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How to use functions in Jmeter?

What is the minimum set of functions one must use to be able to define ALL numbers using ONLY these functions?

  • Can you form all numbers, including transcendentals, without using transcendental coefficients and by means of a set of functions including: 1) only algebraic functions 2) algebraic functions PLUS exponential and logarithmic functions (not natural log) 3) All functions from 2) PLUS trigonometric functions using degrees (not radians) 4) All functions from 3) PLUS other types of functions (which are they?) 5) It is impossible to define all numbers through a finite set of functions.

  • Answer:

    It is in principal impossible the way you want. Why? It has to do with the cardinality of the transcendental numbers being continuum and the cardinality of algebraic numbers being countable. Suppose you had one function that had this desired property that you could obtain all real numbers from. As there is a countable number of algebraic numbers, you get a countable number of transcendental numbers, but you still have only a countable number of things. At the next stage, you may input these transcendental numbers you generated at the last stage. Therefore, even if you do this inductively at any finite level you will still only have countable many numbers. You'd have to do your transfinite induction out to a countable ordinal to get more. So, the way I'm assuming you want the function (only finitely many iterations to get the reals) is clearly out of the question. If you allow infinite iterations, of course you can get all of the transcendentals. Just consider the binary operation of concatenation, then you can build infinite sequence, which have a natural correspondence with reals.

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I would say 5, but it's just a guess. There are infinitely many transcendentals than just pi and e. In fact there are infinitely more transcendentals than there are algebraic numbers (which are also infinite.) (1) is clearly not right since the transcendentals are not the roots of algebraic functions. (2) You cannot define sin/cosine with only exp/log unless you also allow complex numbers. Since the problem doesn't state this explicitly I would assume they are no-go. thats about as much as i know.

Richard

This question isn't entirely clear. Let me state the following as another way to put this question: Let all basic arithmetic operations be allowed, any use of any elementary functions currently known in mathematics, using as variables any algebraic number, is it always possible to express any arbitrary real number with a finite expression? Conversely, does there exist any real number which cannot be? As the other answerer has already pointed out, the cardiality of reals is still higher than all possible values created by such expressions. While it is true that certain math functions will generate transcendental numbers, there is still a mapping from some finite set of algebraic and/or rational numbers to those transcendental numbers generated, so that not ALL transcendental numbers are or can be generated in this fashion. As a matter of fact, I do believe that there are some definite integrals that cannot be expressed in terms of elementary functions, which is another way of saying that there exists some reals that are "outside" of what's expressible, using your suggestions. For example, the Sine Integral function of x isn't expressible by any use of elementary functions. The Sine Integral of x, or Si(x), is itself defined by an definite integral Sin(t)/t from 0 to x. Do we include this as an "elementary function"? And when do we stop including new forms of such "elementary functions"? In fact, Si(x) is considered as a "special function", even though it's a matter of convention which are elementary and which are special. The link to the Wolfram site on the Sine Integral lists only ONE non-infinite value for x in which Si(x) evaulates to any expressible number, namely Si(0) = 0. Usually, a special function will have a number of values that can be expressed in terms of elementary functions of rationals or constants. That should give you some idea. A rigorous proof for this is beyond the scope of this Y!A forum. And a TU for Will for his answer above.

Scythian1950

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