[hamdani yusuf (OP)]

I'm not sure what you meant by this or if you intend to develop the idea.  The differential dx is not a number of any kind (unless we're using something like non-standard analysis and Hyperreal numbers).  For standard analysis, the differential is just shorthand for expressing something about limits.

For example, in equation y=sin(2x). The gradient at the origin is dy/dx =2 when dx is close to 0, for both positive and negative values of dx. But if dx is exactly 0, dy is also exactly 0, which makes dy/dx undefined.

[hamdani yusuf (OP)]

OK.   So you're saying that we can't have exponentiation meaningfully defined just in the Real numbers.  It only makes sense in the Complex plane.  This is a bit controversial but it's ok, let's go with it.

Let's make it fractional exponentiation can't be meaningfully defined just in the Real numbers.

[hamdani yusuf (OP)]

You mentioned in an earlier post that 3^π is not uniquely defined as a Complex number.   It is an infinite set of values,  all with modulus  approx. 31.5    but  arbitray argument.    So we've lost our ability to determine a unique value for most real numbers raised to an irrational exponent.
   The best we can hope for is that a convention is applied for 3^π so that an argument of 0 is assumed - but there is no fundamental value that is more truthful or follows more naturally from the axioms of mathematics.  Is that right?

Right. The reason is consistency with definition for root of unity.

https://en.wikipedia.org/wiki/Root_of_unity
n mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n.

The 5th roots of unity (blue points) in the complex plane

PS I wonder why superscript formats keeps disappearing whenever I quote previous posts. Is it a bug, or is it a feature?

[Eternal Student]

Hi again.

   You've been busy, Hamdani, well done.   Looking through your posts it looks like you are actually trying to find good reasons to disregard some expressions and prioritise others.  I'm quite impressed with the effort, most people would have given up on the idea already.
   It looks like you've decided that a good reason for disregarding an expression is that it doesn't have a limit from the right and the left (and/or these aren't the same).  That's actually quite reasonable.
   
If you're still interested in using that criteria, you should carefully check that there aren't any expressions that could still be a problem.
   I might offer this example as a starting point:



[That doesn't always display well,  it should be:    Limit as x goes to 0   of   (10 raised to -1/x)  all raised to  x  ]   

It has the general limiting form  0⁰ which is what you wanted and it should have a well defined limit from the left and the right.   You can either determine the limit or else plot it with some software or one of the web-sites offering a graph plotter online. 

Best Wishes.

[hamdani yusuf (OP)]

You're right. We need other criteria if we want to define 0^0=1.
In your example, the x cancel each other. So, no matter what the value of x is, the result is the same constant.
It's no longer an exponential function.
lim x->0 (10^(-1/x))^x
= lim x->0 (10^(-x/x))
= lim x->0 (10^(-1))
= 0.1

lim x->99 (10^(-1/x))^x
= 0.1
So, if we want to use limit to prove something, we need to make sure that the value of the limit affects the result. This is a criterion we often take for granted, and only realize when it's being violated.

[Eternal Student]

Hi.

@hamdani yusuf .   
1.   Well done for getting the right limit.    That thing  tends to  1/10 or 0.1
2.   The second part of your post isn't especially clear.  Try explaining what you mean and what you would do another way and see if it still meakes sense.   Then write it down.
    Here's an example, which tries to generalise what you have done:  Suppose a person wants to determine the height of the tower of London.   They try various things and read various books until finally they find a statement like this   "All towers are 100 metres high".   Now you might think that they have their answer but as they consider it a bit more they realise that the height of towers never changes no matter where you are in the world or which tower it is.  So they decide that this was information they really should NOT use.   Since all towers are 100 metres high it just can't be giving any useful information about my tower in London.
    This is the line of reasoning you are proposing for not using  that expression   (10^-1/x) ^ x.   You seem to be saying that because the limit doesn't change when x→anywhere,       it spoils the truth or usefulness of the information for indicating what happens as x→0.

