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0^0 = Owl, no?
Hi.Quote from: hamdani yusuf on 09/03/2022 14:36:32as 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.
as offered by Mascheroni.
In 1752, Euler in Introductio in analysin infinitorum wrote that a0 = 1 and explicitly mentioned that 00 = 1. An annotation attributed to Mascheroni in a 1787 edition of Euler's book Institutiones calculi differentialis offered the "justification"https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero#As_a_value
∀n∩ℜ, logn0 = -∞ so 00 is undefined by this route.
How do you define "undefined" in mathematical sense?
Is logn0 well defined?
Hi.Quote from: hamdani yusuf on 16/03/2022 04:57:04Is logn0 well defined? No, not in the Reals. There's no Real number which is assigned to Logn (0).Quote from: hamdani yusuf on 16/03/2022 04:17:23How 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.
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!
a real number is a value of a continuous quantity that can represent a distance along a line
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.
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.
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.
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.
Quote from: hamdani yusuf on 18/03/2022 10:46:211 * 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.
Hi.Quote from: hamdani yusuf on 18/03/2022 10:46:21I 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.