Beginner's Guide to Precalculus, Calculus and Infinitesimals

[bhobba]

Table of Contents
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[Mark44]

A great improvement would be to use LaTeX in this article instead of things like Xn, x^2, and so on.

 

[bhobba]

Okay, guys, I got the message.

Valid point. It has been a long time since I used LaTex. It will take a while, but I will fix it.

Thanks
Bill

 

[Mark44]

Infinitesimals
Let X be any positive number. Let x be the sequence xn = 1/n. Then, an N can be found such that 1/n < X for any n > N. Hence, by the definition of less than in hyperrational sequences; x < X. Such hyperrational sequences are called infinitesimal. A sequence, x, is infinitesimal if |x| < X for any positive X. If x > 0, x is called a positive infinitesimal. If x < 0, x is called a negative infinitesimal. Normally zero is the only number with that property. Also, we have infinitesimals smaller than other infinitesimals, e.g. 1/n^2 < 1/n, except when n = 1.

This is just a small section of your article that I took a close look at. I haven't looked closely at the other parts. Here are some comments about your work as well as ways that I would write this differently.

Instead of X I would use $\epsilon$, which is typically used in many mathematics textbooks to denote a "reasonably small" but positive real number.

Instead of "Let x be the sequence xn = 1/n." I would define an identifier S like so, using set notation: Let S be the sequence $\{x_n : x_n = \frac 1 n, n \in \mathbb N \}$

Your inequality "x < X" leaves a lot unstated and hurts my head to look at. Namely that a sequence x of numbers (which I'm calling S) is less than a single positive real number X (which I'm calling $\epsilon$. I think you addressed this in another Insights article, but it would be good to also include it here. My version of this would be that $S < \epsilon$ means that all but a finite number of elements in S are less than $\epsilon$.

You wrote: "A sequence, x, is infinitesimal if |x| < X"
You haven't defined the absolute value/norm of a sequence, or if you did, I missed it. If I have a sequence $S = \{1/2, 1/3, 1/4, \dots, 1/n, \dots \}$ and $\epsilon = 0.01$, what's the value of |S| here? And how can I determine whether $|S| < \epsilon$?

Last, I'm not sure that you need to talk about hyperrationals as opposed to hyperreals. All the number in my sequence are rationals, sure enough, but they are also reals. If I search for "hyperrational" on the internet, I don't get a lot of results.