Exploring Sums of Finite/Infinitesimal Numbers: Q29

[rstor]

Homework Statement

ε is a positive infinitesimal and you need to determine whether the given expression is infinitesimal, finite but not infinitesimal, or infinite:

$\frac {\sqrt{4+ε} ~~-2} {ε}$

Relevant Equations

Looking at just the square root in the numerator for now.
I am thinking that 4+ε in the numerator is finite and therefore $\sqrt{4+ε}$ would also be finite. However I am unsure as a video seems to indicate this would be infinitesimal.

Hello All. This is my first post on the Physics Forums. I have started to self-study calculus and based on the feedback from this site and others, I have chosen Elementary Calculus: An Infinitesimal Approach by Jerome Keisler.

I am working through the problems for section 1.5 (page 34/35).
https://people.math.wisc.edu/~keisler/chapter_1b.pdf

I am stuck on question 29 and in the process found that I am not completely sure of what the the sum of a finite and infinitesimal is (i.e. is it finite but not infinitesimal, or is it infinitesimal).

Question 29:
ε is a positive infinitesimal and you need to determine whether the given expression is infinitesimal, finite but not infinitesimal, or infinite:
$\frac {\sqrt{4+ε} ~~-2} {ε}$

Based on the rules of infinitesimal, finite, and infinite numbers (page 30/31) it is my understanding that 4+ε in the numerator is finite and therefore $\sqrt{4+ε}$ would also be finite. However; I am confused as there is a video series on YouTube which follows Keisler's text and indicates that
$\sqrt{1+ε}$ is infinitesimal: https://youtu.be/yw0-wnEuaHc (watch at 9:30 and again at 12:10). Which is correct, and why?

 

[Mark44]

Homework Statement:: ε is a positive infinitesimal and you need to determine whether the given expression is infinitesimal, finite but not infinitesimal, or infinite:

$\frac {\sqrt{4+ε} ~~-2} {ε}$
Relevant Equations:: Looking at just the square root in the numerator for now.
I am thinking that 4+ε in the numerator is finite and therefore $\sqrt{4+ε}$ would also be finite. However I am unsure as a video seems to indicate this would be infinitesimal.

Hello All. This is my first post on the Physics Forums. I have started to self-study calculus and based on the feedback from this site and others, I have chosen Elementary Calculus: An Infinitesimal Approach by Jerome Keisler.

I am working through the problems for section 1.5 (page 34/35).
https://people.math.wisc.edu/~keisler/chapter_1b.pdf

I am stuck on question 29 and in the process found that I am not completely sure of what the the sum of a finite and infinitesimal is (i.e. is it finite but not infinitesimal, or is it infinitesimal).

Question 29:
ε is a positive infinitesimal and you need to determine whether the given expression is infinitesimal, finite but not infinitesimal, or infinite:
$\frac {\sqrt{4+ε} ~~-2} {ε}$

Based on the rules of infinitesimal, finite, and infinite numbers (page 30/31) it is my understanding that 4+ε in the numerator is finite and therefore $\sqrt{4+ε}$ would also be finite. However; I am confused as there is a video series on YouTube which follows Keisler's text and indicates that
$\sqrt{1+ε}$ is infinitesimal: https://youtu.be/yw0-wnEuaHc (watch at 9:30 and again at 12:10). Which is correct, and why?

You can't look at the numerator and denominator separately, but rather, at the fraction as a whole.
What I end up with is that the expression you're working with is approximately $\frac \epsilon 4$, which would make it infinitesimal.
Edit: Boneheaded mistake on my part. The final result is close to $\frac 1 4$, not $\frac \epsilon 4$, which makes the result finite.

What I did, and this is a standard trick for dealing fractions that involve the sum or difference of a square root and some other number, is rationalize the numerator. To do this, multiply the given fraction by the conjugate over itself. IOW, multiply by 1 in the form of $\sqrt{4 + \epsilon} + 2$ over itself.

 

[rstor]

You can't look at the numerator and denominator separately, but rather, at the fraction as a whole.
What I end up with is that the expression you're working with is approximately $\frac \epsilon 4$, which would make it infinitesimal.

What I did, and this is a standard trick for dealing fractions that involve the sum or difference of a square root and some other number, is rationalize the numerator. To do this, multiply the given fraction by the conjugate over itself. IOW, multiply by 1 in the form of $\sqrt{4 + \epsilon} + 2$ over itself.

Thank you Mark. I was looking at part of the numerator separately as I feel I may have a misunderstanding of what $\sqrt{4 + \epsilon}$ would work out to be (would this expression be finite or infinitesimal and why?). Earlier when working on the whole problem I had multiplied the numerator and denominator by $\sqrt{4 + \epsilon} +2$. In your response you mentioned $\sqrt{4 + \epsilon} +4$ -- was this a typo and did you intend to write $\sqrt{4 + \epsilon} +2$ ?

 

[Mark44]

Thank you Mark. I was looking at part of the numerator separately as I feel I may have a misunderstanding of what $\sqrt{4 + \epsilon}$ would work out to be (would this expression be finite or infinitesimal and why?).

