Prelude on functions and graphs

This short video is a prelude to the next near-numbers video proper; it's a quick review of functions, focusing on how to properly use the graphs of real-valued functions. After this video, you in principle know how to apply a function to a near-number as well:

• put your input near-number on the domain axis, 
• push those points up to the graph, 
• push them over to the codomain axis, 
• then let your near-number squeeze to see the result!. 

We'll go through this for a bunch of important examples in the next video (it might take a while to produce, as the animations for functions are a bit involved!), but for now, here's the prelude:

Lesson 2: The Arrow Relation

The learning goals for this video are as follows:

1. to understand what it means for one near-number to "fit into" or "not fit into" another one, both intuitively and in terms of slices, and to be able to determine and explain whether one near-number fits into another or not;
2. to understand what it means for a near-number to converge or diverge, along with the notion of limit, and to be able to determine and explain which of these a given near-number does;
3. to understand what it means for an arrow statement to be indeterminate for a near-number, to be able to explain indeterminacy, and to give examples of it.

Lesson 1: The Basic Near-Numbers 

The learning goals for this video are as follows:

1. to be able to sketch and describe any basic near-number (finite, infinite, or constant);
2. to be able to recognize and distinguish any basic near-number(s) shown to you;
3. to be able to sketch slices of any basic near-number α—both the general picture slice_t α and the slice for any given t.

Further exploration:

• for a given slice of some near-number, use inequalities to express which real numbers x live in that slice (always remember: an inequality often serves as a symbolic description of some picture!).

Online tools

All of the near-number concepts and definitions can be interactively explored, in or out of the classroom, via the Flash animations available at:

http://www.nearnumbers.net

These are old but very functional—being Flash, they won't run on iOS (iPad/iPhone) devices, but they should work on any computer, laptop, or Android phone/device via a browser.  Note that the downloadble .swf files don't run anymore.

If you want to see and play around with any of those listed below (and more), just read the instructions to see the keys to use, and go at it:

• 0 × +∞  or  0⁺ × +∞ 
• +∞ + (−∞)
• sin(+∞)
• +∞/+∞
• e^+∞ or e^+∞

A little history...

The goal for the rest of the posts on this blog will be to systematically present near-numbers in a manner that can be used as a curricular supplement by students and/or teachers of Calculus, and to answer specific questions that come in.  This will take some time to produce!

In the meantime, near-numbers are not a brand new idea; I gave an invited lecture at the MAA Seaway Sectional Meetings at SUNY Plattsburgh in 2010, and I've made an extended YouTube version of this, cut up into segments:

Full playlist (8 videos)

The talk was aimed at the teacher/professor/grad student audience, but several undergrads there said they got a lot out of it, as well.  It's more a pitch for near-numbers than a full presentation of the material, which is the aim of this blog (as well as to refine a few bits).  The videos are as follows:

1. Introduction and basic near-numbers (11:01) This corresponds roughly to the new "Mining near-numbers" video and what will be the next video introducing the basic near-numbers.
2. The arrow relation (9:33), including its logical definition, and the purest picture of indeterminacy.
3. Functions and near-numbers (9:28), including pictures of:
• sin(+∞) ➝ [−1,1]^.
• 1/(+∞) ➝ 0⁺
• 1/0⁺ ➝ +∞
4. Arithmetic and near-numbers (7:42), including pictures of :
• 0 × +∞ ➝ 0^.
• 0⁺ × +∞ ➝ (0,∞)^.
5. Formal treatment of the arrow relation, part 2: functions and arithmetic (8:47), including a picture of:
• +∞ + (−∞) ➝ ℝ^.
6. Key benefits of the near-number approach (5:12)
7. The derivative and integral (6:19) viewed from the near-number perspective, the key point being to see the squeezing involved in convergence for the definitions.
8. Convergence and divergence of sequences (6:51) and conclusion; this includes a description of the proper (as dictated by the definition) way to view the behavior of a sequence: one shouldn't consider one term at a time, but all of the terms, tossing out more and more initial terms—then we see squeezing just as with everything in the near-number context.

What is infinity?

Our first YouTube video is now online:  What is infinity?

In just a few minutes, we show the exact meaning of infinity itself within the context of calculus.  This teaser shows just one pivotal concept of calculus that is often seen as nebulous or fuzzy by calculus students (we'll later see others, such as indeterminacy, limits, etc.)—all of these can be made extremely clear and precise by using the right pictures.

Some readers might find themselves resisting all of this initially, and we welcome feedback; our goal will be to make believers of everyone that takes the time to watch, listen, think, and question what is said here.  As a first step, our next video will peek ahead to a few definitions from calculus as evidence that near-numbers are logically well-founded.

A quick disclaimer about what this video is not about.  First, in interval notation such as (0,∞), the infinity symbol simply means that the interval has no end, which is not subtle and has nothing to do with calculus.  Second, for anyone familiar with the "extended real line," our +∞ would be expressed as ∞^-, squeezing toward ∞ from the left side; this extended real line is not well-suited for calculus, as we no longer have arithmetic working as it should there and it does not help to clarify calculus.

Mining near-numbers

Our second video, Mining near-numbers, is for those that already know Calculus and have doubts whether this is all legit. If you don't know the definition of limit in Calculus yet, don't worry if this video is confusing—it's a peek way ahead and isn't expected to make complete sense yet!

It is important to point out that nothing of the formal logic of Calculus is changed by near-numbers; what they provide is a far more precise and informative way of writing and thinking about the concepts.  By the time we get to some of the hairier bits, such as indeterminate forms and definitions, you'll see just how true this is.

In our next video, we'll start into near-numbers from the ground up, which is the real starting point for the video series.