[Eternal Student]

Hi.

   The original question was:

What's 0 to the power of 0?

    This is actually a very good question and it's something that isn't easily resolved. 
    It also turns out that whenever something like 0⁰ is encountered in science, it has nearly always arisen as a Limt of x^x  as  x→0+    (approaching 0 from the right).   This limit is defined and does equal 1.   As a consequence of this it has become an un-official convention that  0⁰ = 1   and  regrettably many calculators, like the one you were using, will show you that result.

   Too often we start from a false assumption.  It is easy to imagine that just because we can write some mathematical term down it must have some numerical value.  For example, I have never calculated the value of 10067 + 765409 but I might assume it is some Real number.  We also frequently assume that when there are patterns to follow we must be able to extend those patterns.  For example, whatever my answer to that sum might be it should be bigger than the first number, 10067.
  For exponentiation it's actually much safer if we start by assuming nothing at all.  Do not assume a^b defines any function from (a,b) → ℜ   and  don't even assume that the Real numbers exist.  Instead start from more basic assumptions (axioms).   If we do build up the Real Numbers and develop enough real Analysis to construct the exponential series then we will see that 0⁰ was never defined and indeed it cannot be defined in any consistent way as a Real number.

    I could just spit out some chapters from a textbook on Real Analysis or Complex Analysis that talk about the exponential series but I can't do that any better than the textbooks.  Instead let's put out some minor problems to consider, which might help to identify just how complicated it is to raise numbers to an exponent:

   1.    What  is   4^1/2 ?     Why?
   2.    What is    (-4)^1/2 ?   Is there no real solution?
   4.    By the rules of indicies we have (x²)^1/2  = x   for all x.  This seems reasonable but what happens to the LHS and  RHS  when you let x = -3 ?    Don't we obtain  +3 = -3  ?
   5.    3^π  cannot be written as the  integer root of any integer power.  Specifically  3^π ≠ (^a√3)^b  for any inetgers a,b.   So what is the value of  3^π?   Could it be a negative number?    If that's too easy  consider   (-3)^π .

Best Wishes.

PS,  yes I know question 3 was missing.

[Petrochemicals]

Presumably, you meant to say that there are no negative square numbers? (and that is easily fixed with imaginary numbers, but let's stick with real numbers for now).

yes. I imagined something different.

[hamdani yusuf (OP)]

What  is   4^1/2 ?     Why?

y=4^1/2
y=2
or
y=-2
because 2²=4
and (-2)²=4

[hamdani yusuf (OP)]

What is    (-4)^1/2 ?   Is there no real solution?

(-1 . 4)^1/2 = (-1)^1/2 . (4)^1/2
= 2i
or
= -2i
No real solution.

[hamdani yusuf (OP)]

but what happens to the LHS and  RHS  when you let x = -3 ?    Don't we obtain  +3 = -3  ?

No.
I found some calculators treat exponentiation of negative value differently. Some treat the negative sign as part of the base, some others treat it as sign of the result. To avoid ambiguity, we can use brackets.

[hamdani yusuf (OP)]

So what is the value of  3^π?   Could it be a negative number?

A fractional power produces more than one answers. Rational power has finite number of answers. Irrational power has infinite number of answers.
3^π has infinite number of answers in complex plane.
In polar complex plane, the magnitude is around 31.544..., but any angle can be the argument.

[hamdani yusuf (OP)]

If that's too easy  consider   (-3)^π .

(-3)^π = (-1)^π . (3)^π
Since any angle can be the answer, then it's the same as previous question.

[Eternal Student]

Hi.

You're doing well there @hamdani yusuf .
   You've already got enough to see that real numbers raised to some exponenet aren't always equal to a real number.

Sometimes, you can't find any real number that would be suitable.   Sometimes you can find many suitable Real numbers.  It's quite natural to extend the scope of the problem to consider complex numbers but then you can sometimes find an infinite set of numbers as a solution.

You already have enough evidence to recognise that 0⁰  was never required to be a unique Real number.  It's not even required to be a unique Complex number.  You've also presented enough examples to see that it cannot be defined as a real number in any consistent way.   If it has to be anything, it could be a bicycle.  It's just a collection of mathematical symbols we can write down but it's not representative of any numerical value.   ∀6†12  is another set of symbols that doesn't equal or represent any numerical value.

