0

[John Connellan]

curioucity wrote: In Calculus, I only consider 0 as either a very small non-negative or a very small non-positive number, like 0.000000000000000000000000000000000000000000000007

non-negative = positive
non-positive = negative

Make it easier on yourself!!!

 

[malkiri]

non-negative = positive
non-positive = negative

Not true. Non-negative describes both positive numbers and zero. Non-positive describes both negative numbers and zero.

 

[curioucity]

Well, I think I need to refine it...
Actually the reason I consider 0 that way is because many calculus expressions which show 0/0 yield particular numbers (one for each expression)

 

[John Connellan]

Originally posted by malkiri
Not true. Non-negative describes both positive numbers and zero. Non-positive describes both negative numbers and zero.

Not in this case Malkiri!
Since he is trying to consider what zero itself is!!!

 

[malkiri]

Well, by the standard definition of the terms, that's the wrong application. He should say 'nonnegative and nonzero'.

 

[curioucity]

nonnegative and nonzero? You mean that 7 x 10e-30+ I wrote?
I led everyone lost, I see......

 

[thed]

Originally posted by John Connellan
Is 0/0 = 1?

If so, that is strange as well. We get a number out of nothing!

Surprised no one picked up on this.

If you divide anything by 0 the answer is always infinity. I.E. 1/0, 5/0, 1,000,000/0 = ∞. More precisely the limit of m/n as n -> 0 and m an element of the Reals is ∞

0 divided by anything is always zero.

Anything divided by itself is always 1.

So 0/0 has an indetermined answer,

 

[John Connellan]

Very good Thed, thats the proper way of explaining it. Thus zero divided by zero is indetermined.

 

[Dapthar]

Originally posted by John Connellan
Is 0/0 = 1?

If so, that is strange as well. We get a number out of nothing!

I can see how one would come to that assumption, since with limits, one can get something that seems to say that 0/0 is 1. For example.

lim_x→0 x = 0

However, lim_x→0 x/x = lim_x→0 1 = 1.

So to the untrained observer, it would seem like one is getting that 0/0 is 1, when one actually is not getting that result.

Originally posted by thed
Anything divided by itself is always 1.

Except 0.

Another Mathematical curiosity (upon cursory inspection), is that 0⁰ is defined to be 1 in some textbooks. However, this is usually due to the fact that lim_x→0 x^x = 1.

 

[John Connellan]

Originally posted by Dapthar

Anything divided by itself = 1
Except 0.

No, 0/0 can equal 1 however we cannot say it is ONLY equal to 1 becasue it is indetermined (due to what Thed said).

Depends what u mean by "="!!!

 

[thed]

Originally posted by Dapthar

Originally posted by thed
Anything divided by itself is always 1.

Except 0.

Another Mathematical curiosity (upon cursory inspection), is that 0⁰ is defined to be 1 in some textbooks. However, this is usually due to the fact that lim_x→0 x^x = 1.

I'm truly interested in why you say that. It contradicts everything I ever read or learned as an undergraduate. Granted, as Physicist, I did not do much number theory. Please elucidate on what you say, enquiring minds want to know.

 

[James R]

x^x = exp (ln(x^x)) = exp(x ln x)

Lim (x->0) x ln x
= Lim (x->0) ln x/(1/x)
= Lim (x->0) (1/x)/(-1/x²) ....... L'Hopital's rule.
= Lim (x->0) -x
= 0

So lim (x->0) x^x = exp (lim (x->0) x ln x)
=exp(0)
=1

 

[curioucity]

I rather prove it this way

..0
0 (sorry, I just don't know how to use superscript here)
=
..1-1
0 . . .= 0/0

Argue me

 

[James R]

curioucity:

Your argument is circular. It assumes what you are trying to prove. Therefore, the argument is not valid.

 

[Absane]

How is it circular? All he did was rewrite the problem of 0^0 into a form we already understand to not have a set value, 0/0.

 

[James R]

4DHyperCubix

He is trying to prove that 0⁰ = 0, by assuming that 0⁰ = 0. That's circular.

Or maybe he's trying to prove that 0/0 = 0 by assuming that 0⁰ = 0. That's also circular, but with one extra step.

 

[Absane]

Sorry, help me out. Where does he assume 0^0 = 0 or claim that?

 

[curioucity]

?

I just wrote 0^0 as 0/0, that's all...... I wasn't trying to say 0^0=0, nope

 

[Dapthar]

Originally posted by curioucity
I rather prove it this way

0⁰ = (sorry, I just don't know how to use superscript here)

Just use < sup > text you want super scripted < / sup > (without the spaces). For subscripts, just replace sub for sup.

Originally posted by curioucity
0^1-1= 0/0

Sorry, you can't write 0⁰ = 0^1-1 without dividing by zero, thus 0⁰ &ne; 0^1-1.

Originally posted by James R
curioucity:
Your argument is circular. It assumes what you are trying to prove. Therefore, the argument is not valid.

His argument is not circular, but it is still flawed due to division by 0.

Originally posted by 4DHyperCubix
Sorry, help me out. Where does he assume 0^0 = 0 or claim that?

He doesn't. See the above explanation.

Originally posted by curioucity

Did you mean these were the cause?
...
0^-1=1/0
thus not good
?

Yes. division by zero is the error I was referring to earlier.

 

[James R]

Sorry. I think I drew a wrong conclusion from the way he wrote it.

I don't see any real problem with writing 0⁰ = 0/0.

Both of those expressions are indeterminate.