What is 0^0? Your teachers in school probably told you it was undefined on the natural numbers. Among mathematicians, however, the value of 0^0 depends on the context. 0^0 could be undefined, or it could be 1. It would be undefined if you are talking about 0 as a real number. And it would have a value of 1, as a handy shortcut if you are talking about 0 as a natural number for certain applications, e.g. the binomial theorem.

I am a bit troubled by this inconsistency. There are, after all, no inconsistencies between real and natural number arithmetic other than in the case of 0^0. In my very humble opinion, your teachers were right, but they probably didn’t give you the full story. Using simple arithmetic (no equations or proofs), I will try to make the case that 0^0 could be left undefined on the natural numbers, as it is on the real numbers.

Full text: https://www.dcproof.com/ZeroExponentsOnN.htm

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

On Tuesday, March 1, 2022 at 3:25:15 PM UTC+1, Dan Christensen wrote:

> Using simple arithmetic (no equations or proofs), I will try to make the case that 0^0 could be left undefined on the natural numbers [...]

Yeah, we might build cars with just 3 wheels too.

Simple arithmetic looks like

n^0 = 1
n^(m+1) = n * n^m .

for each and every n, m e IN.

See: https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero#Discrete_exponents
and: https://proofwiki.org/wiki/Definition:Power_(Algebra)#Integers

Moreover see:

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

On Tuesday, 1 March 2022 at 11:36:21 UTC-4, Fritz Feldhase wrote:
> On Tuesday, March 1, 2022 at 3:25:15 PM UTC+1, Dan Christensen wrote:
>
> > Using simple arithmetic (no equations or proofs), I will try to make the case that 0^0 could be left undefined on the natural numbers [...]
>
> Yeah, we might build cars with just 3 wheels too.

And why not?

https://monochrome-watches.com/three-wheeled-cars-throughout-history/

On Tuesday, March 1, 2022 at 10:36:21 AM UTC-5, Fritz Feldhase wrote:
> On Tuesday, March 1, 2022 at 3:25:15 PM UTC+1, Dan Christensen wrote:
>
> > Using simple arithmetic (no equations or proofs), I will try to make the case that 0^0 could be left undefined on the natural numbers [...]
>
> Yeah, we might build cars with just 3 wheels too.
>
> Simple arithmetic looks like
>
> n^0 = 1

n^0 = 0 would be just as "simple."

I think any value assigned to 0^0 should be justified in terms of ordinary addition and multiplication. Just like multiplication is justified in terms of addition. This would seem to be impossible. So I think you can justify leaving it undefined.

> n^(m+1) = n * n^m .
>
> for each and every n, m e IN.

I don't find questionable parallel drawn to similar constructs, e.g. empty functions or products to be very convincing.

I must admit, however, that assuming 0^0=1 on N has not resulted in any difficulties with any well-known applications.

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

On 3/1/2022 9:24 AM, Dan Christensen wrote:
> What is 0^0? Your teachers in school probably told you it was undefined on the natural numbers. Among mathematicians, however, the value of 0^0 depends on the context. 0^0 could be undefined, or it could be 1. It would be undefined if you are talking about 0 as a real number. And it would have a value of 1, as a handy shortcut if you are talking about 0 as a natural number for certain applications, e.g. the binomial theorem.
>
> I am a bit troubled by this inconsistency. There are, after all, no inconsistencies between real and natural number arithmetic other than in the case of 0^0. In my very humble opinion, your teachers were right, but they probably didn’t give you the full story. Using simple arithmetic (no equations or proofs), I will try to make the case that 0^0 could be left undefined on the natural numbers, as it is on the real numbers.
>
> Full text: https://www.dcproof.com/ZeroExponentsOnN.htm
>
It seems the way to approach this is to treat this as a function of
either the base number or the exponent, and convert it to a form where
we can use L'Hopital's Rule on it. Either 0/0 or ∞/∞.

It is of the form a^b, if you take the logarithm of it you have b*ln(a).

If this is treated as a^0 (base number as variable, limit as a-->0) you
have 0*ln(a) or 0/(1/ln(a)). Take the derivative as a-->0.

If treated as 0^b (exponent as variable, limit as b-->0) do the same,
you have b*ln(0) as b-->0. Calculate the derivatives of b/(1/ln(0)) or
b/(1/-∞) or b/0 as b-->0.

You can also approach it as x^x or x*ln(x). Single variable but as both
base and exponent.

Once one has a value, take exp(value) for the final answer.

I'll let you do the math yourself...

On 3/1/2022 9:36 AM, Fritz Feldhase wrote:
> On Tuesday, March 1, 2022 at 3:25:15 PM UTC+1, Dan Christensen wrote:
>
>> Using simple arithmetic (no equations or proofs), I will try to make the case that 0^0 could be left undefined on the natural numbers [...]
>
> Yeah, we might build cars with just 3 wheels too.
>
> Simple arithmetic looks like
>
> n^0 = 1
> n^(m+1) = n * n^m .
>
> for each and every n, m e IN.
>
> See: https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero#Discrete_exponents
> and: https://proofwiki.org/wiki/Definition:Power_(Algebra)#Integers
>
> Moreover see:
>
> https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero#Polynomials_and_power_series
>

it depends upon how it, 0^0, is approached.