    Don't get me wrong.... if you are determined then you will eventually find some reasons for disregarding some expressions and only considering limits that arise from other expressions.   However, you need to be a bit guarded against human nature.  Ideally the criteria should be sensible and something that could reasonably have been established BEFORE examining all the possible situations.   AFTER examining all the possibilties, you can certainly find or convince yourself of reasons for only using those expressions.   For example, you can decide to use only those expressions for which the limit will evaluate to +1 (it's certainly going to get the result you wanted).

    Mainstream pure mathematics, especially Real Analysis, went a different way to you.  It's not better, it's just different.  It was possible to define exponentiation based on the exponetial series.   It's actually a very similar idea if you pull it all apart.   In mainstream mathematics they do define  A^B  to be the limit of something involving only whole number exponents.   It's the limit of a suitable sum of terms involving only whole number exponents.

Best Wishes.

[hamdani yusuf (OP)]

It has the general limiting form  0^0 which is what you wanted and it should have a well defined limit from the left and the right.

Except that it doesn't.
lim 10^(-1/0) is not 0. It's undefined.
So, your example represents undefined^0 instead of 0^0 as required. May be next time you can find a better example.

[Eternal Student]

Hi.

lim 10^(-1/0) is not 0. It's undefined.

    You can put modulus signs around x, to have |x|   instead of x  if you're worried about left and right limits.  I probably did miss that out - sorry, it was late and I was rushing and this LaTex equation format takes me ages to get working.  By the time I've spell-checked and read through stuff I rarely get a post finished in less than an hour as it is. 



If, for some reason you don't like a modulus,  replace  with an  x² term to get something like this:



   and in the next step   you can either continue to raise this whole thing to x   (which gives you a differing two sided limit, not a limit of  +1)    or else raise all of the expression to the power of  x²  to get back to something like the original idea.   

May be next time you can find a better example.

   Sadly, I've got some other work to do and I'll probably be leaving this thread (and entire forum) alone for a while.  It's been nice talking to you and I'm glad you've got some interest in mathematics.

Best Wishes and bye for now.

[Petrochemicals]

if we then go to exponents:
3⁷ = 1 x 3 x 3 x 3 x 3 x 3 x 3 x 3 = 2187
3^2.7 = 1 x 3 x 3 x 3^0.7 = 9 x 2.16 = 19.42
3⁰ = 1 x (zero multiplications by three) = 1

To the power of 0 is 1 due to the answer = 1 portion of.

This is  similar to as division where the answer is one portion of, the answer is equated to a singular.

20÷10equated to 1 is 10 singular portions of 20, if this is apparent? It may seem like I am just talking drivel? 20÷0.5 if 0.5 is  equated to a singular the process is to find how many portions of 0.5 are in 1 or how many portions of 10 are in 1.

Vi's a vi, 2 portions of 0.5 in 1 meaning when 10 is the dividend the answer is 2 portions of the dividend or 20.

If you follow a similar method for multiplication  20x 10, 20 is a singular portion, vi's a vi 10 singular portions of 20 is 200.
 
So
 3^2 = 3 x 3 = three singular portions that are 3 each= 9
3^1 = 3x1 = one singular portion that is 3 =3
3^0  =? = nil singular portions that are 3 each=? 0?

[Bored chemist]

if we then go to exponents:
3⁷ = 1 x 3 x 3 x 3 x 3 x 3 x 3 x 3 = 2187
3^2.7 = 1 x 3 x 3 x 3^0.7 = 9 x 2.16 = 19.42
3⁰ = 1 x (zero multiplications by three) = 1

To the power of 0 is 1 due to the answer = 1 portion of.

This is  similar to as division where the answer is one portion of, the answer is equated to a singular.

20÷10equated to 1 is 10 singular portions of 20, if this is apparent? It may seem like I am just talking drivel? 20÷0.5 if 0.5 is  equated to a singular the process is to find how many portions of 0.5 are in 1 or how many portions of 10 are in 1.

