[alancalverd]

∀n∩ℜ, log_n0 = -∞ so 0⁰ is undefined by this route.

[Eternal Student]

Hi.

Looks sensible, @alancalverd .   There might be one symbol out of place but it's not serious.

∀n∩ℜ

  ∀n∈ℜ 

Best Wishes.

[alancalverd]

Agreed, and thank you for the correction!

[Colin2B]

0^0 = Owl, no?

Ah, yes. I see it now said piglet

[hamdani yusuf (OP)]

Hi.

as offered by Mascheroni.

   I hadn't actually heard of him    (The internet says Lorenzo Mascheroni was an Italian mathematician).

I'm not sure how he got from         to    = 1.     It looks suspiciously like some repeated division by (a-a) was done.   
i.d.k. maybe he had something else in mind.

Best Wishes.

The equation is found in Euler's book, which attributes it to Mascheroni.

In 1752, Euler in Introductio in analysin infinitorum wrote that a0 = 1[15] and explicitly mentioned that 0⁰ = 1.[16] An annotation attributed[17] to Mascheroni in a 1787 edition of Euler's book Institutiones calculi differentialis[18] offered the "justification"

https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero#As_a_value

[hamdani yusuf (OP)]

∀n∩ℜ, log_n0 = -∞ so 0⁰ is undefined by this route.

Is log_n0 well defined?

How do you define "undefined" in mathematical sense?

[syhprum]

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

[alancalverd]

Is logn0 well defined?

Yes.Of all the infinities, the real continuum infinity is perhaps the best characterised and the easiest to work with.

[Eternal Student]

Hi.

Is logn0 well defined?

    No, not in the Reals.  There's no Real number which is assigned to  Log_n (0).

How do you define "undefined" in mathematical sense?

    It depends on the context.   In it's simplest usage "undefined" just means  "this thing is not defined".  In the context of this thread I think the phrase "undefined" has most commonly been used to suggest that no real value is assigned to the expression.   However, I'm sure some people have used it in a different context or to convey some other meaning.   It may be best if you identitfy the specific sections where the phrase was used.

Best Wishes.   

[hamdani yusuf (OP)]

Hi.

Is logn0 well defined?

    No, not in the Reals.  There's no Real number which is assigned to  Log_n (0).

How do you define "undefined" in mathematical sense?

    It depends on the context.   In it's simplest usage "undefined" just means  "this thing is not defined".  In the context of this thread I think the phrase "undefined" has most commonly been used to suggest that no real value is assigned to the expression.   However, I'm sure some people have used it in a different context or to convey some other meaning.   It may be best if you identitfy the specific sections where the phrase was used.

Best Wishes.   

In your definition, √(-1) is undefined. It makes i, as well as Euler's identity, e^πi=-1, meaningless.

[alancalverd]

Entirely true, since √-1 is not a real number and is therefore not defined within the set of reals. But Euler uses complex numbers, which include an imaginary part, a multiple of i.

I bow to ES's statement: since you can't assign a numerical value to −∞ it isn't a member of ℜ. But it isn't imaginary either!

[hamdani yusuf (OP)]

Entirely true, since √-1 is not a real number and is therefore not defined within the set of reals. But Euler uses complex numbers, which include an imaginary part, a multiple of i.

I bow to ES's statement: since you can't assign a numerical value to −∞ it isn't a member of ℜ. But it isn't imaginary either!

So, i is defined, even though it's not a real number? It implies that being a real number is not a necessity for something to be mathematically defined.

Can you assign a numerical value to π?
What makes −∞ different?

[alancalverd]

True.

You can assign a real number that is infinintesimally less than π, and one that is greater, so you can define π as "that which lies between..." . You can't assign a real number that is infinitesimally greater or less than -∞, so as ES says, it isn't defined on the real number line.

a real number is a value of a continuous quantity that can represent a distance along a line

i is completely defined as  √-1 but it doesn't lie on the line that passes through 0, 1, - 2.995, π^2.7 , etc. so it isn't a real number.

[Eternal Student]

Hi.

So, i is defined, even though it's not a real number?

   As alancalverd said, that is entirely true.   
   It's not to be given too much importance.  The meaning of a word like "unconstitutional" is defined in some system (we might call that system the English language, for example) but it is not defined in the real number system.
Which leads us to the next comment.   There are a whole load of things that are defined but aren't real numbers.   There's even a whole load of things that are "mathematically defined" but aren't real numbers.