M̶i̶c̶h̶a̶e̶l̶ M̶o̶r̶o̶n̶e̶y̶ w̶r̶o̶t̶e̶:

> O̶n̶ 3/1/2022 9:24 A̶M̶, D̶a̶n̶ C̶h̶r̶i̶s̶t̶e̶n̶s̶e̶n̶ w̶r̶o̶t̶e̶:
>> W̶h̶a̶t̶ i̶s̶ 00? Y̶o̶u̶r̶ t̶e̶a̶c̶h̶e̶r̶s̶ i̶n̶ s̶c̶h̶o̶o̶l̶ p̶r̶o̶b̶a̶b̶l̶y̶ t̶o̶l̶d̶ y̶o̶u̶ i̶t̶ w̶a̶s̶ u̶n̶d̶e̶f̶i̶n̶e̶d̶
>> o̶n̶ t̶h̶e̶ n̶a̶t̶u̶r̶a̶l̶ n̶u̶m̶b̶e̶r̶s̶. A̶m̶o̶n̶g̶ m̶a̶t̶h̶e̶m̶a̶t̶i̶c̶i̶a̶n̶s̶, h̶o̶w̶e̶v̶e̶r̶, t̶h̶e̶ v̶a̶l̶u̶e̶ o̶f̶ 00
>> d̶e̶p̶e̶n̶d̶s̶ o̶n̶ t̶h̶e̶ c̶o̶n̶t̶e̶x̶t̶. 00 c̶o̶u̶l̶d̶ b̶e̶ u̶n̶d̶e̶f̶i̶n̶e̶d̶, o̶r̶ i̶t̶ c̶o̶u̶l̶d̶ b̶e̶ 1. I̶t̶
>> w̶o̶u̶l̶d̶ b̶e̶ u̶n̶d̶e̶f̶i̶n̶e̶d̶ i̶f̶ y̶o̶u̶ a̶r̶e̶ t̶a̶l̶k̶i̶n̶g̶ a̶b̶o̶u̶t̶ 0 a̶s̶ a̶ r̶e̶a̶l̶ n̶u̶m̶b̶e̶r̶. A̶n̶d̶ i̶t̶
>> w̶o̶u̶l̶d̶ h̶a̶v̶e̶ a̶ v̶a̶l̶u̶e̶ o̶f̶ 1, a̶s̶ a̶ h̶a̶n̶d̶y̶ s̶h̶o̶r̶t̶c̶u̶t̶ i̶f̶ y̶o̶u̶ a̶r̶e̶ t̶a̶l̶k̶i̶n̶g̶ a̶b̶o̶u̶t̶ 0
>> a̶s̶ a̶ n̶a̶t̶u̶r̶a̶l̶ n̶u̶m̶b̶e̶r̶ f̶o̶r̶ c̶e̶r̶t̶a̶i̶n̶ a̶p̶p̶l̶i̶c̶a̶t̶i̶o̶n̶s̶, e̶.g̶. t̶h̶e̶ b̶i̶n̶o̶m̶i̶a̶l̶ t̶h̶e̶o̶r̶e̶m̶.
>>
>> F̶u̶l̶l̶ t̶e̶x̶t̶: h̶t̶t̶p̶s̶://w̶w̶w̶.d̶c̶p̶r̶o̶o̶f̶.c̶o̶m̶/Z̶e̶r̶o̶E̶x̶p̶o̶n̶e̶n̶t̶s̶O̶n̶N̶.h̶t̶m̶
>>
> I̶t̶ s̶e̶e̶m̶s̶ t̶h̶e̶ w̶a̶y̶ t̶o̶ a̶p̶p̶r̶o̶a̶c̶h̶ t̶h̶i̶s̶ i̶s̶ t̶o̶ t̶r̶e̶a̶t̶ t̶h̶i̶s̶ a̶s̶ a̶ f̶u̶n̶c̶t̶i̶o̶n̶ o̶f̶ e̶i̶t̶h̶e̶r̶
> t̶h̶e̶ b̶a̶s̶e̶ n̶u̶m̶b̶e̶r̶ o̶r̶ t̶h̶e̶ e̶x̶p̶o̶n̶e̶n̶t̶, a̶n̶d̶ c̶o̶n̶v̶e̶r̶t̶ i̶t̶ t̶o̶ a̶ f̶o̶r̶m̶ w̶h̶e̶r̶e̶ w̶e̶ c̶a̶n̶ u̶s̶e̶
> L̶'H̶o̶p̶i̶t̶a̶l̶'s̶ R̶u̶l̶e̶ o̶n̶ i̶t̶. E̶i̶t̶h̶e̶r̶ 0/0 o̶r̶ ∞/∞.
>
> I̶t̶ i̶s̶ o̶f̶ t̶h̶e̶ f̶o̶r̶m̶ a̶b̶, i̶f̶ y̶o̶u̶ t̶a̶k̶e̶ t̶h̶e̶ l̶o̶g̶a̶r̶i̶t̶h̶m̶ o̶f̶ i̶t̶ y̶o̶u̶ h̶a̶v̶e̶ b̶*l̶n̶(a̶).
>
> I̶f̶ t̶h̶i̶s̶ i̶s̶ t̶r̶e̶a̶t̶e̶d̶ a̶s̶ a̶0 (b̶a̶s̶e̶ n̶u̶m̶b̶e̶r̶ a̶s̶ v̶a̶r̶i̶a̶b̶l̶e̶, l̶i̶m̶i̶t̶ a̶s̶ a̶-->0) y̶o̶u̶
> h̶a̶v̶e̶ 0*l̶n̶(a̶) o̶r̶ 0/(1/l̶n̶(a̶)). T̶a̶k̶e̶ t̶h̶e̶ d̶e̶r̶i̶v̶a̶t̶i̶v̶e̶ a̶s̶ a̶-->0.
>
> I̶f̶ t̶r̶e̶a̶t̶e̶d̶ a̶s̶ 0b̶ (e̶x̶p̶o̶n̶e̶n̶t̶ a̶s̶ v̶a̶r̶i̶a̶b̶l̶e̶, l̶i̶m̶i̶t̶ a̶s̶ b̶-->0) d̶o̶ t̶h̶e̶ s̶a̶m̶e̶, y̶o̶u̶
> h̶a̶v̶e̶ b̶*l̶n̶(0) a̶s̶ b̶-->0. C̶a̶l̶c̶u̶l̶a̶t̶e̶ t̶h̶e̶ d̶e̶r̶i̶v̶a̶t̶i̶v̶e̶s̶ o̶f̶ b̶/(1/l̶n̶(0)) o̶r̶
> b̶/(1/-∞) o̶r̶ b̶/0 a̶s̶ b̶-->0.
>
> Y̶o̶u̶ c̶a̶n̶ a̶l̶s̶o̶ a̶p̶p̶r̶o̶a̶c̶h̶ i̶t̶ a̶s̶ x̶x̶ o̶r̶ x̶*l̶n̶(x̶). S̶i̶n̶g̶l̶e̶ v̶a̶r̶i̶a̶b̶l̶e̶ b̶u̶t̶ a̶s̶ b̶o̶t̶h̶
> b̶a̶s̶e̶ a̶n̶d̶ e̶x̶p̶o̶n̶e̶n̶t̶.
>
> O̶n̶c̶e̶ o̶n̶e̶ h̶a̶s̶ a̶ v̶a̶l̶u̶e̶, t̶a̶k̶e̶ e̶x̶p̶(v̶a̶l̶u̶e̶) f̶o̶r̶ t̶h̶e̶ f̶i̶n̶a̶l̶ a̶n̶s̶w̶e̶r̶.
>
> I̶'l̶l̶ l̶e̶t̶ y̶o̶u̶ d̶o̶ t̶h̶e̶ m̶a̶t̶h̶ y̶o̶u̶r̶s̶e̶l̶f̶...

