CAS: 0/0 is undefined, however 1/0 is infinite?

[sasa]

Since we do not work here with limits but simply dividing, everybody know that dividing by zero is undefined - or not? CAS in Prime and Wolfram Apha claims that 1/0 is infinite...

Mathematics is exact science and clearly define that dividing by zero is undefined, thus this is clearly fundamental bug.

Dr. Parisse?

[Tim Wessman]

https://en.wikipedia.org/wiki/Division_by_zero

Point being "there is a whole article talking about the possible definitions, intricacies and exceptions" saying it is undefined is a perfectly fine definition - especially for an advanced CAS system.

[sasa]

(10-21-2018 09:54 AM)Tim Wessman Wrote:  https://en.wikipedia.org/wiki/Division_by_zero

Point being "there is a whole article talking about the possible definitions, intricacies and exceptions" saying it is undefined is a perfectly fine definition - especially for an advanced CAS system.

Sorry Tim, I'm not going to read any article which try to "prove" obvious nonsense, as here that 1/0=inf.

As mentioned already, in special cases/theory it may be perfectly correct, however that does not mean it should be generalized on plain real numbers, which is absolute nonsense.

I would like to see teacher who explain to students why 1/0 is infinity, since a calculator return such result...

[Tim Wessman]

It's not returning infinity, both are returning "complex infinity" which is a different concept entirely and the most correct thing that could be returned.

Home, which is numeric only and where students should be until they get into specifically looking at CAS type operations just reports an error. The point is that the teacher can choose when and if to talk about the more correct and complex mathematical part.

[sasa]

We talking then about "apples and oranges" here...

My point is that here we do not need any trace of extended complex plane, nor complex infinity sign, nor what it actually means - we work here explicitly with set of real numbers.

Thus CAS should be aware of this through many level of settings, which badly missing here... Once again, CAS basically means to use symbolic evaluation, not enabling by default fully all available extended features and deeper analysis, which user probably do not need at all, if not set explicitly...

I will stop then here, since I believe we miss each other points.

[Tim Wessman]

You will have to talk to the CAS author. His opinion is that it should always be in complex unless the user makes specific assumptions about variables.

However, why do you think it should be only working in the real plane? Which setting are you looking at?

[Joe Horn]

(10-21-2018 10:39 AM)sasa Wrote:  Sorry Tim, I'm not going to read any article which try to "prove" obvious nonsense, as here that 1/0=inf.

Why ask a question if you have already convinced yourself that every reply is "obvious nonsense"?

Quote:As mentioned already, in special cases/theory it may be perfectly correct, however that does not mean it should be generalized on plain real numbers, which is absolute nonsense.

CAS is NOT "generalized on plain real numbers". But Home is. Try 1/0 and 0/0 in Home.

If you want CAS math to be limited to something other than the Reimann Sphere of numbers (in which 1/0=infinity and 0/0 is undefined), then you have to tell it so.

Quote:I would like to see teacher who explain to students why 1/0 is infinity, since a calculator return such result...

Me! Me! <waves hand enthusiastically> Your problem is that you never had me as your math teacher.

1/0 equals infinity not because a calculator says so, nor does the HP Prime (and HP 50g) say so because it is "true", but because it is USEFUL. Is Euclidean Geometry TRUE and are non-Euclidean geometries "absolute nonsense"? No. They are both nothing more than useful tools for certain tasks. Saying that 1/0=infinity is absolute nonsense is like saying that Phillips screwdrivers are designed wrong because they can't turn flathead screws. Prime's CAS is a tool which works by default in the Reimann sphere of numbers. Use it that way. If you want to work in some other number space, either tell it so, or stick to Home view (in which you can enable or disable complex results).

[parisse]

1/0 is infinite (unsigned) for the same reason that sqrt(-1) is i. A CAS computes, it is the role of the math teacher to explain how to use it properly.

[sasa]

(10-21-2018 05:40 PM)Joe Horn Wrote:  Your problem is that you never had me as your math teacher.

Hardly possible. We are probably in the same age... ;-)

Quote:Prime's CAS is a tool which works by default in the Reimann sphere of numbers. Use it that way. If you want to work in some other number space, either tell it so, or stick to Home view (in which you can enable or disable complex results).

Thank you. These are exactly my points. To simplify all maximally:

1. In an early grades, kids will first learn positive natural numbers and concept of zero. WTH are negative numbers?

2. Later will learn what are fractions and real numbers and what is square root . WTH is (i) ? SQRT of negative numbers is not defined in real plane....

3. Learning later about complex plane, calculus... it will learn eventually what the extended complex plane is...

But, there is quite a way until that point!