   Technically,  x^1/n  is defined to be the positive root wherever there was a choice.  So that  9^1/2 is +3 and nothing else.   This is done because it maintains exponentiation as a well defined function for as long as possible.   Anyway, using that it would mean that your answer to q. 4 is wrong.
   If you were given the expression    (x²)^1/2  = x      and then set x= -3    then you would have to deduce that  +3 = -3.   
   Obviously we don't really want anything that silly, so the only possibile resolutions are that we give up on considering a^b  as a well defined function for all a,b ∈ Z   OR ELSE   accept that the given equation   (x²)^1/2  = x  was not valid for all x∈ℜ.
   Mathematics has taken the second option,   the rules of manipulating indicies that we were taught in school do not hold for all real numbers as a base for the exponentiation.  A good teacher might have brought that to the attention of their students but it wouldn't matter much anyway:  As human beings we want to follow patterns and we want to extend these patterns wherever we can, so we would have ignored any warning.
   Thus (x^m)^1/n  = x^m/n  is only a true statement for some values of x.  The original question 4 that I presented was a little misleading.   The most appropriate response should be "Exponentiation cannot follow this rule even though it seems like it should (because otherwise +3 = -3)". 
    I needed this to be considered because there are so many misconceptions and false proofs based on using "rules of indicies" even though these rules do not and cannot hold in the system of mathematics we commonly use.

Best Wishes.

[hamdani yusuf (OP)]

   If you were given the expression    (x²)^1/2  = x      and then set x= -3    then you would have to deduce that  +3 = -3.

No.

sin(0)=sin(π) doesn't imply 0=π
x²=y² doesn't imply x=y
(x²)^1/2=(y²)^1/2 doesn't imply x=y

[Eternal Student]

Hi.

    If you were given the expression    (x2)1/2  = x      and then set x= -3    then you would have to deduce that  +3 = -3.

No.

   I don't know what to say in response to this.   I can see that you don't want to accept it and of course you are free to do as you wish.
    Perhaps we should go back to one of the earlier posts you wrote.

Quote from: Eternal Student on Yesterday at 17:49:40

    What  is   4^1/2 ?     Why?
- - - - - - -
 Hamdani replied:
y=4^1/2
y=2
or
y= -2
because 22=4
and (-2)2=4

    You can't have  y= +2   and also   y= -2   because    equals  (=)  is a transitive relation:   If   y = a  and also y= b  then  a=b.    So   y= +2 and also y= -2  forces the relationship  +2 = -2.
    As a consequence, if you decide  that  4^1/2  is equal to a real number then it can only be taken as one number.
    In order to keep exponentiation as a well defined function for as long as possible  (or if you want to phrase it another way - in order to maintain the consistency of Mathematics so that  =  is transitive),  you are forced to assign only one number  to  4^1/2   or else you must not assign any number to it at all.
    By convention we take  4^1/2  =  positive root of 4.

  This is slightly different to  the example    Sin(0) =  Sin(π).    You can assign more than one value in the domain of a function to the same value in the Range.    However, you cannot assign one value in the domain to more than one value in the Range.     ("Many to One"  is OK     but  "One to Many"  is not).

Best Wishes.

[Zer0]

Thank You for the Corrections.
😊
🙏

Just wanted to show you folks a lil something...


Screenshot_2021-11-04-00-40-56-595_com.miui.calculator.jpg (97.32 kB . 1080x2156 - viewed 2924 times)


Screenshot_2021-11-04-00-41-12-800_com.miui.calculator.jpg (116.43 kB . 1080x2156 - viewed 3000 times)

Ps - I use a Poco Android Mobile(model) by MI(company).
& I find it Strange that a built in default Chinese designed Calculator App is in clear defiance of the all mighty & powerful GooGle.
🤔
(The damm thing won't even Uninstall, & no I'm Not into Rooting so save that advice for someone else)

[Petrochemicals]

Thank You for the Corrections.



Just wanted to show you folks a lil something...




Screenshot_2021-11-04-00-41-12-800_com.miui.calculator.jpg (116.43 kB . 1080x2156 - viewed 3000 times)

Ps - I use a Poco Android Mobile(model) by MI(company).
& I find it Strange that a built in default Chinese designed Calculator App is in clear defiance of the all mighty & powerful GooGle.

