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.