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# Google caculator 1 / 0 = 0?!

## 1 / 0 = undefined. a Bug

#### coolma

3:37 pm on Apr 15, 2005 (gmt 0)

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joined:Apr 11, 2005
posts:20

I love google. so i test all the functions out that provided by google.
But one day my friend asked me what is 1 / 0? I know it but i can not remember how to say it,the teminology. What I was sure is it is undefined or should be error(computer science students always treat it as a runtime error)
and then I searched on Google , its caculator function which is the same place as keyword input textfield, just type 1 / 0. and search it will give you the result. I was supprised that Google got me 1 / 0 = 0, as a science student, I am sure it is wrong. So I e-mail google team to let them know this arithmetic bug. (was excited) After a couple of days I got mail from them with "Thanks" , pretty good aye. But until now it is still not fixed.
I asked my friend who is good at math about the name of this "error".
Then I know It is undefined. anything / 0 = undefined.!

#### kaled

6:14 pm on Apr 28, 2005 (gmt 0)

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joined:Mar 2, 2003
posts:3710

So, all the great mathematicians of the world and of recent history are all wrong. Infinity does not exist as a mathematical concept, instead we must use the term "undefined"....... nonsense.

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.

#### TerrCan123

6:33 pm on Apr 28, 2005 (gmt 0)

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joined:July 26, 2003
posts:87

Who here has ever said infinity doesn't exist? We have been discussing the value of 1/0

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....

#### kaled

8:33 am on Apr 29, 2005 (gmt 0)

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joined:Mar 2, 2003
posts:3710

Anything times zero equals zero, that is basic math.

Clearly, you never progressed past basic maths. I could equally say "inifinity times anything is infinity, that is basic maths", however that would be equally wrong.

and 1/0 is undefined, it isn't reasonable to say it equals infinity.

if 1/0 <> infinity then n * (1/0) <> infinity therefore infinity does not exist.
You said
Who here has ever said infinity doesn't exist?

Clearly, you don't even understand the consequences of your own argument.

Kaled.

#### TerrCan123

6:56 pm on Apr 29, 2005 (gmt 0)

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joined:July 26, 2003
posts:87

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?

#### jomaxx

7:09 pm on Apr 29, 2005 (gmt 0)

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joined:Nov 6, 2002
posts:4768

At the risk of being shot at from both sides, the way I would say it is that 1/0 is infinite.

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.

#### arubicus

10:50 pm on Apr 29, 2005 (gmt 0)

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posts:570

Come on guys everyone knows that 1/0 = remitemadrathigoldigund!

#### kaled

8:54 am on Apr 30, 2005 (gmt 0)

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joined:Mar 2, 2003
posts:3710

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.

Kaled.

#### jim_w

9:40 am on Apr 30, 2005 (gmt 0)

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joined:Feb 19, 2003
posts:695

One clear case of where theory and practical operation DOES NOT go hand-in-hand.

 changed 'practicality' to 'practical operation'[/edit]

#### ciml

8:40 am on May 9, 2005 (gmt 0)

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joined:June 22, 2001
posts:3805

I don't think anyone mentioned this.

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.

;-)

#### 123rock

5:18 am on May 15, 2005 (gmt 0)

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joined:May 15, 2005
posts:4

Unfortunately all of you that claim that division by 0 is undefined have never really truly looked into the matter yourselves but are quoting you're appraised math tutors.

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

#### 123rock

5:37 am on May 15, 2005 (gmt 0)

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posts:4

Also infinity x 0 is not 0. Infinity times 0 is infinity, because infinity is not a number thus it can never be X.

#### kaled

11:31 am on May 15, 2005 (gmt 0)

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joined:Mar 2, 2003
posts:3710

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.

#### ncgimaker

1:20 pm on May 15, 2005 (gmt 0)

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joined:Feb 6, 2005
posts:91

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.

To bring this back on topic, is anyone suggesting that Google is correct in saying that 1/0=0?

Anyone?

#### BillyS

1:41 pm on May 15, 2005 (gmt 0)

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joined:June 1, 2004
posts:3181

Without looking at all these complex equations, three examples, easy to follow:

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.

#### ncgimaker

2:14 pm on May 15, 2005 (gmt 0)

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joined:Feb 6, 2005
posts:91

Interesting:

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...]

#### FromRocky

3:00 pm on May 15, 2005 (gmt 0)

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joined:Dec 31, 2003
posts:1091

After a month of discussion and 8 pages of writing, we did not go anywhere. It's still undefined and infinity.

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?

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.

#### ncgimaker

3:02 pm on May 15, 2005 (gmt 0)

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joined:Feb 6, 2005
posts:91

Even more interesting:

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.

#### ncgimaker

3:07 pm on May 15, 2005 (gmt 0)

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joined:Feb 6, 2005
posts:91

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?

#### kaled

9:13 pm on May 15, 2005 (gmt 0)

#### Senior Member

joined:Mar 2, 2003
posts:3710

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.

This is a valid argument. In previous posts, I have said that you can argue that -infinity=+infinity, however, this yields few useful results in practical mathematics.

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.

#### jomaxx

11:32 pm on May 15, 2005 (gmt 0)

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joined:Nov 6, 2002
posts:4768

Semantics about infinite or infinity or +infinity or indeterminate I don't care much about. But you cannot in principle use the equals sign ("=") to represent 1/0 or 0/0. The equals sign means exactly equal to and if you try to use it in any other context it inevitably leads to absurdities, as has been shown here again and again.

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.

#### BillyS

12:16 am on May 16, 2005 (gmt 0)

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joined:June 1, 2004
posts:3181

I'm amazed how some people would actually like to make up their own definitions of how mathematics works. And without any basis other than "gut" feel that it must be right because it seems right.

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.

0 x 0 = 0

#### 123rock

1:55 am on May 16, 2005 (gmt 0)

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joined:May 15, 2005
posts:4

Infinity times 0 is infinity and that is not nonsence. Infinity is not a point/number, it is a set of numbers, an infinite set of numbers. When you multiply a certain infinite set by 0 you multiply each of its elements by 0, thus you still have an infinite number of elements, even if they are all zeros.

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

#### kaled

9:15 am on May 16, 2005 (gmt 0)

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joined:Mar 2, 2003
posts:3710

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.

#### ncgimaker

10:45 am on May 16, 2005 (gmt 0)

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posts:91

[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 +&#38;infin; or -&#38;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.

#### TerrCan123

6:12 am on May 17, 2005 (gmt 0)

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joined:July 26, 2003
posts:87

I think the proper answer for the Google calculator is no answer, in other words if you ask what 1/0 equals it gives you a blank answer. This is really the only solution based on the possible outcomes. To say it is undefined may not be accurate as it can be defined but with different possible outsomes. So a blank answer would be best IMO.

#### TerrCan123

6:14 am on May 17, 2005 (gmt 0)

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joined:July 26, 2003
posts:87

Sorry for the typo above, outsomes = outcomes.

#### WA_Smith

6:35 am on May 17, 2005 (gmt 0)

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joined:Mar 8, 2005
posts:40

I would rather they explain the value is undefined and provide a form for you to define it ... when i have ran into 1/0 errors i either wanted a 0 or a max value for the simulation to operate as it would in real life.

The best answer to the question if it is not pure math is between 0 and infinity.

#### kaled

8:33 am on May 17, 2005 (gmt 0)

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joined:Mar 2, 2003
posts:3710

I said earlier
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.

This 88 message thread spans 3 pages: 88