true here the limit
does not exist and the value 1/0 is undefined.
<Mathoman> wrote in message news:37988c3...@news.mindspring.com...
cliff...@my-deja.com
Actually, we tend to agree more than we disagree. :-)
You and I both agree that division by zero raises certain issues with
the axiomatic properties of completeness, consistency, and
independence.
But if such an anomaly occurs, then (maybe) there's a different way of
looking at mathematics - from a perspective where it all "fits
together". And this is what I was hinting at.
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Ben
> | theorems. -Paul Erdos |
Paul Erdoes is my hero.