Infinitely large times the infinitesimally small

Mentor note: changed the thread title so readers don't mistake this for a question about verb forms.
What is result of infinitive times infinitive small?
Infinitive?

user079622 said:

What is result of infinitive times infinitive small?

That's not a well-defined question.

PeroK said:

That's not a well-defined question.

How do you mean?

user079622 said:

What is result of infinitive times infinitive small?
Infinitive?

Can be anything. Take for example ##\frac 1 n##. It becomes infinitesimal as ##n \rightarrow \infty##. Now take ##n## times ##\frac 1 n##: ##n \frac 1 n=1##.
##\frac 5 n## also becomes infinitely small, but if you take it ##n## times, you get ##n \frac 5 n=5##.
You could take them ##n^2## times instead and get, say, ##(n^2) \frac 1 n=n \rightarrow \infty##. Etc.

user079622 said:

How do you mean?

Neither "infinitive" nor "infinitely small" are well-defined mathematical terms. Not in standard calculus.

Hill said:

Can be anything. Take for example ##\frac 1 n##. It becomes infinitesimal as ##n \rightarrow \infty##. Now take ##n## times ##\frac 1 n##: ##n \frac 1 n=1##.
##\frac 5 n## also becomes infinitely small, but if you take it ##n## times, you get ##n \frac 5 n=5##.
You could take them ##n^2## times instead and get, say, ##(n^2) \frac 1 n=n \rightarrow \infty##. Etc. have infinitive amount of air that is deflected downward by infinitive small angle. Is upward reaction force infinitive or?

PeroK said:

Neither "infinitive" nor "infinitely small" are well-defined mathematical terms. Not in standard calculus.

I have infinitive amount of air that is deflected downward by infinitive small angle(infinitive wing,airfoil). How much is upward reaction force ?

Hill said:

Take for example ##\frac 1 n##. It becomes infinitesimal as ##n \rightarrow \infty##.

That's not really true. For all natural numbers, ##\frac 1 n## is a well-defined positive number. None of these numbers is "infinitesimal".

user079622 said:

I have infinitive amount of air

If it is fewer than one molecule, then there is no air.

Hill said:

If it is fewer than one molecule, then there is no air.

infinitive long wing use infinitive amount of air

user079622 said:

infinitive long wing use infinitive amount of air

I am sorry for not understanding, because:

Hill said:

I am sorry for not understanding, because:

View attachment 337588

A wing of infinite span will use an infinite amount of air .

user079622 said:

A wing of infinite span will use an infinite amount of air .

Physically, there are no such things as infinitely large wing or infinite amount of air. If your mathematical model allows such things, then the answer depends on your mathematical model, which, as in the examples above, would be different for different mathematical models.

There may be some confusion here. “Infinitive” is a linguistic term about verbs. It has nothing to do with math.

Something that is larger than any real number is called “infinite”. Something that is non-negative but smaller than any positive real number is “infinitesimal”.

There are no infinite or infinitesimal real numbers. But there are both infinite and infinitesimal hyperreal numbers.

In post 4 @Hill discussed limits of sequences of real numbers, but what they said applies for hyperreals also:

user079622 said:

What is result of infinitive times infinitive small?
Infinitive?

Let ##\epsilon## be a positive infinitesimal hyperreal. ##\epsilon^2## is also infinitesimal. ##\omega=1/\epsilon## is infinite. ##\omega^2## is also infinite.

Thus the following are all an infinite times an infinitesimal:

##\omega \epsilon=1## is finite.
##\omega^2 \epsilon=\omega## is infinite.
##\omega \epsilon^2=\epsilon## is infinitesimal.

So indeed:

Hill said:

Can be anything.

And for your follow up question

user079622 said:

I have infinitive amount of air that is deflected downward by infinitive small angle(infinitive wing,airfoil). How much is upward reaction force ?

It could be anything, infinitesimal, finite, infinite. You cannot know simply from the fact that one is infinite and the other is infinitesimal. You have to actually use the relevant equations.