I haven't read the web textbook you're working from, but I would assume that $\sqrt{4 + \epsilon}$ is finite, being slightly larger than 2.

Edit: I've taken a look at the PDF now, and I stand by my reply. $\sqrt{4 + \epsilon}$ is between 2 and 3 (and much closer to 2), which makes this expression finite.
Edit 2: Yes, finite, which is inconsistent with what I said earlier.

Earlier when working on the whole problem I had multiplied the numerator and denominator by $\sqrt{4 + \epsilon} +2$. In your response you mentioned $\sqrt{4 + \epsilon} +4$ -- was this a typo and did you intend to write $\sqrt{4 + \epsilon} +2$ ?

Yes, this was a typo. I have fixed it in my post and in the text that you quoted. I meant $\sqrt{4 + \epsilon} + 2$.

 

[rstor]

I haven't read the web textbook you're working from, but I would assume that $\sqrt{4 + \epsilon}$ is finite, being slightly larger than 2.

Yes, this was a typo. I have fixed it in my post and in the text that you quoted. I meant $\sqrt{4 + \epsilon} + 2$.

My book states that the sum of a finite and infinitesimal is finite. I assumed that taking the square root of this sum will also be finite. I was unsure as the linked video (which is following this book) appears to indicate that the square root of a similar sum is infinitesimal. If we say that $\sqrt{4 + \epsilon}$ is finite then I will show my work and thought process below to show where I am stuck:

Looking at the original question again:
$\frac{\sqrt{4 + \epsilon} ~~- 2}{\epsilon}$

We have $\sqrt{4 + \epsilon}$ which we are saying is finite and -2 is a finite. The sum of a finite and a finite per the book is finite however could also possibly be infinitesimal. If the result in the numerator is infinitesimal, then the quotient of two infinitesimals is undetermined so we need to do further work.

After multiplying the original question by the conjugate $\sqrt{4 + \epsilon} + 2$ I arrive at :

$\frac{\epsilon}{\epsilon\sqrt{4+\epsilon}~~+2\epsilon}$

$\frac{1}{\sqrt{4+\epsilon}~~+2}$

The numerator is finite, the denominator is the sum of two finite numbers again which could possibly be infinitesimal which in that case would result in the answer being infinite. The answer according to the book is finite. I am unsure how to proceed at this point.

 

[Mark44]

It looks like I made another mistake. I'm sorry for misleading you with inconsistent answers...
I neglected to include the $\epsilon$ in the denominator, so in my first post I should have gotten a result close to 1/4, not $\epsilon /4$ as I wrote.
So the answer is that the original expression is finite, which is what the answers indicate

My book states that the sum of a finite and infinitesimal is finite. I assumed that taking the square root of this sum will also be finite.

Yes. Although the examples in the textbook don't show this, one could expand $\sqrt{4 + \epsilon}$ as a Taylor or Maclaurin series, and then use the axioms about adding finite and/or infinitesimal quantities.

I was unsure as the linked video (which is following this book) appears to indicate that the square root of a similar sum is infinitesimal.

I haven't looked at the video, yet, so can't comment on this.

Looking at the original question again:
$\frac{\sqrt{4 + \epsilon} ~~- 2}{\epsilon}$

We have $\sqrt{4 + \epsilon}$ which we are saying is finite and -2 is a finite. The sum of a finite and a finite per the book is finite however could also possibly be infinitesimal. If the result in the numerator is infinitesimal, then the quotient of two infinitesimals is undetermined so we need to do further work.

After multiplying the original question by the conjugate $\sqrt{4 + \epsilon} + 2$ I arrive at :

$\frac{\epsilon}{\epsilon\sqrt{4+\epsilon}~~+2\epsilon}$

$\frac{1}{\sqrt{4+\epsilon}~~+2}$

Yes.

The numerator is finite, the denominator is the sum of two finite numbers again which could possibly be infinitesimal which in that case would result in the answer being infinite. The answer according to the book is finite. I am unsure how to proceed at this point.

You're right on the money. Again, I'm sorry to have misled you and caused confusion.

 

[rstor]

Yes.
You're right on the money. Again, I'm sorry to have misled you and caused confusion.

Thank you for confirming my steps. Though at this point I still have the issue of
$\frac{1}{\sqrt{4+\epsilon} ~~+2}$
having two possible answers. One of them being finite and the other infinite (as the sum of two finite numbers in the denominator could possibly be an infinitesimal and therefore the quotient of 1/infinitesimal = infinite. How do I proceed?

 

[WWGD]

I'm thinking of your expression as the differential quotient for $\sqrt x$ evaluated at $x=4$.

 

[Mark44]

Thank you for confirming my steps. Though at this point I still have the issue of
$\frac{1}{\sqrt{4+\epsilon} ~~+2}$
having two possible answers. One of them being finite and the other infinite (as the sum of two finite numbers in the denominator could possibly be an infinitesimal and therefore the quotient of 1/infinitesimal = infinite. How do I proceed?

