[hamdani yusuf (OP)]

Acording to Mathmetica which I consider the last word on maths 0^0 is inditerminate

What's the consideration behind its answer? Argument from authority is not a strong one in a scientific discussion.

[alancalverd]

Ok. They don't change the result much.

Squeeze the trigger slowly and you'll appreciate the enormous difference between "not much" and "unstable".

[hamdani yusuf (OP)]

1 * 1 is less stable. Perturbing one of the operands with a finite constant does change the result. But multiplying them with a constant infinitesimally close to 1, or adding them with a constant infinitesimally close to 0 doesn't change the result.

Oh yes it does!
(1 * Δ)² = Δ² ≠ 1
(1+Δ) ² = 1 + 2Δ + Δ² ≠ 1.

Ok. They don't change the result much.

Squeeze the trigger slowly and you'll appreciate the enormous difference between "not much" and "unstable".

How can 1*1 unstable?

[alancalverd]

It isn't. Only you suggested it was.

[hamdani yusuf (OP)]

It isn't. Only you suggested it was.

I said 1*1 is less stable than 0*0.

[Eternal Student]

Hi.
   This kind of "stability" idea just isn't worried about too much, to the best of my knowledge.  I think that's why you (Hamdani) aren't getting many good replies about the issue.   The term "stability" is usually applied to other things in mathematics and I'm not sure that you (Hamdani) need to develop a new set of ideas for the stability of basic field operations like addition and multiplication.   There is already a simple idea called continuity which seems to offer all we need.
   However, if you want to develop ideas for this sort of stability, then I don't suppose there's any law against it.

Here's something to think about if you are developing some ideas for stability of basic field operations like + and x on different choices of input numbers:
      You mentioned that 0 x 0  was very stable but that depends on how you measure stability.  Surely you'd want to consider a proportional or percentage change?    A change of  +/- 1 unit in an expected answer of 100 units is nothing to worry about but a change of +/- 1 unit in an answer of  0.01 units   is a    ten-thousand percent error.     Anyway, using this notion of proportional difference,  0 x 0   and   also  0 + 0    isn't stable, it's terrifyingly unstable.  Perturb the input numbers even a tiny amount and you generate a proportional difference that is unbounded.   No other choice of input numbers is that bad.

Best Wishes.

[alancalverd]

A simple example of a trigger instability is the equation y = 1/x. As x passes through zero its value switches between +∞ and -∞.

[hamdani yusuf (OP)]

You mentioned that 0 x 0  was very stable but that depends on how you measure stability.

0 * 0 is extremely stable. The operands can be perturbed heavily without changing the result (e.g. by multiplying them with some constants). Adding one of the operands with a finite constant doesn't change the result. Adding both operands with a finite constant does change the result.

Perturbation by multiplication already implies ratio. Percentage change simply converts perturbation from addition to multiplication.

Anyway, using this notion of proportional difference,  0 x 0   and   also  0 + 0    isn't stable, it's terrifyingly unstable.  Perturb the input numbers even a tiny amount and you generate a proportional difference that is unbounded.   No other choice of input numbers is that bad.

Can you show how?

[hamdani yusuf (OP)]

A simple example of a trigger instability is the equation y = 1/x. As x passes through zero its value switches between +∞ and -∞.

Yes. I understand if that instability makes people think that 1/0 is undefined.
https://www.wolframalpha.com/input?i=1%2F0
Wolframalpha says that 1/0 is complex infinity instead of undefined.
1/0=-1/0=i/0=-i/0=complex infinity. In Riemann's sphere, they are located on the same point.

There are seven indeterminate forms involving 0, 1, and infinity:
https://mathworld.wolfram.com/Indeterminate.html

If complex infinity is allowed as well, then six additional indeterminate forms result:

[Eternal Student]

Hi.

    Anyway, using this notion of proportional difference,  0 x 0   and   also  0 + 0    isn't stable, it's terrifyingly unstable.  Perturb the input numbers even a tiny amount and you generate a proportional difference that is unbounded.   No other choice of input numbers is that bad.

Can you show how?

0 + 0   = 0  = the expected answer for 0 + 0.
Perturb the input values slightly and consider  δ + ε  where  δ, ε are arbitrarily small but positive.
Then  δ + ε   ≠ 0, instead δ + ε =  something small but positive.   The percentage change from the expected answer, 0, is then undefined:
 (something positive)  /   0          x  100%       
 -  the division by 0 is a problem.

Best Wishes.

[hamdani yusuf (OP)]

0 + 0   = 0  = the expected answer for 0 + 0.
Perturb the input values slightly and consider  δ + ε  where  δ, ε are arbitrarily small but positive.
Then  δ + ε   ≠ 0, instead δ + ε =  something small but positive.   The percentage change from the expected answer, 0, is then undefined:
 (something positive)  /   0          x  100%       
 -  the division by 0 is a problem.

Best Wishes.

Perturbation by multiplication already implies ratio. Percentage change simply converts perturbation from addition to multiplication.

[Eternal Student]

Hi.

Perturbation by multiplication already implies ratio. Percentage change simply converts perturbation from addition to multiplication.

   ?  I've no idea what you neant there.

Best Wishes.

[hamdani yusuf (OP)]

The Most Controversial Number in Math

0^0 is highly debated in the mathematical community. Why is 0^0 undefined? Does 0^0=0? Does 0^0=1? In this video I'll address the 0^0 meaning and give you a solid 0^0 proof.

SPOILER ALERT: 0^0=1 proof can be found just using the definition of how we define exponents.

[Bored chemist]

Imagine that, for some reason, I couldn't multiply 3 by 4.
Most of the time it wouldn't matter.
But, if I wanted to cover my 3 by4 metre room, I wouldn't know what area of carpet to buy.

On the other hand, I could get some carpet and cut it to fit the room.
And then I could weigh it and also weigh a square metre of carpet and if I divided one weight be the other I would get 12 which is the product that I somehow couldn't calculate.

I can't (reliably) calculate 0^0.
But if I ever find myself in a situation where I need to know what  0^0 is, I can work it out from the context.
If the value of 0^0 matters to some real-life calculation, you can use reality to calculate 0^0.
If it never occurs in any real-world calculation then it never really matters what the value is, and you can set it equal to the number of angels who can dance on the head of a pin.

[hamdani yusuf (OP)]

The video shows that defining 0^0 is useful in computing some series, and commonly used in numerical method.

[hamdani yusuf (OP)]

0^0: A 200+ Years Debate

Timeline
---
0:00 Opening Quote from Donald Knuth
0:18 The Integer Exponentiation Context
2:36 The Discrete Math Argument
5:57 Examples from Polynomials
7:55 The Abstract Algebra Argument
14:00 Indeterminate Form and The History of 0^0: Team Euler vs. Team Cauchy
17:00 Handling of 0^0 in Programming and Softwares
18:12 Ending words

Shortly, it depends on context.