Forum Moderators: Robert Charlton & goodroi
As I said earlier, a reasonable definition is
infinity = n/0 where n is any value greater than zero. In other words, n is undefined except for being greater than zero.
So, rearranging the equation, infinity * 0 = n. So infinity * 0 is undefined (but positive). This is also what I said earlier.
I don't recall having ever defined infinity as being equal to 1/0.
I can't remember much of it but I do have a subsidiary degree in Maths. I've studied Fourier theory, Nyquist theory, numerical analysis and stuff that I can't even remember the names of. All these require a reasonable definition of infinity. But I guess you know better than everyone else.
Kaled.
and 1/0 is undefined, it isn't reasonable to say it equals infinity.
Also infinity times zero equals zero, that is defined. Anything times zero equals zero, that is basic math. It doesn't equal one, which is what it would have to equal to justify your argument.
I am sure if you asked all the great mathematicians they would tell you this, but never mind little ol me....
Anything times zero equals zero, that is basic math.
and 1/0 is undefined, it isn't reasonable to say it equals infinity.
Who here has ever said infinity doesn't exist?
Kaled.
I say it should say "undefined". You say what? Infinity?
Let us know the answer.....
It's not correct to use the equal sign and say "1/0 = infinity", but the result is essentially infinite and it's not undefined in the same sense that 0/0 is undefined.
My 2/0 cents.
Kaled.
[edit] changed 'practicality' to 'practical operation'[/edit]
[google.co.uk...]
You can force the calculator to try and evaluate an expression by putting an equals sign (=) after it. This only works if the expression is mathematically resolvable. For example, 1-800-555-1234= will return a result, but 1/0= will not.
;-)
If you look at any proof claiming to disprove division by 0 as valid you will notice that it is assumed that 0/0=1 which is of course false:
1. a=1
2. a=b
3. a^2=ab
4. a^2-b^2=ab-b^2
5. (a+b)(a-b)=b(a-b)
6. [(a+b)(a-b)]/(a-b)=b(a-b)/(a-b)
7. a+b=b
8. b+b=b
9. 2=1
If you look at step 6 you see that each side is divided by a-b correct? Well if we rewrite step 6 as:
[(a+b)*0]/0=b*0/0
hopefully we agree that this is the same thing
then comparing this to step 7 we see that:
(a+b)*1=b*1;
So, you're great math teacher decided to go along with some genius' proposal that 0/0=1 and fool everyone into thinking that division by 0 is undefined. As you can see above this is a lie, which proves that 0/0 isn't 1, not that it's invalid. You can prove by this method that 3/2 isn't one, but that doesn't make it invalid.
This is the real proof gents:
0=0
0^1=0
0^(0+1)=0
[0^0]*0=0
0^0=0
0/0=0
Enjoy your answer and may your souls have peace with this 0/0 issue :)
Infinity times 0 is infinity, because infinity is not a number thus it can never be X
Once again, NONSENSE .... read and digest earlier posts.
Kaled.
When X is zero the result is therefore somewhere between -INFINITY and +INFINITY and is thus undefined.
To bring this back on topic, is anyone suggesting that Google is correct in saying that 1/0=0?
Anyone?
1/2 = 0.5
Is the same as 0.5 x 2 = 1 (Agree?)
Well, then if:
1/0 = 0
This is like saying:
0 x 0 = 1 (I don't think so...)
For those arguing about infinity... Let's see that one at work:
1/0 = Infinity
Infinity x 0 = 1 (I don't think so...)
To make all the other rules of mathematics work, any number divided by zero is UNDEFINED.
sqrt(-100) = 10i
2^2000 - no calculator result offered
yet
2^500 offers 3.27339061e150
So I think I know what they've done, there is a signal you get back from the Floating point processor which results in errors on calculators.
Instead of catching that signal, it looks like they might have prechecked the numbers, a negative number in the SQRT function and they make it position and tack an 'i' on (which shows a high level of maths in the programmer), too large a number on ^ results in no error offered.
So the programmer appears to know his maths, but not how errors comes back from the FP processor.
[delorie.com...]
Where is the answer? We may never get one if we try to find an "exact" answer but not with an "approximation".
Let's get back to the topic here, and the Google calculator. What answer should the Google calculator show for 1/0?
I say it should say "undefined". You say what? Infinity?Let us know the answer.....
My answer is "I agree with Google to give 1/0=0". This is the best approximation I can agree with.
My reason:
I like to approach this problem with an approximation.
As ncgimaker mentioned above:
1/X tends to infinity as positive X tends to zero.
1/X tends to -infinity as negative X tends to zero.
When X is zero the result is therefore somewhere between -INFINITY and +INFINITY and is thus undefined.
If we plot this function 1/x with x from -1 to 1, we see that 1/x is broken at x=0. The right side is off down the page while the left side is out of the page from the top.
If we use an approxination, x approaches to zero but not yet zero from both sites: x=10^-100 and x=-10^-100. We got Google in the left site and "Noogle" in the right site. Now we use an average to get 1/x at x=0.
1/x at x~0 = [Google+Noogle]/2 = 0
You can use this approach with any value of x but zero, you will get the same answer. Thus, I think 1/0=0 is the best approximation we can make.
2^(10 * i) = 0.797119617 + 0.603821427 i
They went to all that effort to handle imaginary numbers but yet returned 1/0=0.
