An Introduction to 'sequences' and discrete calculus | Algebraic Calculus One | Wild Egg

One of the serious challenges for any rigorous calculus course is finding suitable definitions of "sequences". These are the main objects of the Discrete Calculus, also sometimes called the Difference Calculus.

Here we introduce the subject informally through the eyes of Thomas Harriot, an early 17th century pioneer in this kind of algebra. The difference and summation operators on sequences are the main topics, and difference and summation tables figure prominently.

There is an arithmetical mistake near the end of Slide 3: the sum 302+132=434, not 334. So that moves through to the result 7^4=2401, not 2301.

Video Contents:
0:00 Intro
0:13 Catalan numbers
1:56 Calculus of differences
4:06 Thomas Harriot
8:47 Harriot's triangular summation table
12:53 Triangular numbers and pyramidal numbers
16:17 Harriot's triangular/binomial "sequences"
20:30 Harriot's Difference theorem

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[tomastamantini]

Very neat and well rounded introduction!

[jehovajah]

I am pleased to hear Harriet given credit as Eallis demanded. Descartes and other Rurioean mathematicians often plagiarised othersvwork .of course Newton mastered this under Barriw and in association with Wallis and studying the methods ofvDesCartes. But he went further as did Leibniz . The disputes clearly obscured this approach giving Berkeley the opportunity to draw a faith analogy ! This is concrete and exactly wherevJustusbGrassmann started with complexions 😍

[keffbarn]

What do you think of dual numbers as a framework for doing calculus?

[twwc]

This is really neat stuff. I'm wondering if you're going to talk about Babbage's Difference Engine which was a mechanical device for computing polynomial approximations to certain functions using essentially the same technique you use on slide 3 to calculate 7^4, which, as has been pointed out in another comment, should be 2401 not 2301.

[joshuaworman]

so i have a question for everyone. what math package (maple, wolfram alpha, etc) would one highest recommend to best practice and explore these concepts? (not sure how best express this question) wich one allows the greatest flexibility and convenience for manipulating algebraic expression with the same flexibility these lessons demand?

[davidjuhasz]

23:03 7th column last number should be 136, not 131

[jehovajah]

I am pleased to hear Harriet given credit as Eallis demanded. Descartes and other Rurioean mathematicians often plagiarised othersvwork .of course Newton mastered this under Barriw and in association with Wallis and studying the methods ofvDesCartes. But he went further as did Leibniz . The disputes clearly obscured this approach giving Berkeley the opportunity to draw a faith analogy ! This is concrete and exactly wherevJustusbGrassmann started with complexions 😍