Vi's a vi, 2 portions of 0.5 in 1 meaning when 10 is the dividend the answer is 2 portions of the dividend or 20.

If you follow a similar method for multiplication  20x 10, 20 is a singular portion, vi's a vi 10 singular portions of 20 is 200.
 
So
 3^2 = 3 x 3 = three singular portions that are 3 each= 9
3^1 = 3x1 = one singular portion that is 3 =3
3^0  =? = nil singular portions that are 3 each=? 0?

Word salad.

[hamdani yusuf (OP)]

So far, the discussion converges into two answers. First, it's 1. Some reasons have been put forward to support it, e.g. usefulness, simplicity, and desciption of exponentiation. Some matemathicians including Euler and Knuth take this position.
The others like Eternal Student said it's undefined.  Some counter examples using limits have been proposed to support it.
The dispute may come from the fact that two zeros in the question give different answers. 0^x=0 for positive x
0^x=undefined for negative x
x^0=1 for both positive and negative x
the question is when x is neither positive nor negative.
Let's campare them with other operations, *, /, + and -.
0*x=x*0=0
for positive and negative x

x/0=undefined
0/x=0

0+x=x+0=x

x-0=x
0-x=-x

For multiplication, addition, and subtraction, there's no dispute that the result is 0 when x=0. It's different for division and exponentiation.

[Petrochemicals]

For multiplication, addition, and subtraction, there's no dispute that the result is 0 when x=0. It's different for division and exponentiation.

There in lies the answer, given multiplication and division are the reverse of each other.

[hamdani yusuf (OP)]

For multiplication, addition, and subtraction, there's no dispute that the result is 0 when x=0. It's different for division and exponentiation.

There in lies the answer, given multiplication and division are the reverse of each other.

What's the reverse of exponentiation?

[Origin]

What's the reverse of exponentiation?

The inverse is the logarithm, just as division is the inverse of multiplication.
Why is this thread still going?  You received your answer in the first page.

[hamdani yusuf (OP)]

What's the reverse of exponentiation?

The inverse is the logarithm, just as division is the inverse of multiplication.
Why is this thread still going?  You received your answer in the first page.

Do you stop questioning things after you get first answer? Don't you want to hear some alternatives which might be a better answer, or provide a deeper understanding?
What's your response to Eternal Student's answer?

[Petrochemicals]

For multiplication, addition, and subtraction, there's no dispute that the result is 0 when x=0. It's different for division and exponentiation.

There in lies the answer, given multiplication and division are the reverse of each other.

What's the reverse of exponentiation?

What is the process first of all?

[Petrochemicals]

0x0 is zero if a pattern is looked to if squares increace by the addition of the prior root and the current root to the prior squareFor example

(2x2)+2+3 =9 =3x3
(3x3)+3+4=16=4x4

And so on

0x0=(-1x-1) +0+ (-1)=0

Whether you can consider -1 to be prior to zero is another matter.

[Eternal Student]

Hi.

Looks ok @Petrochemicals .   

I'm not sure how you intend to connect  0 x 0    with  0^0 = 0⁰   which was the original question in this thread but that's probably not too important.

Best Wishes.

[Petrochemicals]

Hi.

Looks ok @Petrochemicals .   

I'm not sure how you intend to connect  0 x 0    with  0^0 = 0⁰   which was the original question in this thread but that's probably not too important.

Best Wishes.

Short of fiddling in your brain I am powerless to do anything more.

[hamdani yusuf (OP)]

if x=0^0, then
log₀(x)=0
log₀(x)=ln(x)/ln(0)
ln(x)=0
x=1

https://en.wikipedia.org/wiki/Logarithm#Definition
The logarithm of a positive real number x with respect to base b[nb 1] is the exponent by which b must be raised to yield x. In other words, the logarithm of x to base b is the unique real number y such that

The logarithm is denoted "log_b x" (pronounced as "the logarithm of x to base b", "the base-b logarithm of x", or most commonly "the log, base b, of x").

An equivalent and more succinct definition is that the function log_b is the inverse function to the function .