Why 0/0 is undefined? As 1*0 = 1*0 then 1/1 = 0/0 which results in 0/0 = 1.

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onkonovice

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Why 0/0 is undefined?

As
1*0=1*0 then
1/1=0/0
Which results 0/0=1
Please correct me where I am getting wrong.
I am very new to math.
 
Last edited by a moderator:
Thank you stapel for your reply. In the meantime I googled this topic, I found that 0/0 is not undefined rather it's indeterminate.
 
onkonovice said:
Why 0/0 is undefined?

As
1*0=1*0 then
1/1=0/0
Which results 0/0=1
Please correct me where I am getting wrong.
I am very new to math.
Click to expand...
The reason 1/0 is undefined is that there is no value you could give it that would make sense; 1/0 = x would mean that 1 = 0*x, which can never be true.

The reason 0/0 is undefined is different: there are too many values that make sense. In fact, 0/0 = x would mean that 0 = 0*x, which is true for any value of x. So 0/0 doesn't have a unique value, which is necessary in order to define an operation. (This is the same idea as a function.)

And if we defined 0/0 = 1, say, then you could prove (as @stapel has done) that 1 = -3, which would be a disaster for mathematics. So we just can't allow it.
 
onkonovice said:
Thank you stapel for your reply. In the meantime I googled this topic, I found that 0/0 is not undefined rather it's indeterminate.
Click to expand...
There is some confusion about the definitions:

Undefined: This usually means that there is no possible value that can be given, or that there is more than one value that it can logically be given.

Indeterminate: This usually means that there is not enough information to determine what the value should be.

If you are just talking about 0/0, then it is undefined. We can say that 0/0 is any number we like.

What you (apparently) found was an article about limits: when we see that
[imath]\displaystyle \lim_{x \to a} \dfrac{f(x)}{g(x)} \to \dfrac{0}{0}[/imath]

we say this is an "indeterminate form" because the limit does have some value (which may be infinite), we just have to do some work to figure out what it is for this particular limit.

For example:
[imath]\displaystyle \lim_{x \to 0} \dfrac{x^2 + 3x}{2x^2-x} = -3[/imath]

and
[imath]\displaystyle \lim_{x \to 0} \dfrac{sin(x)}{x} = 1[/imath]

despite both of them having the same indeterminate form, 0/0.

-Dan
 
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Thank you topsquark for your reply. I found this article regarding this.
"
what is the value of 0/0?

There's a special word for stuff like this, where you could conceivably give it any number of values. That word is "indeterminate." It's not the same as undefined. It essentially means that if it pops up somewhere, you don't know what its value will be in your case. For instance, if you have the limit as x->0 of x/x and of 7x/x, the expression will have a value of 1 in the first case and 7 in the second case. Indeterminate. "
 
0 times anything is 0. So 0/0 = 0(1/0)=0 (just like 2/3 = 2*(1/3))
On the other hand, when you divide the same numbers you get 1. So 0/0=1.
Since 0/0 equals both 0 and 1, then 0=1 which is not true. This is what 0/0 indetermined.

Here is another argument which I like: Note that in say 10/5=2 we have that 5*2=10. Similarly, in 24/6=4 we have 6*4 = 24. In fact, to figure out what 28/7 equals you can ask yourself 7 times what number equals 28. That number would be 4 and 28/7=4.

Now let's consider 0/0. Let's play the same game and try to figure out 0 times what number equals 0. The problem her is that 0 times ANY NUMBER will give you 0. Got it, so 0/0 can equal any number!(??). That is 0/0=3 and 0/0=7 are both correct. That would imply that 3=7. So if I loan you $3 this week interest free, then I can ask you for $7 next week--after all, 3=7. That is in the end we can't let 0/0 to equal any number and say that it is indertermined.
 
topsquark said:
There is some confusion about the definitions:

Undefined: This usually means that there is no possible value that can be given, or that there is more than one value that it can logically be given.

Indeterminate: This usually means that there is not enough information to determine what the value should be.

If you are just talking about 0/0, then it is undefined. We can say that 0/0 is any number we like.

What you (apparently) found was an article about limits: when we see that
[imath]\displaystyle \lim_{x \to a} \dfrac{f(x)}{g(x)} \to \dfrac{0}{0}[/imath]

we say this is an "indeterminate form" because the limit does have some value (which may be infinite), we just have to do some work to figure out what it is for this particular limit.