completely nonsense.

On 3/1/2022 11:19 AM, sergio wrote:
> On 3/1/2022 9:36 AM, Fritz Feldhase wrote:
>> On Tuesday, March 1, 2022 at 3:25:15 PM UTC+1, Dan Christensen wrote:
>>
>>> Using simple arithmetic (no equations or proofs), I will try to make the case that 0^0 could be left undefined on the natural numbers [...]
>>
>> Yeah, we might build cars with just 3 wheels too.
>>
>> Simple arithmetic looks like
>>
>> n^0 = 1
>> n^(m+1) = n * n^m .
>>
>> for each and every n, m e IN.
>>
>> See: https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero#Discrete_exponents
>> and: https://proofwiki.org/wiki/Definition:Power_(Algebra)#Integers
>>
>> Moreover see:
>>
>> https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero#Polynomials_and_power_series
>>
>
>
> it depends upon how it, 0^0, is approached.
>
>
>
>
>

0 * 0 * 0 *...*0 = 0^k = 0
.
.
.
0 * 0 * 0 * 0 = 0^4 = 0
0 * 0 * 0 = 0 = 0^3 = 0
0 * 0 = 0 = 0^2 = 0
0 = 0 = 0^1 = 0
0^0 = 0

Michael Moroney wrote:

> On 3/1/2022 9:24 AM, Dan Christensen wrote:
>> What is 0^0? Your teachers in school probably told you it was undefined
>> on the natural numbers. Among mathematicians, however, the value of 0^0
>> depends on the context. 0^0 could be undefined, or it could be 1. It
>
> It seems the way to approach this is to treat this as a function of
> either the base number or the exponent, and convert it to a form where
> we can use L'Hopital's Rule on it. Either 0/0 or ∞/∞.

Amazing, for what your corrupt fake money government, and the lying TV and
mainstream media, is making you endure a war, hunger, lack of heat and
oil, and even expect a nuclear war. Wake up, friends.

A Reality Check on the NWO Lies About Russia vs Ukraine.
https://www.bitchute.com/video/2GmMZqZ97fD3/

Zelensky Actor_Dancer - At least Reagan Just Appeared With a Monkey
https://www.bitchute.com/video/cKx0h5uejQHb/

On Tuesday, March 1, 2022 at 8:46:38 PM UTC+1, sergio wrote:

> 0^0 = 0

Nope.

"Define" n^m (n, m e IN) as

n * n * ... n * 1
-----------------
m-times "n *"

Then

n^0 = 1, for each and every n e IN (even for n = 0).

I guess you KNOW that the "empty product" is 1, right?

-------------------------------

Btw. we should "define" m * n the same way,

n + n + ... n + 0
-----------------
m-times "n +"

to get 0 * n = 0 for each and every n e IN.

(The "empty sum" is 0.)

Recursively:

0 * n = 0
(m+1) * n = n + m * n.

n ^ 0 = 1
n ^ (m+1) = n * n ^ m.

On 3/1/2022 3:53 PM, Duane Hume wrote:
> Michael Moroney wrote:
>
>> On 3/1/2022 9:24 AM, Dan Christensen wrote:
>>> What is 0^0? Your teachers in school probably told you it was undefined
>>> on the natural numbers. Among mathematicians, however, the value of 0^0
>>> depends on the context. 0^0 could be undefined, or it could be 1. It
>>
>> It seems the way to approach this is to treat this as a function of
>> either the base number or the exponent, and convert it to a form where
>> we can use L'Hopital's Rule on it. Either 0/0 or ∞/∞.
>
> Amazing, for what your corrupt fake money government, and the lying TV and
> mainstream media, is making you endure a war, hunger, lack of heat and
> oil, and even expect a nuclear war. Wake up, friends.
>
> A Reality Check on the NWO Lies About Russia vs Ukraine.
> https://www.bitchute.com/video/2GmMZqZ97fD3/
>
> Zelensky Actor_Dancer - At least Reagan Just Appeared With a Monkey
> https://www.bitchute.com/video/cKx0h5uejQHb/

Do we have two different nymshifters here? The Unicode abuser and this
one, the bitchute fan? Both responded to my post in completely different
ways. Did we witness a shift change of those posting using the
nymshifter account? Are the nymshifters being kept in a dungeon below
the lowest basement level of the Kremlin? I have to check which one, if
not both, came from the slums of a former Yugoslavian city, who wound
up, starving, on the streets of Moscow, before becoming a slave of the
Russian secret service.

