What makes a good teacher? What is a formula?

[Gomoto]

Math for fun? Project Euler!

[AloneAgainstAll]

"Point for having formula is to leap from known to unknown" - can you use precise mathematical language? Is this prove that my formula isnt formula? How well defined function can be not a formula? Can you provide any example that well defined funtion is not formula?

Is

1.f(x)=x formula?
2 f(x)= Floor(x) formula?
3 f(x)= x+x^2 formula?

If you answer yes for this 3 questions, then you cant deny my formula being formula. If you deny, then which of these you negate?


Which expression you want me to explain? You mean explain formula i showed? I am not sure which expression you refer to.


If we define formula exactly to ban this "formula" (well, it wouldnt be formula then) for being solution to problem, then sure. But thats not math anymore.

The only true weak point of this formula is that its useless for computations. Even F(10) needs already good machine. But using floor function and sum is perfectly ok, and thats not only my view, but general consensus of mathematicians. I wonder what Paulo Ribenboim would think if he heard that ppl thinks that formula he invented is not a formula .

[Kirby]

AloneAgainstAll, I’d consider the example you gave to be a formula. But like you said, it may not be that useful due to the computation it requires to calculate. It would be a lot better computationally if it were expressed, eg, as a polynomial without the summations. But I think that kind of polynomial might be what doesn’t exist.

Anyway, I don’t see a problem with calling what you shared a formula.

[Kirby]

For example here:

http://mathworld.wolfram.com/Formula.html

In mathematics, a formula is a fact, rule, or principle that is expressed in terms of mathematical symbols.

The example given fits this definition, whether or not it is useful.

Is it possible that the professor in question meant to say that you can’t express this type of function as a non-constant *polynomial*? That seems like an easier claim to make.

By the way, this still has little to do with Shin Jinseo - maybe I should change the thread title...

[jlt]

So I have a much shorter formula that computes f(n):

f(n) = f(n).

[Bill Spight]

"Point for having formula is to leap from known to unknown" - can you use precise mathematical language?

The question of whether "formula" is ambiguous or not is a lingusitic question.

Is this prove that my formula isnt formula? How well defined function can be not a formula? Can you provide any example that well defined funtion is not formula?

Sure, but not by your definition of formula. That cannot be the definition of formula your elementary school teacher used (or that of the mathematicians who made the same claim), because she was well aware that there is a well defined function, f(n) = Pn, where n is a natural number and Pn is the nth prime. The ancient Greeks knew how to find the nth prime. So, from her perspective, a well defined function is not necessarily a formula.

Anyway, it's your turn.

Please explain the meaning of the expression you posted.

[Kirby]

So I have a much shorter formula that computes f(n):

f(n) = f(n).

Seems like a true equality. But AloneAgainstAll‘s formula describes something about the nature of f, albeit computationally expensive.

[Kirby]

Think of it this way: if somebody has no idea how to compute the nth prime, AloneAgainstAll has a formula that is descriptive.

[AloneAgainstAll]

Yes f(n)=f(n), but that does not give n-th prime number or whatever, it just give f(n).

Kirby, you can extract post which should not belong to this thread to another thread (you can even cut my posts and keep parts, and move another parts i am perfectly fine with that). Thank you that at least one person here can admit that i was right (well, not me but the guy who inveted it, the more i think about it, the more i respect him).

No, Greeks didnt defined functions at all (i am not sure when the first correct definition of function was provided, but it was well after the Greeks and even Gauss). Erastotenes knew algorithm to get all prime numbers, but algorith is not always function. You seems to think that formula, function etc has "floating" meaning, thats not true.

I asked you which expression you want me to explain. You didnt said, but keeps asking me to explain. I will not play this dumb game anymore. Thats enough of rudness i can endure.

[jlt]

f(n) is the n-th prime number, so you can't say it doesn't give it.

If you prefer, f(n)=max{m>1 | for all 1 < a(1) < ... < a(n) < m, there exist i, x, y such that a(i)=xy and x>1 and y>1}.

That's a formula according to Kirby's definition.

P.S. Bill was referring to the expression viewtopic.php?p=253915#p253915

[AloneAgainstAll]

If f(n) is n-th prime number, all you said is that n-th prime number equals n-th prime number. Thats not the same as providing forumla that makes f(n)=n-th prime number.

[jlt]

"f(n)=max{m>1 | for all 1 < a(1) < ... < a(n) < m, there exist i, x, y such that a(i)=xy and x>1 and y>1}"

is expressed by mathematical symbols, and tells how to compute f(n), so is a formula that computes f(n) according to Kirby's definition.

[Kirby]

I don’t understand why folks are so reluctant to call the given expression a formula. It seems we largely agree on its limitations. So it’s just a matter of definition. But isn’t it at least agreeable that “formula” is a generally used term? We have more specific words like “polynomial”, etc., to express a more constrained body of expressions.

Anyway, I’ll split the thread soon - I’m not at a place where it’s easy to do at the moment.

[jlt]

So it’s just a matter of definition.

That's what I've said several times. The teacher has a definition of "formula". AloneAgainstAll has another definition in mind and concludes that his teacher was wrong, and is so bad that he wouldn't recommend her to his enemies. I found that judgment too hard.

[AloneAgainstAll]

"f(n)=max{m>1 | for all 1 < a(1) < ... < a(n) < m, there exist i, x, y such that a(i)=xy and x>1 and y>1}"

is expressed by mathematical symbols, and tells how to compute f(n), so is a formula that computes f(n) according to Kirby's definition.

OK - my english seems to be really weak, and when i tried to use this formula i failed. Can you give me step by step description how you calculate f(5)?