I don't see that $\frac{1}{\sqrt{4+\epsilon} ~~+2}$ would have two possible answers, relative to finite vs. infinitesimal. $4 + \epsilon$ is finite. Per the nomenclature of your textbook, $st(4 + \epsilon) = st(4) + st(\epsilon) = 4 + 0 = 4$ , and
$st((4 + \epsilon)^{1/2}) = (st(4 + \epsilon))^{1/2} = (4 + 0)^{1/2} = 2$, which is finite.

So the fraction you wrote above is infinitesimally close to 1/4, hence is finite.

 

[rstor]

I don't see that $\frac{1}{\sqrt{4+\epsilon} ~~+2}$ would have two possible answers, relative to finite vs. infinitesimal. $4 + \epsilon$ is finite. Per the nomenclature of your textbook, $st(4 + \epsilon) = st(4) + st(\epsilon) = 4 + 0 = 4$ , and
$st((4 + \epsilon)^{1/2}) = (st(4 + \epsilon))^{1/2} = (4 + 0)^{1/2} = 2$, which is finite.

So the fraction you wrote above is infinitesimally close to 1/4, hence is finite.

I haven't reached to the next section (1.6) as yet which deals with standard parts. The reason I came to how there could be two possible answers is based on page 31 where it outlines a list of rules. It says that b and c are hyperreal numbers that are finite but not infinitesimal. It then indicates under the "Sums:" section that "b+c is finite (possibly infinitesimal)". An example can be seen here

Another example is from the same problem set on page 34, question 9 where you need to determine if the given expression is finite, finite but not infinitesimal, or infinite. $\epsilon$ and $\delta$ are positive infinitesimals :

$(3 + \epsilon)(4 + \delta)~-12$
At first glance one might conclude that each expression in the parenthesis is finite, and the product of two finite numbers is finite, and therefore the sum of this finite product and another finite, -12, is also finite. However the answer is infinitesimal. So if one takes the complete rule under consideration, that the sum of two finites could possibly be infinitesimal, one is forced to do further work, expand, and then see that the result is really infinitesimal (as some of the constant finite numbers cancel).

So I used the same logic for other problems, that is, where ever I encounter the sum of two finite numbers, not to automatically conclude that the result is finite. If I did automatically make this conclusion then when starting on the original problem (question 29):
$\frac{\sqrt{4 + \epsilon} ~~- 2}{\epsilon}$
I might have said that $\sqrt{4 + \epsilon}$ is finite and -2 is finite and therefore the sum of two finite numbers is finite. Therefore the quotient of a finite and an infinitesimal is infinite. However the answer is finite. It is only when I applied the complete rule that the sum of two finites could possibly be infinitesimal (resulting in the quotient of two infinitesimals which is undetermined) that made me do further work (take the conjugate) to see if I could reduce the expression to obtain a single result. However even after doing so, I still have an issue in the denominator:
$\frac{1}{\sqrt{4+\epsilon} ~~+2}$
as the sum of two finites in the denominator could be infinitesimal and therefore 1/infinitesimal is infinite.

 

[Orodruin]

I might have said that 4+ϵ is finite and -2 is finite and therefore the sum of two finite numbers is finite.

Finite, yes, but you cannot rule out infinitesimal. To do that you need to look at the square root and conclude that the finite but not infinitesimal parts of the square root and the -2 cancel and sum up to zero. What remains must therefore be infinitesimal (or zero) so you need to look at the infinitesimal part of the square root.

However even after doing so, I still have an issue in the denominator:
14+ϵ +2
as the sum of two finites in the denominator could be infinitesimal and therefore 1/infinitesimal is infinite.

No, it could not be infinitesimal because the non-infinitesimals do not sum to zero.

 

[Office_Shredder]

I think a general rule of thumb, if you have a sum of finite things, and you plug in 0 for all the infinitesimal parts, if you get a nonzero number then the sum will be fine, not infinite.

 

[Mark44]

It says that b and c are hyperreal numbers that are finite but not infinitesimal. It then indicates under the "Sums:" section that "b+c is finite (possibly infinitesimal)".

The example he gives is one where b is positive and finite while c is negative and finite. If you add two positive, finite numbers, you get a (positive) finite number.

 

[rstor]

I believe I now see my mistake: I was blindly following the rule of adding finite hyperreals without fully understanding why their sum could possibly be infinitesimal. From what I understand from Orodruin's reply: When looking at the original question, the possibility of the numerator being infinitesimal exists as it is not readily apparent if the finite constant numbers cancel in the numerator leaving us with an infinitesimal in the numerator.

After multiplying by the conjugate and then reaching the form of
$\frac{1}{\sqrt{4+\epsilon} ~~+2}$
it is my understanding from Mark and Orodruin's reply that we can now tell that the sum of finites in the denominator result in a finite as we are adding positive finite values and there is no chance of the finite constants canceling (that could leave us with an infinitesimal).