On Tuesday, March 1, 2022 at 5:45:31 PM UTC+1, Dan Christensen wrote:
> On Tuesday, March 1, 2022 at 10:36:21 AM UTC-5, Fritz Feldhase wrote:
> >
> > Simple arithmetic looks like
> >
> > n^0 = 1
> >
> n^0 = 0 would be just as "simple."

But it would not allow for the recursive defintion I mentioned, idiot.

> > n^(m+1) = n * n^m .
> >
> > for each and every n, m e IN.

This would not work with n^0 = 0. (*sigh*)

Why are you so dense, Dan?

On 3/1/2022 2:12 PM, Duane Hume wrote:
> M̶i̶c̶h̶a̶e̶l̶ M̶o̶r̶o̶n̶e̶y̶ w̶r̶o̶t̶e̶:
>
>> O̶n̶ 3/1/2022 9:24 A̶M̶, D̶a̶n̶ C̶h̶r̶i̶s̶t̶e̶n̶s̶e̶n̶ w̶r̶o̶t̶e̶:
>>> W̶h̶a̶t̶ i̶s̶ 00? Y̶o̶u̶r̶ t̶e̶a̶c̶h̶e̶r̶s̶ i̶n̶ s̶c̶h̶o̶o̶l̶ p̶r̶o̶b̶a̶b̶l̶y̶ t̶o̶l̶d̶ y̶o̶u̶ i̶t̶ w̶a̶s̶ u̶n̶d̶e̶f̶i̶n̶e̶d̶
>>> o̶n̶ t̶h̶e̶ n̶a̶t̶u̶r̶a̶l̶ n̶u̶m̶b̶e̶r̶s̶. A̶m̶o̶n̶g̶ m̶a̶t̶h̶e̶m̶a̶t̶i̶c̶i̶a̶n̶s̶, h̶o̶w̶e̶v̶e̶r̶, t̶h̶e̶ v̶a̶l̶u̶e̶ o̶f̶ 00
>>> d̶e̶p̶e̶n̶d̶s̶ o̶n̶ t̶h̶e̶ c̶o̶n̶t̶e̶x̶t̶. 00 c̶o̶u̶l̶d̶ b̶e̶ u̶n̶d̶e̶f̶i̶n̶e̶d̶, o̶r̶ i̶t̶ c̶o̶u̶l̶d̶ b̶e̶ 1. I̶t̶
>>> w̶o̶u̶l̶d̶ b̶e̶ u̶n̶d̶e̶f̶i̶n̶e̶d̶ i̶f̶ y̶o̶u̶ a̶r̶e̶ t̶a̶l̶k̶i̶n̶g̶ a̶b̶o̶u̶t̶ 0 a̶s̶ a̶ r̶e̶a̶l̶ n̶u̶m̶b̶e̶r̶. A̶n̶d̶ i̶t̶
>>> w̶o̶u̶l̶d̶ h̶a̶v̶e̶ a̶ v̶a̶l̶u̶e̶ o̶f̶ 1, a̶s̶ a̶ h̶a̶n̶d̶y̶ s̶h̶o̶r̶t̶c̶u̶t̶ i̶f̶ y̶o̶u̶ a̶r̶e̶ t̶a̶l̶k̶i̶n̶g̶ a̶b̶o̶u̶t̶ 0
>>> a̶s̶ a̶ n̶a̶t̶u̶r̶a̶l̶ n̶u̶m̶b̶e̶r̶ f̶o̶r̶ c̶e̶r̶t̶a̶i̶n̶ a̶p̶p̶l̶i̶c̶a̶t̶i̶o̶n̶s̶, e̶.g̶. t̶h̶e̶ b̶i̶n̶o̶m̶i̶a̶l̶ t̶h̶e̶o̶r̶e̶m̶.
>>>
>>> F̶u̶l̶l̶ t̶e̶x̶t̶: h̶t̶t̶p̶s̶://w̶w̶w̶.d̶c̶p̶r̶o̶o̶f̶.c̶o̶m̶/Z̶e̶r̶o̶E̶x̶p̶o̶n̶e̶n̶t̶s̶O̶n̶N̶.h̶t̶m̶
>>>
>> I̶t̶ s̶e̶e̶m̶s̶ t̶h̶e̶ w̶a̶y̶ t̶o̶ a̶p̶p̶r̶o̶a̶c̶h̶ t̶h̶i̶s̶ i̶s̶ t̶o̶ t̶r̶e̶a̶t̶ t̶h̶i̶s̶ a̶s̶ a̶ f̶u̶n̶c̶t̶i̶o̶n̶ o̶f̶ e̶i̶t̶h̶e̶r̶
>> t̶h̶e̶ b̶a̶s̶e̶ n̶u̶m̶b̶e̶r̶ o̶r̶ t̶h̶e̶ e̶x̶p̶o̶n̶e̶n̶t̶, a̶n̶d̶ c̶o̶n̶v̶e̶r̶t̶ i̶t̶ t̶o̶ a̶ f̶o̶r̶m̶ w̶h̶e̶r̶e̶ w̶e̶ c̶a̶n̶ u̶s̶e̶
>> L̶'H̶o̶p̶i̶t̶a̶l̶'s̶ R̶u̶l̶e̶ o̶n̶ i̶t̶. E̶i̶t̶h̶e̶r̶ 0/0 o̶r̶ ∞/∞.
>>
>> I̶t̶ i̶s̶ o̶f̶ t̶h̶e̶ f̶o̶r̶m̶ a̶b̶, i̶f̶ y̶o̶u̶ t̶a̶k̶e̶ t̶h̶e̶ l̶o̶g̶a̶r̶i̶t̶h̶m̶ o̶f̶ i̶t̶ y̶o̶u̶ h̶a̶v̶e̶ b̶*l̶n̶(a̶).
>>
>> I̶f̶ t̶h̶i̶s̶ i̶s̶ t̶r̶e̶a̶t̶e̶d̶ a̶s̶ a̶0 (b̶a̶s̶e̶ n̶u̶m̶b̶e̶r̶ a̶s̶ v̶a̶r̶i̶a̶b̶l̶e̶, l̶i̶m̶i̶t̶ a̶s̶ a̶-->0) y̶o̶u̶
>> h̶a̶v̶e̶ 0*l̶n̶(a̶) o̶r̶ 0/(1/l̶n̶(a̶)). T̶a̶k̶e̶ t̶h̶e̶ d̶e̶r̶i̶v̶a̶t̶i̶v̶e̶ a̶s̶ a̶-->0.
>>
>> I̶f̶ t̶r̶e̶a̶t̶e̶d̶ a̶s̶ 0b̶ (e̶x̶p̶o̶n̶e̶n̶t̶ a̶s̶ v̶a̶r̶i̶a̶b̶l̶e̶, l̶i̶m̶i̶t̶ a̶s̶ b̶-->0) d̶o̶ t̶h̶e̶ s̶a̶m̶e̶, y̶o̶u̶
>> h̶a̶v̶e̶ b̶*l̶n̶(0) a̶s̶ b̶-->0. C̶a̶l̶c̶u̶l̶a̶t̶e̶ t̶h̶e̶ d̶e̶r̶i̶v̶a̶t̶i̶v̶e̶s̶ o̶f̶ b̶/(1/l̶n̶(0)) o̶r̶
>> b̶/(1/-∞) o̶r̶ b̶/0 a̶s̶ b̶-->0.
>>
>> Y̶o̶u̶ c̶a̶n̶ a̶l̶s̶o̶ a̶p̶p̶r̶o̶a̶c̶h̶ i̶t̶ a̶s̶ x̶x̶ o̶r̶ x̶*l̶n̶(x̶). S̶i̶n̶g̶l̶e̶ v̶a̶r̶i̶a̶b̶l̶e̶ b̶u̶t̶ a̶s̶ b̶o̶t̶h̶
>> b̶a̶s̶e̶ a̶n̶d̶ e̶x̶p̶o̶n̶e̶n̶t̶.
>>
>> O̶n̶c̶e̶ o̶n̶e̶ h̶a̶s̶ a̶ v̶a̶l̶u̶e̶, t̶a̶k̶e̶ e̶x̶p̶(v̶a̶l̶u̶e̶) f̶o̶r̶ t̶h̶e̶ f̶i̶n̶a̶l̶ a̶n̶s̶w̶e̶r̶.
>>
>> I̶'l̶l̶ l̶e̶t̶ y̶o̶u̶ d̶o̶ t̶h̶e̶ m̶a̶t̶h̶ y̶o̶u̶r̶s̶e̶l̶f̶...
>
> completely nonsense.

