How can I know how many zeros dose the function have and how can I find the location of zero in the function?

Function

[Marc Jakobi]

Please provide more detail.

[Rena Berman]

(Answers dev) Unable to restore question.

Answer 1

[Image Analyst]

Perhaps try the roots() function. If not, then supply info on your specific function.

Answer 2

[Walter Roberson]

Let p(x) be 1 if a certain property is true about the number x, and 0 if the property is false. Let q(x) be 1 - p(x), so q(x) is 0 if the property is true for the number x, and 1 if the property is false for x.

If we had a way of knowing how many zeros an arbitrary function has, then we could apply that method to q(x), and since that hypothetical method tells us the number of zeros the function has, it would tell us the number of times q(x) was 0, which would tell us the number of times p(x) was 1, which would tell us the number of times that some number, x, has the property p.

This hypothetical method could then be applied with respect to any property that can be determined in isolation, and through application to q(x) would therefore tell us how many numbers exist for which the property was true.

Now let the property p(x) be "x is odd and the sum of the unique positive integer factors of x (including 1) is equal to x itself". That is, let p(x) be the property that x is an Odd Perfect Number. Then, by applying the hypothetical method of predicting the number of zeros, we would then know the number of Odd Perfect Numbers that exist. This would be very interesting to number theorists, as over hundreds of years of effort, number theorists have not been able to find any Odd Perfect Numbers and have also been unable to prove that none exist. (It has been proven that there is an infinite number of Even Perfect Numbers; there is is an Even Perfect Number that can be derived from every Mersenne Prime)

This argument in itself does not prove that no such method of predicting the number of zeros of all arbitrary functions can exist at all, but people have in fact proven that to be true.

The proof that no such method of predicting the number of zeros can exist for arbitrary functions, is a proof over arbitrary functions. There are classes of functions where the number of zeros can be predicted; for example for polynomials, the number of complex roots can be completely predicted.

[Walter Roberson]

Suppose that you had a function NZ, that given a function F, could tell you how many zeros F had. Let NZB = @(F) NZ(F) > 0 -- that is, whether it has any zeros or not. What is NZB applied to NZB ? What is NZB applied to the function (1-NZB) ?

If you look at the way The Halting Problem is constructed, then you can use the same techniques to establish that you can force a contradiction by applying functions to their opposites -- you can construct a function that returns 1 if and only if it does not return 1. This line of reasoning establishes that it is not possible to construct a function that can tell you how many zeros any arbitrary function has.