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Full Version: 0^0 = 1?
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(04-24-2017 04:26 AM)Thomas Okken Wrote: [ -> ]No, it's because they don't change the subject by introducing transformations that cause you to end up looking at a completely different problem.

How was the subject changed? How is it a completely different problem? What "transformations" were introduced?

(04-24-2017 04:26 AM)Thomas Okken Wrote: [ -> ]Consider lim (x->0) x/x. I don't think it's controversial to say that that is 1. Now take lim (x->0) f(x)/g(x), where f(x) and g(x) are nice, well-behaved functions, both zero at x=0, both with a nonzero derivative at x=0, continuously differentiable, etc. etc. etc.

Suddenly you can make that limit come out to whatever you want. Does this lead to some deep insight? No, because you're no longer looking at the same problem.

The point above isn't clear. Can you clarify? I was with you until "Does this lead..."

Making the limit come out to whatever is desired want is one of the properties of dealing with indeterminate forms, whether dealing with 0^0 or 0/0 or 1^inf or etc. This property is why indeterminate forms do not have assigned values in general (only in some special cases).
"Now take lim (x->0) f(x)/g(x), where f(x) and g(x) are nice, well-behaved functions..."

Actually, making these assumptions is changing the problem. The original problem is the behaviour of x^y near (x=0, y=0) and whether that justifies defining 0^0 to be 1. No f's, no g's and certainly no assumptions that these function need to be so nice that they behave like polynomials around (0, 0).

The examples (even just considering those with both x and y positive real numbers and choosing principal values) clearly show that (uniquely) defining 0^0 is a bad choice.

The only justification given so far for choosing 0^0 to be 1 comes from a context in which y is limited to integers, e.g. looking at power series (including polynomials) and making it easy to write the general term with a variable exponent without making an exception for an exponent of 0. This is certainly not a numerical context but an algebraic one and so it seems irrelevant when we are talking about numerical evaluations.
The 0^0 issue has puzzled me for quite a while. So like the extensive discussion. Here are my two cents.

Major commercial math programs do not agree on the handling. Mathematica returns an "Indeterminate", Matlab a "1". The latter was suggested as "useful" by Donald Knuth in his seminal book. IEEE seems to agree. The seminal NIST Handbook on Mathematical Functions does not.

So basically there seems no real answer, just useful conventions.

Enclosed are my notes showing that calculators take different approaches. I also listed some references from both sides of the aisle.

Would be fun to collect more on these results. Please let me know if you find errors in my notes.

Felix

Edit: Attachment attached
(04-25-2017 03:00 PM)Felix Gross Wrote: [ -> ]Enclosed are my notes showing that calculators take different approaches. I also listed some references from both sides of the aisle.

I'm not sure what machines/programs in your results do have an integer mode and may, eventually, return different answers in both real/integer (even complex) modes. The more I read about this topic, the more convinced I am about the "correctness" of the 1 result when dealing with integers.
(04-25-2017 03:00 PM)Felix Gross Wrote: [ -> ]The 0^0 issue has puzzled me for quite a while. So like the extensive discussion.

... "1". The latter was suggested as "useful" by Donald Knuth in his seminal book. IEEE seems to agree...

Enclosed are my notes ...
Excellent notes! I'm with Knuth and the IEEE.

Quote:So basically there seems no real answer, just useful conventions.

For me, that's mathematics! As an undergraduate, our courses in pure mathematics were fairly strictly divided between Algebra and Analysis. The two answers we're seeing here seem to reflect that division well.

(There was also a greater division between pure and applied. And numerical methods sat somewhere nearby as a third area, closer to applied.)
The WP 34S conforms to ISO/IEC 9899:2011 for real and complex arguments.

Simplistically, for reals x0 = 1 for all x, including NaNs. For complex numbers xy = ey.log(x) for all x and y.

- Pauli
(04-25-2017 03:00 PM)Felix Gross Wrote: [ -> ]Enclosed are my notes showing that calculators take different approaches. I also listed some references from both sides of the aisle.

Would be fun to collect more on these results. Please let me know if you find errors in my notes.

Nice document!