See my other post on the two nymshifters. This one is the Unicode abuser
nymshifter. I wonder how many others there are.

extreme crank Michael Moroney wrote:

>> Amazing, for what your corrupt fake money government, and the lying TV
>> and mainstream media, is making you endure a war, hunger, lack of heat
>> and oil, and even expect a nuclear war. Wake up, friends.
>>
>> A Reality Check on the NWO Lies About Russia vs Ukraine.
>> https://www.bitchute.com/video/2GmMZqZ97fD3/
>> Zelensky Actor_Dancer - At least Reagan Just Appeared With a Monkey
>> https://www.bitchute.com/video/cKx0h5uejQHb/
>
> D̶o̶ w̶e̶ h̶a̶v̶e̶ t̶w̶o̶ d̶i̶f̶f̶e̶r̶e̶n̶t̶ n̶y̶m̶s̶h̶i̶f̶t̶e̶r̶s̶ h̶e̶r̶e̶? T̶h̶e̶ U̶n̶i̶c̶o̶d̶e̶ a̶b̶u̶s̶e̶r̶ a̶n̶d̶ t̶h̶i̶s̶
> o̶n̶e̶, t̶h̶e̶ b̶i̶t̶c̶h̶u̶t̶e̶ f̶a̶n̶? B̶o̶t̶h̶ r̶e̶s̶p̶o̶n̶d̶e̶d̶ t̶o̶ m̶y̶ p̶o̶s̶t̶ i̶n̶ c̶o̶m̶p̶l̶e̶t̶e̶l̶y̶ d̶i̶f̶f̶e̶r̶e̶n̶t̶
> w̶a̶y̶s̶. D̶i̶d̶ w̶e̶ w̶i̶t̶n̶e̶s̶s̶ a̶ s̶h̶i̶f̶t̶ c̶h̶a̶n̶g̶e̶ o̶f̶ t̶h̶o̶s̶e̶ p̶o̶s̶t̶i̶n̶g̶ u̶s̶i̶n̶g̶ t̶h̶e̶
> n̶y̶m̶s̶h̶i̶f̶t̶e̶r̶ a̶c̶c̶o̶u̶n̶t̶? A̶r̶e̶ t̶h̶e̶ n̶y̶m̶s̶h̶i̶f̶t̶e̶r̶s̶ b̶e̶i̶n̶g̶ k̶e̶p̶t̶ i̶n̶ a̶ d̶u̶n̶g̶e̶o̶n̶ b̶e̶l̶o̶w̶
> t̶h̶e̶ l̶o̶w̶e̶s̶t̶ b̶a̶s̶e̶m̶e̶n̶t̶ l̶e̶v̶e̶l̶ o̶f̶ t̶h̶e̶ K̶r̶e̶m̶l̶i̶n̶? I̶ h̶a̶v̶e̶ t̶o̶ c̶h̶e̶c̶k̶ w̶h̶i̶c̶h̶ o̶n̶e̶, i̶f̶
> n̶o̶t̶ b̶o̶t̶h̶, c̶a̶m̶e̶ f̶r̶o̶m̶ t̶h̶e̶ s̶l̶u̶m̶s̶ o̶f̶ a̶ f̶o̶r̶m̶e̶r̶ Y̶u̶g̶o̶s̶l̶a̶v̶i̶a̶n̶ c̶i̶t̶y̶, w̶h̶o̶ w̶o̶u̶n̶d̶
> u̶p̶, s̶t̶a̶r̶v̶i̶n̶g̶,
> o̶n̶ t̶h̶e̶ s̶t̶r̶e̶e̶t̶s̶ o̶f̶ M̶o̶s̶c̶o̶w̶, b̶e̶f̶o̶r̶e̶ b̶e̶c̶o̶m̶i̶n̶g̶ a̶ s̶l̶a̶v̶e̶ o̶f̶ t̶h̶e̶ R̶u̶s̶s̶i̶a̶n̶ s̶e̶c̶r̶e̶t̶
> s̶e̶r̶v̶i̶c̶e̶.

you stupid extreme crank troll.