Dale said:

There may be some confusion here. “Infinitive” is a linguistic term about verbs. It has nothing to do with math.

Something that is larger than any real number is called “infinite”. Something that is non-negative but smaller than any positive real number is “infinitesimal”.

There are no infinite or infinitesimal real numbers. But there are both infinite and infinitesimal hyperreal numbers.

In post 4 @Hill discussed limits of sequences of real numbers, but what they said applies for hyperreals also: Let ##\epsilon## be a positive infinitesimal hyperreal. ##\epsilon^2## is also infinitesimal. ##\omega=1/\epsilon## is infinite. ##\omega^2## is also infinite.

Thus the following are all an infinite times an infinitesimal:

##\omega \epsilon=1## is finite.
##\omega^2 \epsilon=\omega## is infinite.
##\omega \epsilon^2=\epsilon## is infinitesimal.

So indeed: And for your follow up question It could be anything, infinitesimal, finite, infinite. You cannot know simply from the fact that one is infinite and the other is infinitesimal. You have to actually use the relevant equations.

But 1/ ∞=0, not infinitely small

user079622 said:

But 1/ ∞=0, not infinitely small

The expression ##\frac 1 \infty## is meaningless. The symbol ##\infty## cannot be used in an arithmetic expression.

Mark44 said:

The expression ##\frac 1 \infty## is meaningless. The symbol ##\infty## cannot be used in an arithmetic expression.

Dont understand

user079622 said:

But 1/ ∞=0, not infinitely small

What exactly is the symbol ∞ in that claim?

If ∞ is an infinite hyperreal number then the claim is not true.

There is no infinite real number, ∞ is not a real number. So what is it? Don’t just throw symbols around, identify explicitly what they mean.

Dale said:

What exactly is the symbol ∞
in that claim?

If ∞ is an infinite hyperreal number then the claim is not true.

There is no infinite real number, ∞ is not a real number. So what is it? Don’t just throw symbols around, identify explicitly what they mean.

We learned in school that number divided by ∞ is zero...

user079622 said:

A wing of infinite span will use an infinite amount of air .

You treat that as a 2D problem, where all quantities are essentially per unit length along the wingspan.

pasmith said:

You treat that as a 2D problem, where all quantities are essentially per unit length along the wingspan.

Yes 2D airfoil is wing with infinite span

user079622 said:

We learned in school that number divided by ∞ is zero...

This a simplification, taught to children. It is not actually true.
The appropriate way to deal with this is via limits, which is beyond (most) children.

As the denominator of a fraction is increased without bound, the fraction's value approaches zero.

user079622 said:

We learned in school that number divided by ∞ is zero...

That isn’t what I asked. I asked what that symbol means.

By the way, shame on your teachers for teaching you that.

Dale said:

By the way, shame on your teachers for teaching you that.

There is no evidence that any teacher taught the OP that.

PeroK said:

There is no evidence that any teacher taught the OP that.

Other than the OP’s report that they learned it in school.

Dale said:

Other than the OP’s report that they learned it in school.

Whether that counts as evidence or not, I'm reluctant to criticise some unknown teacher on the basis of it.

Dale said:

By the way, shame on your teachers for teaching you that.

If that is what happened. There are other options:

1. The teacher taught correctly but the student misunderstood
2. This never happened and the student made it up
3. We are being trolled.

Given that many replies happen immediately after the previous message, I cannot believe the OP is carefully considering them, so I am disinclines to blame the teacher when there are alternatives.

Fair points, both @PeroK and @Vanadium 50 make sense.

Without criticizing any teachers, @user079622 has a clear misconception. Whether they were actually taught this or not, it is wrong.

I asked the OP, what is the symbol ##\infty## in the expression ##1/\infty=0##? Usually division is defined between two numbers, so that would imply that the symbol ##\infty## is a number. If it is something else then the OP should identify exactly what it is, as I asked.

If it is a number then what kind of number is it? If the symbol ##\infty## is intended to represent a real number then it is false. There is no real number for which that statement holds.

