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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.

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.

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?

https://en.wikipedia.org/wiki/Root_of_unityn 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

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.

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

May be next time you can find a better example.

if we then go to exponents:3^{7} = 1 x 3 x 3 x 3 x 3 x 3 x 3 x 3 = 21873^{2.7} = 1 x 3 x 3 x 3^{0.7} = 9 x 2.16 = 19.423^{0} = 1 x (zero multiplications by three) = 1

Quote from: chiralSPO on 02/11/2021 12:56:32if we then go to exponents:3^{7} = 1 x 3 x 3 x 3 x 3 x 3 x 3 x 3 = 21873^{2.7} = 1 x 3 x 3 x 3^{0.7} = 9 x 2.16 = 19.423^{0} = 1 x (zero multiplications by three) = 1To 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= 93^1 = 3x1 = one singular portion that is 3 =33^0 =? = nil singular portions that are 3 each=? 0?

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

Quote from: hamdani yusuf on 27/11/2021 22:59:25For 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?

Quote from: hamdani yusuf on 28/11/2021 10:55:29What'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.

Quote from: Petrochemicals on 28/11/2021 01:16:36Quote from: hamdani yusuf on 27/11/2021 22:59:25For 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?

Hi.Looks ok @Petrochemicals . I'm not sure how you intend to connect 0 x 0 with 0^0 = 0^{0} which was the original question in this thread but that's probably not too important.Best Wishes.

https://en.wikipedia.org/wiki/Logarithm#DefinitionThe 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 .