A friend of mine was trying to tell me that X/0 is equal to infinity (As opposed to the null set). I know there has got to be a proof to prove this wrong. Here is his reasoning:
"I based my theory purely on the fact that when you divide a number by a number very close to x (10^-100), you get an increasingly larger number as the divisor reaches closer to zero"
HellPh...@iname.com wrote in message <3805463A.58E1B...@iname.com>... >A friend of mine was trying to tell me that X/0 is equal to infinity (As >opposed to the null set). I know there has got to be a proof to prove >this wrong. Here is his reasoning:
>"I based my theory purely on the fact that when you divide a number by a >number very close to x (10^-100), you get an increasingly larger number >as the divisor reaches closer to zero"
>Can anyone help me prove him wrong?
Tell him that 1) 1/0 is _UNDEFINED_. and 2) Infinity is _not_ a number.
Another reason to give him is that behavior for a few examples is not a proof, is not even close to a proof (unless of course the examples are a base case and the induction step for an inductive proof). Tell him to check with negative values of x as well.
What if x is identically equal to zero? Then, no matter how small the number is that you are dividing it by, the numerator is smaller than the denominator which means that the result is less than one. In fact, for any denominator not equal to zero itself, the result is obviously zero.
But division by zero itself is undefined. The reason it is undefined is that there is no one definition that anyone has ever devised than can be uniformly applied to all cases to yield meaningful results.
HellPh...@iname.com wrote in message <3805463A.58E1B...@iname.com>... >A friend of mine was trying to tell me that X/0 is equal to infinity (As >opposed to the null set). I know there has got to be a proof to prove >this wrong. Here is his reasoning:
>"I based my theory purely on the fact that when you divide a number by a >number very close to x (10^-100), you get an increasingly larger number >as the divisor reaches closer to zero"
HellPh...@iname.com wrote: > A friend of mine was trying to tell me that X/0 is equal to infinity > (As > opposed to the null set). I know there has got to be a proof to prove > this wrong.
It is neither. Division by zero is undefined. x/0 isn't anything; it's illegal.
> "I based my theory purely on the fact that when you divide a number by > a > number very close to x (10^-100), you get an increasingly larger > number > as the divisor reaches closer to zero"
He is saying lim{x -> 0+} (a/x) = oo -- it's a statements about limits, and not even in general (the limit from the right is positive infinity; the limit overall does not exist). That doesn't have any bearing on whether or not a/0 = oo.
-- Erik Max Francis | icq 16063900 | whois mf303 | email m...@alcyone.com Alcyone Systems | irc maxxon (efnet) | web http://www.alcyone.com/max/ San Jose, CA | languages en, eo | icbm 37 20 07 N 121 53 38 W USA | Wed 1999 Oct 13 (13%/950) | &tSftDotIotE __ / \ The great artist is the simplifier. \__/ Henri Amiel
In article <3805463A.58E1B...@iname.com>, <HellPh...@iname.com> wrote: > A friend of mine was trying to tell me that X/0 is equal to infinity (As > opposed to the null set). I know there has got to be a proof to prove > this wrong.
What number system was he using? In "the real number system" X/0 is undefined. In a number system known as "the Riemann sphere" 1/0 is defined and equal to an element called infinity. I do not know of any system where X/0 is the null set.
-- Gerald A. Edgar ed...@math.ohio-state.edu Department of Mathematics telephone: 614-292-0395 (Office) The Ohio State University 614-292-4975 (Math. Dept.) Columbus, OH 43210 614-292-1479 (Dept. Fax)
We can define the reciprocal function (x^-1) via description theory.
(1) (x^-1) defined [the y(x*y=1 & x =/= 0)]
If x=0 or x*y =/= 1, then (x^-1) becomes [the y(contradiction)].
~Ez(z=[the y(contradiction)]), ie. [the y(contradiction)] does not exist, is a theorem in description theory.
:. (x^-1) does not exist when x=0.
It is false to say that 1/0 is equal to anything. 1/0 is equal to (the null set) {}, is false. 1/0 is equal to (the infinite number) oo, is false. (even if infinity was a number)
~Ez(z=(0^-1)). There is no z such that it is equal to (0^-1), including itself.
~(1/0=1/0). 1/0=1/0 is false.
There is no predicate that can be said truthfully about (0^-1) as subject.
<HellPh...@iname.com> wrote in message news:3805463A.58E1B732@iname.com... > A friend of mine was trying to tell me that X/0 is equal to infinity (As > opposed to the null set). I know there has got to be a proof to prove > this wrong. Here is his reasoning:
> "I based my theory purely on the fact that when you divide a number by a > number very close to x (10^-100), you get an increasingly larger number > as the divisor reaches closer to zero"
In article <3805463A.58E1B...@iname.com>, HellPh...@iname.com wrote: > A friend of mine was trying to tell me that X/0 is equal to infinity (As > opposed to the null set). I know there has got to be a proof to prove > this wrong. Here is his reasoning:
> "I based my theory purely on the fact that when you divide a number by a > number very close to x (10^-100), you get an increasingly larger number > as the divisor reaches closer to zero"
> Can anyone help me prove him wrong?
First of all, X/0 for real numbers is not defined at all, so it is not equal to anything. But here are a few things that your friend should think about:
1. What if X is -1 ? By your friends argument, the result should be -infinity. So at least you should distinguish between X >= 0 and X <= 0.
2. What if X is 0? By your friends argument, the result should be 0. By other arguments the results should be 1 (X/X is always one, therefore 0/0 should be one). So X = 0 seems to be weirder than other cases. At least you need to distinguish the different cases X > 0, X = 0, and X < 0
3. What if you divide by a number very close to - 10^(-100)? Then the sign is reversed. If X = 1 for example, X / (- 10^ (-100)) is a negative number of very large size, so I would argue that 1 / 0 = -infinity. My argument is just as good as your friends. So there are two equally good arguments that X/0 = infinity and X/0 = -infinity.
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