Giac/Xcas updates ported to HP Prime?
02-21-2017, 02:42 PM
Post: #14
 Joe Horn Senior Member Posts: 2,002 Joined: Dec 2013
RE: Giac/Xcas updates ported to HP Prime?
(02-21-2017 01:59 PM)retoa Wrote:  a^0=1 for a<>0 (1)

0^n=0 for n<>0 (2)

0^0=? that gives 1 if you apply (1) and 0 if you apply (2), both of which are correct, so 0^0 is undefined.
You can not set it as 1 per default, it would be in contradiction with (2) , nor you can set it to 0, as it would be in contradiction with (1).

Easy but interesting math...

I disagree with that reasoning, and here's why. I teach my students that "Y to the X power" means, "Start with an accumulator of 1, then multiply the accumulator by Y, X times." So 3^4 means start with an accumulator of 1, then multiply the accumulator by 3, 4 times.

0^n (where n<>0) means "Start with 1, then multiply the accumulator by 0, n times." The result will always be 0, as you said.

n^0 (where n<>0) means "Start with 1, then multiply the accumulator by n, zero times." The result will be 1, as you said.

0^0 means "Start with 1, then multiply the accumulator by 0, 0 times." The result is 1. And that's why 0^0=1, QED.

One objection to the above is, "But why do you start with 1? Isn't that kinda arbitrary?" There are two answers to this. (a) Starting with any other number gets the wrong answer for non-zero values of X and Y. (b) It's a perfect parallel to the definition of multiplication, which says "X times Y" means "Start with an accumulator of 0 (the additive identity), then add X to the accumulator Y times." Just as multiplication is defined as repeated addition, starting with the additive identity (0), so too powers are defined as repeated multiplication, starting with the multiplicative identity (1). And that's why 0.^0. returns 1 in RPL.

Another objection is, "But 2^3 doesn't mean 1*2*2*2; it simply means 2*2*2." To which I reply, So you want to define Y^X as "Start with Y, then multiply by Y, X-1 times"? No good; that leaves Y^0 undefined, unlike my definition above.

Although arguments by authority should not be as convincing as arguments by logic, here are a few just to illustrate that I'm not alone in my opinion.

"Some textbooks leave the quantity 0^0 undefined, because the functions x^0 and 0^x have different limiting values when x decreases to 0. But this is a mistake. We must define x^0=1 for all x, if the binomial theorem is to be valid when x = 0 , y = 0 , and/or x = -y . The theorem is too important to be arbitrarily restricted! By contrast, the function 0^x is quite unimportant."
-- Concrete Mathematics p.162 (R. Graham, D. Knuth, O. Patashnik)

"The number of mappings from the empty set to the empty set is obviously 1, but mathematically it's 0^0. Therefore 0^0=1." -- sci.math FAQ

"0^0=1" -- Leonhard Euler

"Zero raised to the zero power is one. Why? Because mathematicians said so. No really, it’s true." -- http://www.askamathematician.com/2010/12...-disagree/

<0|ɸ|0>
-Joe-
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