For example:
[imath]\displaystyle \lim_{x \to 0} \dfrac{x^2 + 3x}{2x^2-x} = -3[/imath]

and
[imath]\displaystyle \lim_{x \to 0} \dfrac{sin(x)}{x} = 1[/imath]

despite both of them having the same indeterminate form, 0/0.

-Dan
Click to expand...
Dan,
I need to look up what you're saying as I have always thought that 1/0 was undefined while 0/0 was indetermined.
Steve
 
Steven G said:
Dan,
I need to look up what you're saying as I have always thought that 1/0 was undefined while 0/0 was indetermined.
Steve
Click to expand...
These are the definitions I have learned. I think they are good, but various people do use them in different ways. Regardless, if someone is going to use the words, they need to have a definition set out. The OP apparently came across "indeterminate form" so I thought I'd lay out some definitions to clear up the confusion. Your definitions aren't necessarily wrong if they don't match with mine.

-Dan
 
Steven G said:
Dan,
I need to look up what you're saying as I have always thought that 1/0 was undefined while 0/0 was indetermined.
Steve
Click to expand...
In my understanding, "indeterminate" properly applies only to limits, as a form. ("Indetermined" is not a word.) It's been borrowed from that calculus usage and applied at a lower level, but this is where it came from.

When applied to an expression like 0/0, it just means that there are different justifiable values, and so we can't assign a specific value. That is, "indeterminate" is one reason an expression can be "undefined". Both 1/0 and 0/0 are undefined, just for different reasons. At least that's how I see it.
 
Ok let me clear why the question arise? In cricket when a batsman score 2 runs in one ball his strike rate is 200, if he score 1 run in 1 ball his strike rate is 100, when he score 6 runs in 1 ball his strike rate is 600. But problem arise when he get out without facing any ball i.e. 0 run in 0 ball. What will be his strike rate?
As 0/0=x, where x can be any value. (0,1,2,3....) but not infinity because ♾️*0=1.
 
onkonovice said:
Ok let me clear why the question arise? In cricket when a batsman score 2 runs in one ball his strike rate is 200, if he score 1 run in 1 ball his strike rate is 100, when he score 6 runs in 1 ball his strike rate is 600. But problem arise when he get out without facing any ball i.e. 0 run in 0 ball. What will be his strike rate?
As 0/0=x, where x can be any value. (0,1,2,3....) but not infinity because ♾️*0=1.
Click to expand...
You just can't talk about a strike rate in that case.

And, no, it is not necessarily true that [imath]\infty\times0=1[/imath]. That, too, is an indeterminate form!
 
onkonovice said:
Isn't it true that 1/0=♾️?
Hence I say that ♾️*0=1.
Click to expand...
In some contexts, we can say informally that 1/0 is infinity. But infinity is not a number, and it doesn't follow the rules. That's why we properly say that it's simply undefined.

In particular, if you say that [imath]1/0=\infty[/imath], then you also have to say that [imath]2/0=\infty[/imath]. But that in turn implies that [imath]0\times\infty=2[/imath]. So we've shown that 1 = 2, which, again, would be disastrous for all of mathematics. So we simply can't say that 1/0 is infinity, if we are treating infinity as a number. We can only use infinity as an informal description, meaning here that 1 divided by a very small number gives a very large number.

Infinity is very slippery. The concept of limits was invented essentially as a set of special gloves with which to handle it!
 
Thanks for your reply dr.peterson. I saw the link of indeterminate form. Where I found that there are 7 types of indeterminate form. 0/0 and ♾️*0 both are on that list.
 
onkonovice said:
Thanks for your reply dr.peterson. I saw the link of indeterminate form. Where I found that there are 7 types of indeterminate form. 0/0 and ♾️*0 both are on that list.
Click to expand...
Yes, but all of those indeterminate forms are used in limits, not what you would plug into a calculator. (Okay, there are calculators that will do that, but you know what I mean!) 0/0 can have any value we want it to be. The indeterminate form 0/0 has a value that is determined by the limit it came from.

They have the same written symbol, but they are very different things.

-Dan
 
onkonovice said:
Ok let me clear why the question arise? In cricket when a batsman score 2 runs in one ball his strike rate is 200, if he score 1 run in 1 ball his strike rate is 100, when he score 6 runs in 1 ball his strike rate is 600. But problem arise when he get out without facing any ball i.e. 0 run in 0 ball. What will be his strike rate?
As 0/0=x, where x can be any value. (0,1,2,3....) but not infinity because ♾️*0=1.
Click to expand...
Acinttya, mon ami, acinttya. Thereabouts. Go Wiki.
 
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