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Rationality, and MS Excel (and Other Calculators)

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Rationality, and MS Excel (and Other Calculators)

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This morning, Mathieu had a nice experience in his course on computational method in actuarial science. But let us start with some mathematical formal definitions.

First, recall that http://latex.codecogs.com/gif.latex?y^x is – somehow – a standard expression. No one should be surprised to see such an expression. Generally (as explained in http://en.wikipedia.org/… ), this function is defined only when http://latex.codecogs.com/gif.latex?y\in\mathbb{R}_+. The idea is that the definition of http://latex.codecogs.com/gif.latex?y^x is that


And it is a definition. Such a function exists only if http://latex.codecogs.com/gif.latex?y\in\mathbb{R}_+ (maybe excluding http://latex.codecogs.com/gif.latex?0). This would be a standard definition in real-analysis.

Now, this ‘power’ function appears also in complex analysis, when dealing with unit roots. From instance, if http://latex.codecogs.com/gif.latex?z=y^{\frac{1}{k}}e^{i%20\frac{2n\pi}{k}}, where http://latex.codecogs.com/gif.latex?y\in\mathbb{R}_+ and http://latex.codecogs.com/gif.latex?k\in\mathbb{N}_\star, for some http://latex.codecogs.com/gif.latex?n\in\mathbb{N}, then http://latex.codecogs.com/gif.latex?z^k=y. Thus, in complex-analysis it might be more complex to define properly http://latex.codecogs.com/gif.latex?y^x since it might not be unique. But we can relate (sometimes, when http://latex.codecogs.com/gif.latex?x is the inverse of an integer, or maybe a rational number ?) with roots of polynomial functions. So far, nothing new…

Let us get back to Mathieu’s problem. Actually, in his course, he wanted to compute http://latex.codecogs.com/gif.latex?(-8)^{\frac{1}{3}}. With a French version of Excel, entering

you do get http://latex.codecogs.com/gif.latex?-2. If you look at the ‘help’ window, you have some more details

It looks like this hat function can be used to define objects such as http://latex.codecogs.com/gif.latex?y^x. But with

you get

(meaning that this is a problem…). It is also possible to use the power (puissance in French) function of Excel,

Here, you also get

The weird part here is that, in the ‘help’ window, you can read that this power function can be used with any number in http://latex.codecogs.com/gif.latex?\mathbb{R}.

Another point… what about  ? Somehow, it is just the square of the previous one (with the fraction)… Here, typing

you get

(similarly with the power function). So clearly, it is not that simple to use this power function. Now, if you use Google (which is now my new online calculator when I am in class, when I cannot use R), if the power is a fraction (or to be more specific the inverse of an integer), then it works as Excel


you get

 But if you type (which should be close, from a continuity property of the power function)


you get

and similarly

On Wolfram Mathworld, enter

Mathematica does recognize that we try to deal with unit roots: the result is here

with – as expected – a numerical approximation

With Matlab, Mathieu did obtain the same as Mathematica (its decimal approximation). And to conclude, with R, Mathieu did obtain

> (-8)^(1/3)
[1] NaN
> (-8)^(.333333333333333)
[1] NaN

So for R, you cannot use this hat function on negative numbers.

Now, how can we interpret those outputs ?

1) My understanding is that clearly, with MS Excel, http://latex.codecogs.com/gif.latex?x^{ab}\neq%20\left(x^a\right)^bsince


which is problematic. For instance, in insurance, with monthly discounts, we do have functions like http://latex.codecogs.com/gif.latex?u^{\frac{k}{12}}. What if


2) The problem comes – probably (MS Excel is not an open software, so it might be hard to check) -  from the fact that http://latex.codecogs.com/gif.latex?y^{\frac{1}{n}} is interpreted as an inverse of a (possibly) bijective function. To be more specific, http://latex.codecogs.com/gif.latex?x=y^{\frac{1}{n}}means that http://latex.codecogs.com/gif.latex?x^n=y. When http://latex.codecogs.com/gif.latex?n is an odd integer, then (in real-analysis) there is a unique inverse, and thus, http://latex.codecogs.com/gif.latex?y^{\frac{1}{n}} is uniquely defined, since http://latex.codecogs.com/gif.latex?x\mapsto%20x^n is a bijective http://latex.codecogs.com/gif.latex?\mathbb{R}\rightarrow\mathbb{R} function. This is what MS Excel (and Google) is doing: http://latex.codecogs.com/gif.latex?x\mapsto%20x^3 is a bijective http://latex.codecogs.com/gif.latex?\mathbb{R}\rightarrow\mathbb{R} function, so http://latex.codecogs.com/gif.latex?(-8)^{\frac{1}{3}} means that we need to find theunique (real) value http://latex.codecogs.com/gif.latex?x such that http://latex.codecogs.com/gif.latex?x^3=-8. Thus, somehow, it makes sense to return http://latex.codecogs.com/gif.latex?-2.

3) There is still a problem with Google, and Mathematica. That is fine to return unit roots in http://latex.codecogs.com/gif.latex?\mathbb{C}. But how comes there is only one value ? I mean, yes http://latex.codecogs.com/gif.latex?1+\sqrt{3}%20\%20i is a possible answer, since


but one can also observe that , and similarly, http://latex.codecogs.com/gif.latex?(-2)^3=-8 and


One can check with

With R, since we do not deal with power function here, but with roots, if we want to find http://latex.codecogs.com/gif.latex?x such that http://latex.codecogs.com/gif.latex?x^3=-8, the function is

> polyroot(c(8,0,0,1))
[1]  1+1.732051i -2+0.000000i  1-1.732051i

Which is different… Weird isn’t it ?

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