Infinitely large times the infinitesimally small

[user079622]

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?

 

[PeroK]

What is result of infinitive times infinitive small?

That's not a well-defined question.

 

[user079622]

That's not a well-defined question.

How do you mean?

 

[Hill]

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.

 

[PeroK]

How do you mean?

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

 

[user079622]

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?

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 ?

 

[PeroK]

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".

 

[Hill]

I have infinitive amount of air

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

 

[user079622]

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

infinitive long wing use infinitive amount of air

 

[Hill]

infinitive long wing use infinitive amount of air

I am sorry for not understanding, because:

 

[user079622]

I am sorry for not understanding, because:

View attachment 337588

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

 

[Hill]

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.

 

[Dale]

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:

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:

Can be anything.

And for your follow up question

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.

 

[user079622]

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

 

[Mark44]

But 1/ ∞=0, not infinitely small

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

 

[user079622]

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

Dont understand

 

[Dale]

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.

 

[user079622]

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...

 

[pasmith]

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.

 

[user079622]

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

 

[DaveC426913]

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.

 

[Dale]

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.

 

[PeroK]

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

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

 

[Dale]

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

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

 

[PeroK]

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.

 

[Vanadium 50]

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.

 

[Dale]

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.

 

[DaveE]

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.

 

[jbriggs444]

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.

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.

 

[DaveC426913]

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."