(The damm thing won't even Uninstall, & no I'm Not into Rooting so save that advice for someone else)

Appalling grammer, it should be cannot, not "can't". You also cannot divide by 1 or anything beneath

[Bored chemist]

Appalling gramme, it should be cannot, not "can't". You also cannot divide by 1 or anything beneath

Wrong on so many levels.
I can divide by a half perfectly well.
Five divided by half is ten.
(five divided in half is rather less than ten).

I can divide by one very easily indeed.
x/1=x

There's nothing wrong with "can't" as a word.

And it looks like you can't spell grammar- though it may be that you can't abbreviate "programme"


Presumably, you consider these statements of fact to be "repetitive antagonism"

[hamdani yusuf (OP)]

https://www.wolframalpha.com/input/?i=4%5E0.5
For fractional power, we get a principal root and one or more other roots. But the selection of principal root is just a convention. It's neither more nor less true than the other roots.

Here's another example related to the original question here.
https://www.wolframalpha.com/input/?i=%28-0.01%29%5E-0.01

[hamdani yusuf (OP)]

This is slightly different to  the example    Sin(0) =  Sin(π).    You can assign more than one value in the domain of a function to the same value in the Range.    However, you cannot assign one value in the domain to more than one value in the Range.     ("Many to One"  is OK     but  "One to Many"  is not).

Try arcsin(x) = 0

[Eternal Student]

Hi again.

But the selection of principal root is just a convention.

   Yes and no.   It is a convention but it is a widely used and widely recognised one.  I think it would be hard to find any credible source in any country of the world that didn't define a fractional power to be the positive or principal root.   So that  4^1/2 = +2   and never -2.
    There's another good reason for making this decision.  It turns out that we're only going to get the positive root if we use the exponetial series.

    The exponential series,  Exp(x) has all the properties we want for e^x.
   Exp(x) = 

   I mentioned earlier that if you build up the Real numbers from first principles and develop some Real Anlysis then this series is found to be convergent for all x∈ℜ.  We can take the definition further and apply it in the Complex plane and then it turns out the Exp(x) function is just fantastic  (it has an infinite radius of convergence,  it's holomorphic throughout C, it's just ideal and very well behaved).
   The key point is that it has the properties we want for exponentiation,   in particular  Exp(x) . Exp (y) = Exp (x+y).  From which all the other usual properties of exponents will follow   Exp(-x) = 1/Exp(x)     and   for rational  numbers a/b    we will have  Exp(a/b) =   (^b√e)^a

     It is therefore possible to define exponentiation with this series as your prototype.  So it's not necessary to imagine that   a^b   means  that you start with the base a multiplied by itself a few times,  take a suitable root and sometimes invert the final answer.   Instead we have an analytical definition for  a^b  it is the number you obtain as the limit of this series     Exp ( b. Ln a)   and it will be well defined  whenever  Ln(a)  exists.
    (The advantage of making the limit of this series your definition of a^b is that it remains consistent with algebriac definitions for rational exponents but also allows you to evaluate irrrational exponents and, if you extend into complex analaysis, arbitrary complex exponents).
   Anyway, we get two results from this identification.  Firstly, Exp(1/2) =  sum of terms all of which are positive, so we get a positive answer.  This means that Exp(1/2) is finding the positive square root of e and not the negative root.  So if we're going to use this analytical definition for exponentiation then 4^1/2 = +2 and not -2.
   The second item of interest is that, despite all the advantages of defining exponentiation based on this series, it's surprisingly uselss for evaluating 0 raised to a power and we have to fall back to algebraic definitions for this.  (This post is already long, so I'm stopping here).

You also mentioned this:

Try arcsin(x) = 0

  ArcSin is only a well defined function when you restrict the domain of the parent Sine function.


Best Wishes.

[hamdani yusuf (OP)]

ArcSin is only a well defined function when you restrict the domain of the parent Sine function.

The original question doesn't restrict the path to the answer to be a function. There's no restriction either for the solution to be in real number.

[hamdani yusuf (OP)]

it will be well defined  whenever  Ln(a)  exists.

When does Ln(a) not exist?

[hamdani yusuf (OP)]

Here's (-0.3)^-0.3 according to Wolfram Alpha.
https://www.wolframalpha.com/input/?i=%28-0.3%29%5E-0.3

Position in the complex plane

I'm not sure why that particular point is selected as the principal root.

Plot of all roots in the complex plane

[Eternal Student]

Hi.

I'm losing track of what was being discussed here. 
This bit I can do:

When does Ln(a) not exist?

   When a = 0.

Best Wishes.