Here's a few extra datapoints:
• HP 48SX - 1 (you can remove "Emulator" from the HP 48)
• TI-82 - ERR:DOMAIN
• Excel 2007 (Windows) - #NUM!
• Windows 7 Calculator - 1
Maple V3 returns:

Error, 0^0 is undefined
Would any of you posting on this thread mind if I move it to General, as Gerald suggested in Issues and Administration?
That might be where someone would first look to review this in the future, but it did fit in "not quite hp" to start with.
Addendum: moved & title modified per the two following posts from Bob and emece67
Maybe change title to 'How do different machines represent 0^0" when you move it, since the discussion went well beyond only the wp34s, and a casual reader may skip it if they are not interested in the wp34s.
(04-27-2017 03:35 PM)Den Belillo (Martinez Ca.) Wrote: [ -> ]Would any of you posting on this thread mind if I move it to General, as Gerald suggested in Issues and Administration
That might be where someone would first look to review this in the future, but it did fit here to start with.

As the TP, I think it is a reasonable and desirable move.

(04-27-2017 04:28 PM)rprosperi Wrote: [ -> ]Maybe change title to 'How do different machines represent 0^0" when you move it, since the discussion went well beyond only the wp34s, and a casual reader may skip it if they are not interested in the wp34s.

Simply removing the "(wp34s)" from the title may be enough.

Regards.
Thanks for the new datapoints. Will check other calculators over the next days.

Updated the document.

Felix
(04-27-2017 08:01 PM)Felix Gross Wrote: [ -> ]Thanks for the new datapoints. Will check other calculators over the next days.

Updated the document.

Felix

HP12C-- Error 0
These are the results from the following calculators (not emulators)
HP-41C "DATA ERROR"
HP-25 "Error"
HP-11c "Error 0"
HP-48GX "1"
HP-32SII "INVALID yX"
HP-35s "INVALID yX"
HP-42S "Invalid Data"
HP 48GX -> 1
HP 30b -> 1
TI-85 -> ERROR 04 DOMAIN
Sharp PC-1475 in CALC mode -> E 0
Sharp PC-1475 in BASIC mode (single & double precision) -> ERROR 2
Kaufhof Astor 1 -> E 0
Windows XP CALC -> 1

EDIT: Casio fx-85MS -> Math ERROR
HP 200LX (D) -> "Fehlermeldung: 0 kann nicht mit 0 potenziert werden"

(EDIT: NB regarding the "Astor 1" - this calculator comes in a case and finish/color set identical to that of the Casio fx-570, but the functions and key assignments are different, so it must be a Casio OEM model of around the same time frame.)
How zeros (and ones) are treated depends on use. The most common definition for 0^0 by mathematicians is 1. Basically this comes from combinatorial arguments. The only case where another definition would make sense is in looking at 0^0 with both the base and exponent as reals. Then there is a path-dependent answer.

To get the binomial coefficients to come out right, one should define 0^0 as 1. Some computer languages define 0^0 as 1 if both zeros are integers and zero if the exponent is zero and the base is real. An real exponent is undefined as there are many branch points to the exponential.

I'd go for 1 in general; it's rare that another number would be more useful. Similarly the sum of an empty list should be 0 and the product of an empty list should be 1.
I'm puzzled by the fact that 0^0 has no mainstream convention (seeing the responses of various calculators) . Since mathematics, once set axioms and conventions, is pretty consistent.

edit, the sharp 506w: error.
(04-28-2017 04:57 AM)ttw Wrote: [ -> ]How zeros (and ones) are treated depends on use. The most common definition for 0^0 by mathematicians is 1. Basically this comes from combinatorial arguments. The only case where another definition would make sense is in looking at 0^0 with both the base and exponent as reals. Then there is a path-dependent answer.

To get the binomial coefficients to come out right, one should define 0^0 as 1. Some computer languages define 0^0 as 1 if both zeros are integers and zero if the exponent is zero and the base is real. An real exponent is undefined as there are many branch points to the exponential.

I'd go for 1 in general; it's rare that another number would be more useful. Similarly the sum of an empty list should be 0 and the product of an empty list should be 1.

Actually, most mathematician regard 0^0 as an Indeterminate Form. They do not regard it as 1 (or 0) unless there are additional constraints.
(04-28-2017 04:50 PM)Chris Dreher Wrote: [ -> ]Actually, most mathematician regard 0^0 as an Indeterminate Form. They do not regard it as 1 (or 0) unless there are additional constraints.

0^0 is indeed indeterminate and takes on different values depending on the context. Defining 0^0 = 1 for various applications is neither "wrong" nor "right" -- it is merely a choice that makes the subsequent mathematical results useful.
My HP35 1143S gives blinking zero for every base value ^ 0.
But e.g. 10^0 = 1 for sure.
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