In the sum up to 2ⁿ.



But then why not allow the Sieve of Eratosthenes? See above.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

If she assumed such a restriction she should express it beforehand, and thats true for any restrictions.

And such a restriction is nonsense, if we assume such a restriction we cant solve problem: "using Erastotens algorithm find all primes below 100" because only number 100 is given! And if you somehow get number 1, 2 and others, you can get same way numbers 101, 102 and all others natural numbers.

As i said - we must assume that whole set of natural numbers is given, cuz its needed to define prime numbers.
Sieve of Erastotenes is not a formula nor function, its algorithm. Very big difference.

Transforming algorithm to function is sometimes very hard.

A sum is a sum. We know how to sum.




If you can express it with math symbols like that, why not allow it? I don't see a problem. It's a formula.

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be immersed

I'll try to summarize my stance:

1. "Formula" often doesn't have a rigorous definition
2. Folks often use "formula" casually, and can be just some way of expressing some rule or fact mathematically
3. Other things like "polynomial" have much stricter definitions. There are constraints in there that you don't always have when using the word "formula"
4. The teacher was probably thinking of a particular constrained definition of "formula" when it was being used
5. It would have been less ambiguous to use a less ambiguous term, which expressed the constraints that teacher was trying to convey

If you just use "formula", you have folks like me and AloneAgainstAll, which find the statement to be false. If you're going to make a sweeping statement like that, you should use a precise definition.

This is not to say that the teacher is a bad teacher. I like Knotwilg's point that, if nothing else, this has caused us all to think about the problem on our own!

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be immersed

Kirby he means that we can sum only up to given number by his assumption that only first n-th prime numbers ar given, and only those we can use in our formula. Its sad that if we assume that we cant give formula for volume of ball (and other standard formulas), even if we have radius, but thats his definition.

For me, formula is equal to function. If we assume Jlt definition we have problems i showed. If we assume Bill's definition we have even more problems.

But whatever we beat from this "formula" discuss its completely offtopic - i expressed explicitly what formula meant for teacher in the problem. Maybe we should stop beating this dead from the beginning horse?

Personally, I think the definition of "function" is more strict than the definition of "formula".

But anyway, yeah... The horse is dead. RIP

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be immersed

The problem arises when the unknown, which you are trying to find, is part of the formula. Yes. there is such a thing as recursion, but then you can use subscripts.

As for the Sieve of Eratosthenes, presumably the teacher had shown the elementary students how to tell whether a natural number is a prime or not, which in turn implies the Sieve of Eratosthenes, even if she did not mention it. Whatever formula she had in mind, if it included the Sieve, it could not have been part of the formula.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Indeed. Formula is an informal term.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Indeed, there are recursive formulas

But this is not recursive. It's iterative. You can often convert recursive formulas to iterative ones, which is what is done here.



If so, it cannot be used to make strong mathematical claims.

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be immersed

Ah so all that fuss was beacuse you assumed that formula is informal term?? Why the hell you didnt said it when you said teacher was correct?

You indeed cant give formula for literally ANTYHING if term formula is not formally well defined!

Isnt that called strawman in english?

Consider your audience. Why complicate matters for a group of elementary students? She used formula in an informal, non-precise way, as it is used in ordinary language.

There is a similar problem, BTW, in teaching go to adults. For instance, some people are told that you need two eyes for life. Later, these people see a seki and are confused. And the question of what is an eye is a mess. See discussions on Sensei's Library. It is probably not a good idea to tell raw beginners about Two Headed Dragons, which they may never see in real life. OTOH, why lie to them and say that a false eye cannot help you make life? My solution has been to define a false eye, not by its local shape, but by the fact that eventually the opponent can put at least one stone that forms it into atari. What to tell beginners, or elementary school students, is not easy to say.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

It is not my definition, thank you very much. My definition has nothing to do with balls. I have reasons for thinking that the teacher had a restricted sense of formula in mind.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

You think that the teacher was making a strong mathematical claim to a bunch of elementary school students? I think she was talking to them using normal speech.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

But she (is it a she?) used that imprecise term to say to make a bold statement about non-existence.

Sometimes when people are referring to numbers, they are talking strictly about the natural numbers. Using casual, ordinary language, someone might claim that there aren't numbers less than zero. Obviously, folks who are considering negative numbers might take issue with that.

Then you can get into a 100+ post argument about how a number is defined. When really, there are different ways of specifying numbers (natural numbers, reals, extended reals, etc.).

It'd be much safer for the teacher to say that there are no *natural numbers* less than zero. Because that's the class of things she's talking about.



So the teacher had better not say that!

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be immersed

Normal speech that was wrong.

Anyway, I already paid my respects to the dead horses. I've expressed my thoughts on this enough. I'm out.

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be immersed

I believe the formula in question is not a closed form formula: https://en.wikipedia.org/wiki/Closed-form_expression.

Summation notation is problematic, because given an initial set of values given by elementary (and some non-elementary functions), the number of such function values you must compute grows arbitrarily large as n goes to infinity.

The distinction between a closed-form-expression and others is quite important. I'm not sure how formula is used, but I think the worst you could accuse your teacher of doing is being slightly less precise.

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Occupy Babel!

Consider her audience. Math was not taught as well when I was in grade school as it is now, but nearly all of our ten years olds understood number to mean a non-negative integer. Why confuse them with the term, natural number? Maybe counting number would have been OK, though. But then the teacher would open the door to questions about non-counting numbers before she was ready to teach about them.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

I didnt wrote that you definition has sth to do with balls, i said that using this definition we cant give formula for ball volume. Btw, who "invented" this definition of formula?

But really whatever, as i said, if formula is not defined then we cant give formula for anything. Case closed.

Well, then, your complaint was irrelevant.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

What complain? I didnt made any complain.

Irrevelant is discussion about what definition of formula assumed teacher.