M̴̡̺̀͗͐͒̃ͅi̶̢͑̄̈́̾̾̅̕c̷̯̬̮̼͆͊͋̆̊͝h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕à̶̜̲̈́͂̓̎͗̽e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝l̷̡̧̢̙͈͍͈͈̉̄͋̔̉̆͝ M̴̡̺̀͗͐͒̃ͅỏ̵̝̫̹̱̲̻̗͉̓͒͘̕r̸̡̹͚̣͔͍̮̩͌͐͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝y̷̢̛̪͈͒̆͝ ẉ̸̣̘͙̍̕r̸̡̹͚̣͔͍̮̩͌͐͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅe̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝:

>>> b̶a̶s̶e̶ a̶n̶d̶ e̶x̶p̶o̶n̶e̶n̶t̶.
>>>
>>> O̶n̶c̶e̶ o̶n̶e̶ h̶a̶s̶ a̶ v̶a̶l̶u̶e̶, t̶a̶k̶e̶ e̶x̶p̶(v̶a̶l̶u̶e̶)
>>> f̶o̶r̶ t̶h̶e̶ f̶i̶n̶a̶l̶ a̶n̶s̶w̶e̶r̶.
>>>
>>> I̶'l̶l̶ l̶e̶t̶ y̶o̶u̶ d̶o̶ t̶h̶e̶ m̶a̶t̶h̶ y̶o̶u̶r̶s̶e̶l̶f̶...
>>
>> completely nonsense.
>
> S̴̡͈͎̬͍̥͈̣͑̾͋e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ m̷̢͈̣̠̃̔̓́͛͒͂̽y̷̢̛̪͈͒̆͝ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝ p̷̛̻̺̝͂̓̎́̕͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅẉ̸̣̘͙̍̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜y̷̢̛̪͈͒̆͝m̷̢͈̣̠̃̔̓́͛͒͂̽s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕i̶̢͑̄̈́̾̾̅̕f̴̡̱̣͇̙̹͆́́͘͝͠t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅe̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕. Ṫ̸̡̤̉̐̽h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕i̶̢͑̄̈́̾̾̅̕s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜e�
�̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ i̶̢͑̄̈́̾̾̅̕s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ U̷̢̱͒̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜i̶̢͑̄̈́̾̾̅̕c̷̯̬̮̼͆͊͋̆̊͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ḑ̶̙͇̰̗̦̠̽e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ à̶̜̲̈́͂̓̎͗̽ḇ̷̛̠͂̾̄̓u̸̲͓̫̙̘͉̾̈̎̇̃̽̍̃͝͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝
> n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜y̷̢̛̪͈͒̆͝m̷̢͈̣̠̃̔̓́͛͒͂̽s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕i̶̢͑̄̈́̾̾̅̕f̴̡̱̣͇̙̹͆́́͘͝͠t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅe̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝. I̴̡̛̲͙͖̭͑͛͐ ẉ̸̣̘͙̍̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜ḑ̶̙͇̰̗̦̠̽e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝ h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ẉ̸̣̘͙̍̕ m̷̢͈̣̠̃̔̓́͛͒͂̽à̶̜̲̈́͂̓̎͗̽n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜y̷̢̛̪͈͒̆͝ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ à̶̜̲̈́͂̓̎͗̽r̸̡̹͚̣͔͍̮̩͌͐͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝.

you are too stupid to talk. Come back with a functional brain. Idiot.

On 3/1/2022 3:14 PM, Fritz Feldhase wrote:
> On Tuesday, March 1, 2022 at 8:46:38 PM UTC+1, sergio wrote:
>
>> 0^0 = 0
>
> Nope.
>
> "Define" n^m (n, m e IN) as
>
> n * n * ... n * 1
> -----------------
> m-times "n *"
>
> Then
>
> n^0 = 1, for each and every n e IN (even for n = 0).
>
> I guess you KNOW that the "empty product" is 1, right?
>
> -------------------------------
>
> Btw. we should "define" m * n the same way,
>
> n + n + ... n + 0
> -----------------
> m-times "n +"
>
> to get 0 * n = 0 for each and every n e IN.
>
> (The "empty sum" is 0.)
>
> Recursively:
>
> 0 * n = 0
> (m+1) * n = n + m * n.
>
> n ^ 0 = 1
> n ^ (m+1) = n * n ^ m.

what if I pull out a 0 ? 0^0 = 0 * 0^(-1) (just adding exponents)

then 0^0 = 0 * 0^(-1) = 0/0

and we all know that 0/0 = any constant, not just 1, including 0

0/0 = k then 0 = 0 * k for any k including 0

[oh, I just multiplied both sides by 0, hmmm....]

this one is much more interesting;

I like y = n^n as n approaches 0 from the right, say n = +2 to 0

there is a minimum at n =1/e and it goes to 1 at n = 0

as it crosses over into -n you get complex numbers with poles

there is a 3D graph here of the complex plane

https://mathworld.wolfram.com/GammaFunction.html

Anyhow we expect two solutions that disagree for 0^0, 1 and 0, it depends upon the approach

