JMM Baltimore 2019

If anyone's around the Baltimore area and has near-number-related questions, I'm around for the math meetings this week, so hit me up! Workshop on Saturday if you or anyone you know will be there!

Expressions and limits

The video on expressions and limits is now online!  It begins by demonstrating the more refined statements near-numbers allow and connects back to limit notation, with some examples illustrating the shortfall of the limit-centric pedagogy (and how near-numbers fill in the gaps).  Key learning goals are:

• to understand how near-numbers allow different (but consistent) results when substituted into  algebraically "equal" expressions,
• to get an introduction to techniques for avoiding ☆-forms,
• and to see how near-numbers relate to limit notation used in the classical calculus pedagogy.

The third video on near-number arithmetic is now online, discussing ☆-forms for both arithmetic and powers, including the way in which they all stem from the most basic one: +∞ + –∞.  Key learning goals are:

• to review the meaning of the near-number ☆ and what we mean by a "☆-form" as a dead-end in a computation,
• to derive all basic ☆-forms for arithmetic and powers, noting the way in which they're all connected,
• and to clear up misconceptions about "indeterminacy" and why "∞ – ∞" not being zero doesn't run at all against the fact that for any real number a, we do have a – a = 0.

Near-number arithmetic, part two: multiplication and division

The second of a three-part lesson on near-number arithmetic is now online, covering multiplication and division; next will come a video discussing and connecting all of the ☆-forms.  Key learning goals are:

• how binary arithmetical operations operate on near-numbers (one pair of slices at a time, on all pairs of numbers, one from each slice);
• familiarity with multiplication of basic near-numbers: finite × finite, finite × infinite, and infinite × infinite, and division via multiplying by the reciprocal;
• ☆-forms for multiplication and division: 0⁺× +∞, +∞/+∞, and 0⁺/0⁺
• The number zero times anything—including +∞—is simply zero.

Near-number arithmetic, part one: addition and subtraction

The first of a three-part lesson on near-number arithmetic is now online!  This first video covers addition and subtraction; next will come a video on multiplication and division, and finally a video discussing and connecting all of the ☆-forms.  Key learning goals are:

• how binary arithmetical operations operate on near-numbers (one pair of slices at a time, on all pairs of numbers, one from each slice);
• familiarity with addition of basic near-numbers: finite + finite, finite + infinite, and infinite + infinite, and subtraction via adding the negative;
• ☆-forms for addition and subtraction, including the fundamental ☆-form, +∞ + –∞.

Near-numbers and functions!

The full video on near-numbers and functions (10:54) is finally online, including looks at the following functions:

• reciprocation,
• squaring,
• the square root,
• negation,
• the exponential,
• the natural logarithm, and
• the sine function.

The key learning goal of the video is to see how to use functions with near-numbers, and to note that functions undefined at some point remain so—we mark the occurrence of actual undefined results with an asterisk (*) on the near-number result, though we should seldom (if ever) need to do so.

The examples go at full speed, so it could be useful to slow playback to 75% speed on your first watching, or to be ready to pause and/or rewind!

Functions and near-numbers (part 1!)

The first 4:15 of the video on functions and near-numbers is online! It goes through all of the basics, as well as some examples of reciprocation; the full video will include several more function examples, and it should be online in a few weeks.

The learning goals are:

• to be able to use graphs of functions to apply those functions to near-numbers, remembering that what matters is what happens when the near-numbers keep squeezing, and
• to remember that functions undefined at some point (such as when we try to divide by zero by plugging 0 into the reciprocation function x ↦1/x) are still undefined and that we mark near-number results by an asterisk (*) when input values are lost due to the function being undefined.

Note that this video shows the answers to 1/+∞, 1/–∞, and all of those shown in the video's thumbnail!