Let back to the main point... Making CAS more customizable, should prevent student in certain learning phase to ask himself WTH questions. The same as it is in this thread - I would like to see ability to set limitation up to desired plane, not to be forced to get unexpected results in real and standard complex plane.

Some of my personal doubt about CAS limited settings in some cases (some of them I already pointed in an previous thread) during testing latest public beta and reading related PDF "User Guide" are following:

1. What is the meaning of "Simplify" setting in the Prime when calculator always return unsimplified result after derivation regardless the setting?

2. What is the point of "Complex" setting at all?

Furthermore, as an advanced tester of CAS in Prime, I would like to know what various levels of "Simplify" will exactly perform, not just description:
"
Minimum—do basic simplifications (default)
Maximum—always try to simplify
"

As most of PDF on Dr. Parisse site are wrote in France language, I would appreciate suitable references and links to documents in English...

Etc.

P.S. As someone self-learned English (it is my second foreign language, though), my mental English vocabulary is mostly limited to technical content and for other social skills sentences may look quite harsh sometimes (and always full of syntax and semantic errors, I'm trying to correct in every post, if I even spot it)..

But be convinced they are not, it is only lack of finding more proper words. The purpose of all this is to point on some issues which corrections may make the product even better or more customizable for certain target group.

Thank you for your understanding.

[parisse]

minimum calls the regroup command, maximum calls the simplify command.
Programming a system with a lot of settings like you would like is not the same as programming a CAS. And anyway I don't think you can replace a math teacher.

[CyberAngel]

(10-22-2018 04:41 AM)parisse Wrote:  minimum calls the regroup command, maximum calls the simplify command.
Programming a system with a lot of settings like you would like is not the same as programming a CAS. And anyway I don't think you can replace a math teacher.

but 0^0 = 1
and....it's very useful.

[Voldemar]

(10-22-2018 08:05 PM)CyberAngel Wrote:  but 0^0 = 1

I prefer 0^0 is undefined. As Prime.

[sasa]

(10-22-2018 08:25 PM)Voldemar Wrote:  

(10-22-2018 08:05 PM)CyberAngel Wrote:  but 0^0 = 1

I prefer 0^0 is undefined. As Prime.

If we mix "apples and oranges" any result is true and false in the same time - depending on personal perspective of a beholder...

And that is exactly the reason why establishing strict boundaries (settings) are necessary in CAS. I doubt it is not possible, but I will not going to read someone full source code in order to suggest exact code change. If Dr. Parisse is willing to reconsider all upper, that is the only way.

[compsystems]

My question is to assume one or another thing, in what it affects the logic of the internal subroutines (cas kernel) and the user program?.

[ijabbott]

(10-22-2018 08:05 PM)CyberAngel Wrote:  but 0^0 = 1
and....it's very useful.

But not always true!

[parisse]

Exactly. 0^0 is most of the time 1 as a limit (if basis and exponent are both analytic) but it is sometimes undefined as a limit therefore I must keep it undefined. 1/0 is always infinite (unsigned or complex) as a limit.

[ijabbott]

Here is an example where 0^0 is not 1, courtesy of blackpenredpen:

[sasa]

(10-23-2018 07:45 AM)ijabbott Wrote:  Here is an example where 0^0 is not 1, courtesy of blackpenredpen:

Definition: Any number powered by 0 is 1. [
Definition: ln(x) is defined for any argument except 0 (if we include complex plane). Graphically, If argument x tend to zero, it will never touch y ordinate.

Infinity is not a number it is a concept and as well we do not prove axioms, we also do cannot manipulate with 0 for prove!

[Albert Chan]

(10-23-2018 11:07 AM)sasa Wrote:  Definition: Any number powered by 0 is 1.

Let prove first that 0^0 is 1 :
x^0 = 1
0 * ln(x) = ln(1)
0 * ln(x) = 0
But we have a problem now...

ln(0) = -infinity
0 * (-infinity) = 0
(-infinity) = 0/0
(-infinity)= undef

I do not follow why you are allowed to divide 0 from both side, getting:

ln(x) = 0/0 = undef ?

Following this logic, not only 0^0 ≠ 1, for any x, x^0 ≠ 1 ...

[sasa]

(10-23-2018 01:33 PM)Albert Chan Wrote:  I do not follow why you are allowed to divide 0 from both side, getting:
ln(x) = 0/0 = undef ?

Because we know that 0/0 is undef. So we "smartly" get what we need...
Similar "prove" is for 2+2=5.

Quote:Following this logic, not only 0^0 ≠ 1, for any x, x^0 ≠ 1 ...

Exactly! Since we have the axiom x^0 = 1, where x is any number, we do not need to prove anything!