It implies that being a real number is not a necessity for something to be mathematically defined.

     Yes, entirely true.   There are many axiom systems, or many different systems of mathematics.  It is quite possible for something to be "mathematically defined"  but not be a real number.   The set of all rotations in 3-dimensional space is a structure called a group under the group operation •   which is just the composition of two rotations.   It's a perfectly good mathematical structure, all the objects in it are mathematically defined etc.  - you can do things with that structure that we would all consider as being mathematics.     However, none of the objects in that structure are real numbers.
   I should think it is possible to make everything perfectly well defined in SOME system but that system isn't always useful.  (Actually this is debatable -  if the system is built from standard set theory then there are problems defining things like the set of all sets which do not contain themselves  -  but let's just assume that in general you can make everything well defined in some system of mathematics).   Let's just bring this back to the main topic of the thread:  There may be some system in which the expression 0⁰ is defined, if that's what you wanted to know.   However, it's not the conventional real number system and where the symbol 0  and the notation for exponentiation a^b is used  to produce the expression 0⁰.  In the real number system, the expression 0⁰ is deliberately left "undefined":  It is a meaningless expression, it's just a collection of symbols put together.    There is a convention that 0⁰ =1 which is very useful for things like writing out power series  in the form  because you don't have to keep writing the constant term a₀ separately from the sigma notation when x could be 0.   (But that's just a convention for ease of notation).

   Why is it deliberately left undefined?  See earlier discussions.   There are problems maintaining all the properties we may want from the exponential function (like continuity) and/or all the usual rules of manipulating indices in an algebraic expression.
   Can you just add the definition 0⁰ =1 ?   Sure you can, just declare it and the job is done.  It doesn't make the aforementioned problems go away but there's no law that says you can't extend the definition of a^b like that and no special squad of mathematicians, let's call them the continuity police, that will come knocking on your door to punish you for that decision.   
    However, since the problems haven't actually just gone away, you're going to have to be very careful when working with expressions like 0⁰ and not assume all the usual rules of algebraic manipulation and/or analytical conditions like continuity apply.  So you're still almost always going to have to treat the appearance of an expression like 0⁰ as a special or separate case from the appearance of most other exponential expressions like a^b.  You gained nothing useful just by adding the definition 0⁰ =1.  What we would really want is that all the properties of exponents we've come to expect for a^b  continue to hold when a and b are both 0 and that does seem to be impossible (it produces inconsistencies) in the Real Number system.   You can't have all the usual properties holding you have to pick and choose just a few.   I agree that setting 0⁰ = 1 probably allows the greatest amount of these properties to hold but the important thing is that it's not all of them. 

Best Wishes.

[hamdani yusuf (OP)]

I agree that setting 0⁰ = 1 probably allows the greatest amount of these properties to hold but the important thing is that it's not all of them.

OK, deal.

I see the reluctance from accepting to define a value of a mathematical operation is related to stability of the result when the operands are perturbed. Here are my findings for mathematical operations with defined results:
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.
1 * 0 is very stable. The first operand can be perturbed heavily without changing the result. Multiplying the second operand with a finite constant doesn't change the result, but adding it with a finite constant does change the result.
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.

[alancalverd]

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.

[Eternal Student]

Hi.

I see the reluctance from accepting to define a value of a mathematical operation is related to stability of the result when the operands are perturbed.

    I don't really know what you mean by "stability".   It sounds like you're considering continuity when you consider infinitessimal perturbations.

Best Wishes.

   

[alancalverd]

Not sure that stability is ever a problem when dealing with real numbers since their functions are inherently continuous. It can be  a problem when modelling a continuous function with discretised variables but that's computing, not maths!

[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.

[hamdani yusuf (OP)]

Hi.

I see the reluctance from accepting to define a value of a mathematical operation is related to stability of the result when the operands are perturbed.

    I don't really know what you mean by "stability".   It sounds like you're considering continuity when you consider infinitessimal perturbations.

Best Wishes.

   

Yeah it does.
0^0 is more stable than 1*1 when the perturbation is in form of multiplication with the operands.
The first operand is stable against addition, but the second operand is not.