On 3/1/2022 3:24 PM, Duane Hume wrote:
> M̴̡̺̀͗͐͒̃ͅi̶̢͑̄̈́̾̾̅̕c̷̯̬̮̼͆͊͋̆̊͝h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕à̶̜̲̈́͂̓̎͗̽e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝l̷̡̧̢̙͈͍͈͈̉̄͋̔̉̆͝ M̴̡̺̀͗͐͒̃ͅỏ̵̝̫̹̱̲̻̗͉̓͒͘̕r̸̡̹͚̣͔͍̮̩͌͐͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝y̷̢̛̪͈͒̆͝ ẉ̸̣̘͙̍̕r̸̡̹͚̣͔͍̮̩͌͐͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅe̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝:
>
>
>>>> b̶a̶s̶e̶ a̶n̶d̶ e̶x̶p̶o̶n̶e̶n̶t̶.
>>>>
>>>> O̶n̶c̶e̶ o̶n̶e̶ h̶a̶s̶ a̶ v̶a̶l̶u̶e̶, t̶a̶k̶e̶ e̶x̶p̶(v̶a̶l̶u̶e̶)
>>>> f̶o̶r̶ t̶h̶e̶ f̶i̶n̶a̶l̶ a̶n̶s̶w̶e̶r̶.
>>>>
>>>> I̶'l̶l̶ l̶e̶t̶ y̶o̶u̶ d̶o̶ t̶h̶e̶ m̶a̶t̶h̶ y̶o̶u̶r̶s̶e̶l̶f̶...
>>>
>>> completely nonsense.
>>
>> S̴̡͈͎̬͍̥͈̣͑̾͋e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ m̷̢͈̣̠̃̔̓́͛͒͂̽y̷̢̛̪͈͒̆͝ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝ p̷̛̻̺̝͂̓̎́̕͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅẉ̸̣̘͙̍̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜y̷̢̛̪͈͒̆͝m̷̢͈̣̠̃̔̓́͛͒͂̽s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕i̶̢͑̄̈́̾̾̅̕f̴̡̱̣͇̙̹͆́́͘͝͠t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅe̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕. Ṫ̸̡̤̉̐̽h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕i̶̢͑̄̈́̾̾̅̕s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ i̶̢͑̄̈́̾̾̅̕s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ U̷̢̱͒̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜i̶̢͑̄̈́̾̾̅̕c̷̯̬̮̼͆͊͋̆̊͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ḑ̶̙͇̰̗̦̠̽e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ à̶̜̲̈́͂̓̎͗̽ḇ̷̛̠͂̾̄̓u̸̲͓̫̙̘͉̾̈̎̇̃̽̍̃͝͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝
>> n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜y̷̢̛̪͈͒̆͝m̷̢͈̣̠̃̔̓́͛͒͂̽s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕i̶̢͑̄̈́̾̾̅̕f̴̡̱̣͇̙̹͆́́͘͝͠t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅe̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝. I̴̡̛̲͙͖̭͑͛͐ ẉ̸̣̘͙̍̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜ḑ̶̙͇̰̗̦̠̽e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝ h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ẉ̸̣̘͙̍̕ m̷̢͈̣̠̃̔̓́͛͒͂̽à̶̜̲̈́͂̓̎͗̽n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜y̷̢̛̪͈͒̆͝ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ à̶̜̲̈́͂̓̎͗̽r̸̡̹͚̣͔͍̮̩͌͐͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝.
>
>
> you are too stupid to talk. Come back with a functional brain. Idiot.
what online tool are you using ?

Fritz Feldhase <franz.fritschee.ff@gmail.com> wrote:
> On Tuesday, March 1, 2022 at 5:45:31 PM UTC+1, Dan Christensen wrote:
>> On Tuesday, March 1, 2022 at 10:36:21 AM UTC-5, Fritz Feldhase wrote:

>> > Simple arithmetic looks like

>> > n^0 = 1

>> n^0 = 0 would be just as "simple."

> But it would not allow for the recursive defintion I mentioned, idiot.

>> > n^(m+1) = n * n^m .
>> >
>> > for each and every n, m e IN.

> This would not work with n^0 = 0. (*sigh*)

> Why are you so dense, Dan?

Even on a forum like sci.math, I protest at the above. Dan is anything
but dense, and throwing such insults around is going to have the effect
of degrading one of the few sensible threads down to the level of the
typical thread.

Please don't do this again.

--
Alan Mackenzie (Nuremberg, Germany).

sergio wrote:

>>>>> I̶'l̶l̶ l̶e̶t̶ y̶o̶u̶ d̶o̶ t̶h̶e̶ m̶a̶t̶h̶ y̶o̶u̶r̶s̶e̶l̶f̶...
>>>>
>>>> completely nonsense.
>>>
>>> S̴̡͈͎̬͍̥͈̣͑̾͋e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ m̷̢͈̣̠̃̔̓́͛͒͂̽y̷̢̛̪͈͒̆͝ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝ p̷̛̻̺̝͂̓̎́̕͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅẉ̸̣̘͙̍̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜y̷̢̛̪͈͒̆͝m̷̢͈̣̠̃̔̓́͛͒͂̽s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕i̶̢͑̄̈́̾̾̅̕f̴̡̱̣͇̙̹͆́́͘͝͠t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅe̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕. Ṫ̸̡̤̉̐̽h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕i̶̢͑̄̈́̾̾̅̕s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜
e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ i̶̢͑̄̈́̾̾̅̕s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ U̷̢̱͒̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜i̶̢͑̄̈́̾̾̅̕c̷̯̬̮̼͆͊͋̆̊͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ḑ̶̙͇̰̗̦̠̽e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ à̶̜̲̈́͂̓̎͗̽ḇ̷̛̠͂̾̄̓u̸̲͓̫̙̘͉̾̈̎̇̃̽̍̃͝͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝
>>> n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜y̷̢̛̪͈͒̆͝m̷̢͈̣̠̃̔̓́͛͒͂̽s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕i̶̢͑̄̈́̾̾̅̕f̴̡̱̣͇̙̹͆́́͘͝͠t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅe̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝. I̴̡̛̲͙͖̭͑͛͐ ẉ̸̣̘͙̍̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜ḑ̶̙͇̰̗̦̠̽e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝ h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ẉ̸̣̘͙̍̕ m̷̢͈̣̠̃̔̓́͛͒͂̽à̶̜̲̈́͂̓̎͗̽n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜y̷̢̛̪͈͒̆͝ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ à̶̜̲̈́͂̓̎͗̽r̸̡̹͚̣͔͍̮̩͌͐͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝.
>>
>>
>> you are too stupid to talk. Come back with a functional brain. Idiot.
>
> what online tool are you using ?

you must be kidding, made by himself, and I still see some buffer overflow
in it. Use your brain etc.