If it is a hyperreal then it is also false. I don’t know of any number system where ##1/\infty=0## is true, but I don’t know all number systems. The closest that I can think of is if ##\infty## is an infinite hyperreal then ##\mathrm{st}(1/\infty)=0## but in general ##\mathrm{st}(x)\ne x##.

I suspect that what the OP actually meant to write is $$\lim_{x\rightarrow \infty} \frac {1}{x}=0$$ This statement is true, but doesn’t contradict anything I said in post 13.

Dale said:

I suspect that what the OP actually meant to write is limx→∞1x=0

I doubt that, this is a general math forum, after all. I think it's what you thought he should have written. I also suspect that it does nothing to explain this to the OP at a level he can understand. The rest of the crowd here already knows this.

Dale said:

If it is a hyperreal then it is also false. I don’t know of any number system where ##1/\infty=0## is true, but I don’t know all number systems.

IEEE floating point contains two signed infinities and is algebraicly complete. Although I have not been able to Google up an explicit statement, this strongly implies that ##\frac{1}{+\text{inf}}## yields a result of 0.

https://en.wikipedia.org/wiki/IEEE_754#Design_rationale said:

The special values such as infinity and NaN ensure that the floating-point arithmetic is algebraically complete: every floating-point operation produces a well-defined result and will not—by default—throw a machine interrupt or trap. Moreover, the choices of special values returned in exceptional cases were designed to give the correct answer in many cases.

Note, however, that IEEE floating point is not an algebraic field. It has behaviors that violate other axioms that the real numbers (for instance) adhere to.

PeroK said:

There is no evidence that any teacher taught the OP that.

If the OP was taught that as part of an elementary curriculum, where they do not teach limits, I can see the OP taking it as gospel. Had he pursued a math curriculum into high school he would have been disabused of the notion.

The technique is called "Lies to Children" *. (I didn't make it up, and it's not as cynical as it sounds.)

It's sort of similar to teaching Newtonian physics, when Einsteinian physics will be taught later. A student who stops with Newton might likewise assume a rocket can l accelerate to infinity.

* "Educators who employ lies-to-children do not intend to deceive, but instead seek to 'meet the child/pupil/student where they are', in order to facilitate initial comprehension, which they build upon over time as the learner's intellectual capacity expands."

jbriggs444 said:

IEEE floating point contains two signed infinities and is algebraicly complete. Although I have not been able to Google up an explicit statement, this strongly implies that ##\frac{1}{+\text{inf}}## yields a result of 0.

Note, however, that IEEE floating point is not an algebraic field. It has behaviors that violate other axioms that the real numbers (for instance) adhere to.

So when I asked the OP what ##\infty## is then one reasonable response would have been that it is the IEEE floating point signed infinity. And if you are right then the statement would be true there.

But then the problem is that the question in the OP cannot be asked because there is no infinitesimal number in the IEEE floating point.

user079622 said:

Mentor note: changed the thread title so readers don't mistake this for a question about verb forms.
What is result of infinitive times infinitive small?
Infinitive?

Rearrange A*B to A/(1/B) then try L'Hopital's Rule.

jbriggs444 said:

IEEE floating point contains two signed infinities and is algebraicly complete. Although I have not been able to Google up an explicit statement, this strongly implies that ##\frac{1}{+\text{inf}}## yields a result of 0.

Note, however, that IEEE floating point is not an algebraic field. It has behaviors that violate other axioms that the real numbers (for instance) adhere to.

This is really all a bit of a non-sequiter The set of numbers used in floating point arithmetic is defined to deal with values that are encountered in 'real' life. in my Maths Analysis course we were told to be scrupulous about open and closed intervals and the use of continuous and differentiable.

To get this properly sorted out, you need to go much deeper into 'proper' Maths. You can't rxpect a group of Engineers to come up with a working rule that behaves itself outside the range of experience. The term +inf is pragmatic. Without it, they. would still be arguing about what zero and infinity mean. That's the job of Mathematicians. The concept of the limit is not too hard to get on top of and it gets you through a lot of Maths but there are many processes that can't be dealt with so simply.