Contrast this with Yahoo,
2^1100 = INF .
2^(10*i) Offers no suggestion
1001/(10-10) Offers no suggestion
Google's precision shows they're using standard FP maths.
log(1 - 0.9999999999999999) = -15.9545898 (should be 16)
log(1 - 0.99999999999999999) = no result offered (really 17)
I think thats 'double' precision binary representation.
Interesting stuff.
If we use an approxination, x approaches to zero but not yet zero from both sites: x=10^-100 and x=-10^-100. We got Google in the left site and "Noogle" in the right site. Now we use an average to get 1/x at x=0. 1/x at x~0 = [Google+Noogle]/2 = 0
Actually I wish I could edit my post because its not *between* -INFINITY and +INFINITY, it is a *choice* *between* -INFINITY and +INFINITY and thus undeterminable.
No matter, there is an IEEE standard for this see the link I put in the later post. I'll settle for their error signal as the correct answer (and its not their INF signal).
1/0= IEEE Error signal.
Fair enough?
1/X tends to infinity as positive X tends to zero.
1/X tends to -infinity as negative X tends to zero.When X is zero the result is therefore somewhere between -INFINITY and +INFINITY and is thus undefined.
I still believe that it is more accurate to state that n/0 = infinity rather than undefined, because some calculations do genuinely yield undefined values.
Kaled.
In other words a simple sentence explaining what 1/0 represents and why it can't be calculated is fine, but you can't properly say 1/0=0 or 1/0=infinity any more than you can say 1/0=my Aunt Fanny.
I don't care about calculus and infinately small slices of an object.
We learn in grade school that:
One Googzillions times Zero is always = Zero
If you believe this, then you cannot believe that:
1/0 = 0 (Cuz thats like saying Zero times Zero = One)
It is UNDEFINED because to define it means you need to rewrite a s*itload of other rules that are affected by this new definition.
So for all of those that believe 0/1 = 0, please tell me your theory for 0 x 0 = 1.
And for those that believe Google is using some kind of advanced logic, even Google agrees:
0 x 0 = 0
Again, the two equations 0/0=x and x*0=0 are not the same. If you do not agree then show me any valid mathematical method that gives you either equation from the other. I will spare you the brain power and say that you cannot since:
1. multiplying each side by 0 to get x*0=0 from 0/0=x means that you assumed that 0/0=0 because you only get the equation 0/0=x*0
2. Dividing each side by 0 to get x=0/0 means that you assumed that 0/0=1 because x*0/0=0/0 is the equation you get.
Thus your 1/2=2 then what equals x times 0 quetion is invalid. It may as well be another imaginary value.
As for the guy who restated his limits theory for the 15th time, there is no such thing as positive or negative infinity, what you are talking about is logarithmic functions and just because the LIMIT of 1/x is increasing as x decreases doesn't mean that 1/0 is infinite or indeterminate.
1/x tends to both negative infinity as it gets close to zero? I think you're a bit confused about that, that's saying x is two values at the same time.
1/x tends to your "positive" infinity while 1/-x tends to the "negative" one, then if you actually draw it you see that the infinity that they meet is the same. Again this is a limit, and infinity cannot be represented by a fraction, thus it cannot be said it is either positive nor negative because of the sign of the fraction that APPROACHES it.
The only reason that 0/0 isn't 0 is because there is no physical application of it. In fact it does not exist in nature. However if you take any equation and multiply it by 0/0 and replace 0/0 by 0 or vice versa you will see that you get no wrong results.
If you haven't had enough fun with 0/0 take a look at this:
1/0=0/0
0/0=0/0
1-1/0=0/0
1/0 - 1/0=0/0
[1/0]*[2/2] - 1/0=0/0
2/0 - 1/0=0/0
1/0=0/0 and thus x/0=0/0
Infinity times 0 is infinity and that is not nonsence
Since you seem to be mathematical genius (but one who can't spell nonsense) I suggest you consider the issue of areas under curves and convergence, etc.
Calculating the area under a curve is called integration. Integration is fundamental to vast tracts higher mathematics.
Take the family curves y=1/xn in the range 1 < x < infinity.
When 0 < n <= 1, the area is infinite
When n > 1, the area is finite.
Therefore, any argument that does not allow for the area under an infinite curve to be finite, must be nonsense.
Kaled.
[mathworld.wolfram.com...]
(Are we allowed to link to Mathword?)
+INFINITY is an unbounded number greater than every real number
-INFINITY is an unbounded number less then every real number
The other link I gave to the IEEE 754 spec shows how they handle divide by zero. They return a signal 'invalid operation' into a matherr handler of type 3:
`Division by Zero'
This exception is raised when a finite nonzero number is divided by zero. If no trap occurs the result is either +&infin; or -&infin;, depending on the signs of the operands.
It even tells you how they fix it up if you don't catch it, they substitute the token -Infinity or +infinity depending on the signs.
It looks like Google ignored the Floating point 'matherr' signal, maybe they just sprintf'd it and then looked in the string for NaN (= Not a number) or something.
The best answer to the question if it is not pure math is between 0 and infinity.
For a basic calculator that can display text, the correct output should be something like "Cannot divide by zero". An advanced calculating engine that can handle algebraic equations might be smart enough to distinguish between undefined and infinity but that does not describe the Google calculator.
Does anyone disagree?
Kaled.