Where are the specifications and proofs?
If 0^0 were really undefined in DC Proof, then
you could prove that this is undefined as well:

forall x (x e R => 0^(x-x) is undefined)
Can you prove that? I doubt. How do you express
in DC Proof that an expression is undefined?
What every mathematician and text book

can do, cannot be done in DC Proof.

Dan Christensen schrieb am Dienstag, 1. März 2022 um 15:25:15 UTC+1:
> What is 0^0? Your teachers in school probably told you it was undefined on the natural numbers. Among mathematicians, however, the value of 0^0 depends on the context. 0^0 could be undefined, or it could be 1. It would be undefined if you are talking about 0 as a real number. And it would have a value of 1, as a handy shortcut if you are talking about 0 as a natural number for certain applications, e.g. the binomial theorem.
>
> I am a bit troubled by this inconsistency. There are, after all, no inconsistencies between real and natural number arithmetic other than in the case of 0^0. In my very humble opinion, your teachers were right, but they probably didn’t give you the full story. Using simple arithmetic (no equations or proofs), I will try to make the case that 0^0 could be left undefined on the natural numbers, as it is on the real numbers.
>
> Full text: https://www.dcproof.com/ZeroExponentsOnN.htm
>
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

On Wednesday, March 2, 2022 at 8:22:44 AM UTC-5, Mostowski Collapse wrote:

> Dan Christensen schrieb am Dienstag, 1. März 2022 um 15:25:15 UTC+1:
> > What is 0^0? Your teachers in school probably told you it was undefined on the natural numbers. Among mathematicians, however, the value of 0^0 depends on the context. 0^0 could be undefined, or it could be 1. It would be undefined if you are talking about 0 as a real number. And it would have a value of 1, as a handy shortcut if you are talking about 0 as a natural number for certain applications, e.g. the binomial theorem.
> >
> > I am a bit troubled by this inconsistency. There are, after all, no inconsistencies between real and natural number arithmetic other than in the case of 0^0. In my very humble opinion, your teachers were right, but they probably didn’t give you the full story. Using simple arithmetic (no equations or proofs), I will try to make the case that 0^0 could be left undefined on the natural numbers, as it is on the real numbers.
> >
> > Full text: https://www.dcproof.com/ZeroExponentsOnN.htm
> >

> Where are the specifications and proofs?

If you want formal proofs, see my blog posting at https://dcproof.wordpress..com/2014/09/17/infinity-the-story-so-far/

> If 0^0 were really undefined in DC Proof, then
> you could prove that this is undefined as well:
>
> forall x (x e R => 0^(x-x) is undefined)
>

To prove, for example, that f(x) is undefined, you simply have to prove that x is not in the domain of f.

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

On Tuesday, March 1, 2022 at 4:14:18 PM UTC-5, Fritz Feldhase wrote:
> On Tuesday, March 1, 2022 at 8:46:38 PM UTC+1, sergio wrote:
>
> > 0^0 = 0
>
> Nope.
>
> "Define" n^m (n, m e IN) as
>
> n * n * ... n * 1
> -----------------
> m-times "n *"
>
> Then
>
> n^0 = 1, for each and every n e IN (even for n = 0).
>
> I guess you KNOW that the "empty product" is 1, right?
>
> -------------------------------
>
> Btw. we should "define" m * n the same way,
>
> n + n + ... n + 0
> -----------------
> m-times "n +"
>
> to get 0 * n = 0 for each and every n e IN.
>
> (The "empty sum" is 0.)
>
> Recursively:
>
> 0 * n = 0
> (m+1) * n = n + m * n.
>
> n ^ 0 = 1
> n ^ (m+1) = n * n ^ m.

On Tuesday, March 1, 2022 at 4:17:52 PM UTC-5, Fritz Feldhase wrote:
> On Tuesday, March 1, 2022 at 5:45:31 PM UTC+1, Dan Christensen wrote:
> > On Tuesday, March 1, 2022 at 10:36:21 AM UTC-5, Fritz Feldhase wrote:
> > >
> > > Simple arithmetic looks like
> > >
> > > n^0 = 1
> > >
> > n^0 = 0 would be just as "simple."
> But it would not allow for the recursive defintion I mentioned

As I have shown, there are infinitely many binary functions on N that satisfy

1. x^2 = x*x
2. x^(y+1) = x^y * x

The cool thing is, they agree on every combination of base and exponent value EXCEPT for 0^0, for which any value including 1 will work.

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

On Wednesday, March 2, 2022 at 10:31:48 AM UTC+1, Alan Mackenzie wrote:

> Please don't do this again.

What can I say? Fuck you?

On Wednesday, March 2, 2022 at 3:22:08 PM UTC+1, Dan Christensen wrote:

> As I have shown, there are infinitely many binary functions on N that satisfy
>
> 1. x^2 = x*x
> 2. x^(y+1) = x^y * x

But we are not interested in these infinitely many variants of "power".

Actually, we would like to define ^ (on IN) such that it is total. (Just like + and *.)

Why are you so dense, Dan?